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Qualitative and Quantitative Analysis of Population Models with Variable Qualitative and Quantitative Analysis of Population Models with Variable

Carrying Capacity Carrying Capacity

Diny Zulkarnaen

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Qualitative and Quantitative Analysis of PopulationModels with Variable Carrying Capacity

Diny Zulkarnaen

This thesis is presented as part of the requirements for the conferral of the degree:

Doctor of Philosophy

Supervisor:Dr. Marianito R. Rodrigo

Co-supervisor:Associate Professor Annette Worthy

The University of WollongongSchool of Mathematics and Applied Statistics

October 22, 2021

This work © copyright by Diny Zulkarnaen, 2021. All rights reserved.

No part of this work may be reproduced, stored in a retrieval system, transmitted, in any form orby any means, electronic, mechanical, photocopying, recording, or otherwise, without the priorpermission of the author or the University of Wollongong.

This research has been conducted with the support of the Indonesia Endowment Fund for Educa-tion (LPDP) Scholarship.

Declaration

I, Diny Zulkarnaen, declare that this thesis is submitted in partial fulfilment ofthe requirements for the conferral of the degree Doctor of Philosophy, from theUniversity of Wollongong, is wholly my own work unless otherwise referencedor acknowledged. This document has not been submitted for qualifications atany other academic institution.

Diny Zulkarnaen

October 22, 2021

Abstract

Population models are established to understand the description of population

growth dynamics in an area or ecosystem. The use of maximum (carrying) capac-

ity as a variable is deemed to be more realistic than a constant since the maximum

population size may change because of some factors such as technology, economy

and so on. Until now, many researchers have proposed various forms of popu-

lation models, while the variable carrying capacity has begun to be widely used.

However these variations, either for population or carrying capacity growth rate,

have similar characteristics. Therefore qualitative and quantitative analysis can

be done to the population models in a general form.

The purpose of this thesis is to examine population models involving a single or

a system of ordinary differential equations, where the carrying capacity is set to

be one of the state variables. Qualitative and quantitative solutions are analysed

here. For human population, several carrying capacity models are introduced

to verify population dynamics against actual data collected from the Food and

Agricultural Organisation (FAO). In an ecological environment, Kolmogorov’s

general population models with given assumptions are used to find exact solu-

tions. Then a population harvesting term is added to these models to inspect

steady state behaviour as a function of the harvesting values. The population

models are also implemented in fisheries management. Fish population is har-

vested by a control effort variable in order to gain maximum net profit. Then

the model is modified by specifying the fish carrying capacity as a food source,

where it is also harvested with a different control effort variable.

iv

Acknowledgements

I would like to thank my supervisor, Dr. Marianito Rodrigo for his support, pro-

vided me with invaluable knowledge from teaching me the core of mathematics

to helping me how to solve more advanced mathematics problems. Also Profes-

sor Annette Worthy for the advice given.

I would like to acknowledge my colleagues, Dong for being a good friend during

the coursework, Muhammad Al Balwy for introducing me to all department fa-

cilities when I started having my own room, and Salman for keeping me spirited

by having great discussions.

I am also deeply grateful to my wife, my children and my parents who never stop

supporting and encouraging me to keep going on my study and cheering me up

when unexpected situations occured.

I wish to thank the Indonesia Endowment Fund for Education (LPDP) for the fi-

nancial support. I never would have stepped my foot in Australia without this

support which led to so many priceless experiences including academic enrich-

ment and living with people and communities in such diversity and harmony.

Finally I would like to acknowledge the Indonesian muslim community in Illawa-

rra, the Indonesian Student Assosiation in Wollongong and the IQRO Foundation

in Sydney who always gave the best warm welcome whenever my family and I

attended every event they held.

v

Contents

Abstract iv

1 Introduction 1

1.1 Population models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Population per capita growth rate . . . . . . . . . . . . . . . . . . . . 6

1.3 Variable carrying capacity . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Basic Theory 22

2.1 Human carrying capacity and food availability . . . . . . . . . . . . 22

2.2 Integration-based parameter estimation method . . . . . . . . . . . 25

2.3 Population models with harvesting . . . . . . . . . . . . . . . . . . . 30

2.3.1 Constant harvesting rate . . . . . . . . . . . . . . . . . . . . . 31

2.3.2 Variable harvesting rate . . . . . . . . . . . . . . . . . . . . . 33

2.4 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6 Optimal control in harvesting . . . . . . . . . . . . . . . . . . . . . . 53

2.6.1 Variational approach . . . . . . . . . . . . . . . . . . . . . . . 56

2.6.2 Hamiltonian method . . . . . . . . . . . . . . . . . . . . . . . 60

3 Modelling Carrying Capacity as Food Availability 63

3.1 Population model with one ODE . . . . . . . . . . . . . . . . . . . . 64

vi

3.1.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . 64

3.1.2 Integration-based parameter estimation method . . . . . . . 66

3.1.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 70

3.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.2 Population model with two ODEs . . . . . . . . . . . . . . . . . . . 79

3.2.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . 80

3.2.2 Integration-based parameter estimation method . . . . . . . 81

3.2.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 84

3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 Analytical Solution of a General Population Model with Variable Car-

rying Capacity 94

4.1 Population model formulation . . . . . . . . . . . . . . . . . . . . . 95

4.2 Analytical solution procedure . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Special case: proportional per capita growth rates . . . . . . . . . . 105

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Qualitative Behaviour of a General Harvesting Population Model 112

5.1 Preliminary result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Steady state analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.1 Carrying capacity per capita growth rate depends on pop-

ulation size only . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.2.2 Carrying capacity per capita growth rate depends on carry-

ing capacity only . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6 Optimisation of A Harvesting Model 146

6.1 Model of one harvesting effort . . . . . . . . . . . . . . . . . . . . . 146

6.2 Model with two harvesting efforts . . . . . . . . . . . . . . . . . . . 153

6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

vii

7 Summary and Future Directions 165

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Bibliography 173

A Convex Theory 184

B Programming Code: Carrying Capacity as Food Availability 186

B.1 One parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . 186

B.2 Two parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . 189

viii

Chapter 1

Introduction

Population studies help us understand the processes regarding the size, growth

and distribution of populations in an area or ecosystem. Such studies can use

mathematical models to provide a better explanation of how complex interactions

in nature and processes work.

A population model was first introduced in the 18th century by Malthus [77]

with the aim of investigating the dynamics of a population growth. He estab-

lished the population model in the form of a differential equation of a constant

per capita population growth rate, the rate at which the population size changes

per individual in the population. The result was that the population increases ex-

ponentially and indefinitely when the constant value is set to be positive which is

unrealistic. The model was further developed and is widely used, formulated by

Verhulst [114] in 1838, which is known as the logistic model. This model has been

extensively used to study the cause and effect relationship between the so-called

‘carrying capacity’ (i.e. the population size that available resources can support)

and the population size (see, for instance, the seminal papers by Verhulst [114]

and Pearl & Reed [92], as well as the references by Gotelli [46], Pastor [91] and

Brauer & Castillo-Chavez [16] with their comprehensive references). For certain

initial conditions, this model has an S-shape pattern in population growth, so

1

that if it grows or decreases, it will reach a limiting equilibrium value. This is why

the logistic population model is now widely used and developed by researchers

so that it can solve the more complicated problems, such as population growth

fluctuating due to periodic seasonality [71] or population dynamics that behave

stochastically [43]. Although in some cases discrete-time models are more appro-

priate for non-overlapping populations [62, 78, 87], most population dynamics

are described as continuous-time models.

1.1 Population models

The following is the differential equation of the well-known logistic model writ-

ten in the form of an initial value problem as

dN

dt= rN

(1− N

K

), N(0) = N0. (1.1)

The parameter r denotes the constant growth rate that is assumed a positive

value, whereas N(t) represents the population size at time t and K > 0 is the

constant carrying capacity that is the limiting value of population that can be

sustained. By solving (1.1), the population size over the time can be obtained

as [42]

N(t) =N0K

N0 + (K −N0)e−rt.

Figure 1.1 shows the population dynamics with respect to time governing the lo-

gistic model in (1.1). As can be seen in the dynamics of the solid graph in that fig-

ure, the population grows towards the limiting value of carrying capacity which

creates a sigmoidal or S-shape pattern when initial value N0 is chosen less than

K/2. Initially the population size increases faster, but after it reaches a certain

value the growth becomes slower until it reaches the equilibrium or steady state.

At this stage the population attains the carrying capacity K, although it actually

never reaches it, but approaches this value asymptotically.

2

Figure 1.1: The population dynamics of logistic model that reaches the carryingcapacity.

Another model that can be used to study population dynamics is the model

introduced by Gompertz [44]. He formulated a function called the Gompertz

curve or the Gompertz function with the aim of understanding human mortality.

Then, in 1932, Winsor [116] described Gompertz function as a growth curve and

made a comparison with the logistic curve. The result is that the two curves

have the similar characteristic of having a sigmoidal shape curve as in the logistic

model. As a consequence, some researchers started to use the Gompertz function

to model and analyse population dynamics, one of which is tumour growth [33,

90]. The following is the Gompertz differential equation, expressed as

dN

dt= rN ln

(KN

). (1.2)

Since then, population models have been increasingly developed by many scien-

tists. They tried to explore and analyse population dynamics, then modified the

models in order to have one that closely describes the real life situation of a spe-

cific species, for example the mass culture of Daphnia Magna which is proposed

3

by Smith [110], where the mass growth rate is governed by

dN

dt= rN

K −N

K + aN. (1.3)

The parameter a in this model is positive and represents the ratio between the

rate of growth and replacement per unit time, or one can write a = r/c with

c denoting the replacement rate, where metabolic loss and dead organisms are

included.

Gilpin & Ayala investigated the dynamics of the fly species of Drosophila,

modelled by [41]dN

dt= rN

[1−

(NK

)θ], (1.4)

where θ > 0.

The population dynamics for all of these population models as described

in (1.1), (1.2), (1.3) and (1.4) are illustrated in Figure 1.2. As we can observe,

for a certain values of parameter and initial population, all the models have the

agreement of the sigmoidal shape, where the populations tend toward the same

limiting value of the carrying capacity K. With this behaviour, the models can

Figure 1.2: The population dynamics when the model (1.1), (1.2), (1.3) and (1.4)are applied.

4

then be formulated generally. In other words, all of these four models can be

grouped as a class of population growth rate models expressed as

dN

dt= F (N). (1.5)

Thus the general function F (N) can be assumed to have the properties

limN→0

F (N) = 0, F (K) = 0, F (N) > 0 for 0 < N < K, F (N) < 0 for N > K,

to satisfy the growth behaviour for these four models.

With the general population model in (1.5), the steady state behaviour can be

analysed, for example, using the linearisation process [16]. This linearisation pro-

cess is essential since it describes the same behaviour of the original one in (1.5)

but only near the steady state N∞.

Theorem 1.1.1. If all solutions of the linearised form

u′(t) = F ′(N∞)u(t), where u(t) = N(t)−N∞ (1.6)

at N∞ tend to zero as t → ∞, then all solutions of (1.5) with N(0) sufficiently close to

N∞ tend to the steady state N∞ as t→ ∞.

A sufficient condition that all solutions u(t) of the linearisation tend to zero

is F ′(N∞) < 0. Therefore, for steady state N∞ with F ′(N∞) < 0 we must have

F (N) > 0 for N < N∞ and F (N) < 0 for N > N∞, where N is sufficiently close to

N∞. These solutions, shown by direction fields, are illustrated in Figure 1.3. Here,

we can see that a solution with N(0) > N∞ is monotone decreasing and bounded

below atN∞ such that when t tends to infinity the solution tends to limiting value

N∞ if there are no other steady state between N(0) and N∞. A similar argument

applies to the case N(0) < N∞ where a solution tends to N∞ as t tends to infinity

since it increases monotonically to N∞ (if there are no other steady state between

5

Figure 1.3: The direction field of the differential equation in (1.5).

N(0) and N∞). With the steady state given in Theorem 1.1.1, we can define the

asymptotically stable of a steady state as follows

Corollary 1.1.2. A steady state N∞ of the model in (1.5) with F ′(N∞) < 0 is asymptot-

ically stable, while a steady state N∞ with F ′(N∞) > 0 is unstable.

In case we obtain F (N∞) = 0, there is no conclusion that can be drawn and further

work needs to be done to find out the steady state behaviour.

1.2 Population per capita growth rate

A population model can also be represented in terms of the population per capita

growth rate instead of population growth rate as given earlier in (1.5). This has

been explored by Brauer & Castillo-Chavez [16] by formulating a general popu-

lation model asdN

dt= Nf(N), (1.7)

where the function f = f(N) is defined as the population per capita growth rate.

Using this model, they examined population qualitative behaviour and consid-

ered three types based on population sustainability:

6

(i) Compensation model.

A compensation model describes a population that can be maintained in

size. This type of models assumes f(N) to be nonnegative and decreases

for 0 ≤ N ≤ K, or we can write these assumptions as

f(N) ≥ 0, f ′(N) ≤ 0 for 0 ≤ N ≤ K.

One example that satisfies these assumptions is f(N) = 1 − N/K which

forms the logistic model (see equation (1.7)).

(ii) Depensation model.

Depensation means a decrease in population leads to a reduction in its pro-

duction. Thus, a model is said to be depensation when f(N) is increasing

for smallN , and the following are the assumptions which satisfy the depen-

sation model.

f(N) ≥ 0, f ′′(N) ≤ 0 for 0 ≤ N ≤ K,

f ′(N) > 0 for 0 < N < K∗,

f ′(N) < 0 for K∗ < N < K,

where K∗ denotes the value that maximises f(N). Notice that the function

f(N) = N(K − N) fulfils these assumptions with K∗ = K/2 as the max-

imiser of f(N).

(iii) Critical depensation model.

This type of model occurs in the case that a depensation (reduction) level

is high enough that the population is no longer able to maintain itself. In

terms of population per capita growth rate, the value of f(N) is actually

negative for small N . Here are two assumptions for the critical depensation

7

model.

Nf(N) < 0 for 0 < N < K0, Nf(N) ≥ 0 for K0 ≤ N ≤ K,

where 0, K0 and K are steady states that appear in a model of this type. A

function f(N) = (1−N/K)(N/K0 − 1) can be used as an example here.

Now let the population in an ecosystem consist of two species, for which

these two species have to compete with each other to survive, or one species

posits as prey while the other one is the predator, and so on. A well-known class

of models is due to Kolmogorov [70].

Kolmogorov models

Suppose there are two species in an area, with sizes N and P , whose per capita

growth rate of each species is specified generally by a function, namely f(N,P )

for one species and g(N,P ) for the other one. Thus a model that describes this

situation can be expressed in the general form

dN

dt=Nf(N,P ),

dP

dt=Pg(N,P ).

(1.8)

This system of differential equations is sometimes called a Kolmogorov model

(see [16] to know more).

Next we also suppose that (1.8) describes the two species dynamics where

one is posited as prey N of the other species P , or we can say that it describes a

prey-predator model with the following assumptions:

(i) D2f(N,P ) < 0, D1g(N,P ) > 0, D2g(N,P ) ≤ 0.

(ii) For some prey carrying capacity K > 0, f(K, 0) = 0 and f(N,P ) < 0 if

N > K; for some minimum prey population to support predator L > 0, we

have g(L, 0) = 0.

8

Here, Dj in the assumption denotes the partial derivative with respect to the in-

dependent variable in the jth position. The Rosenzweig–MacArthur model [102]

is one of the models that meets these assumptions with f(N,P ) = f(N)−Pϕ(N)

and g(N,P ) = cϕ(N) − d, where c, d > 0 and ϕ(N) describes prey caught per

predator per unit time. Here, we can use the Holling type II functional response

ϕ(N) = aN/(1 + asN) with a, s > 0. Then, we have D2f(N,P ) = −ϕ(N) < 0,

D1g(N,P ) = cϕ′(N) > 0 and D2g(N,P ) = 0. The carrying capacity K is deter-

mined by f(K) = 0, while L by cϕ(L) = d.

The Kolmogorov model (1.8) can also be implemented in another type of in-

teraction, for instance, two species interaction with mutual benefit. This type of

interaction may be characterised as facultative (two species could survive sep-

arately), or obligatory (each species will go extinct with no assistance from the

other). Thus the assumptions for a Kolmogorov model given earlier change due

to the different interaction, so now the assumptions for mutual interaction in the

Kolmogorov model (1.8) are specified as

(i) D1f(N,P ) < 0, D2f(N,P ) ≥ 0 for all N,P ≥ 0.

(ii) D1g(N,P ) ≥ 0, D2g(N,P ) < 0 for all N,P ≥ 0.

(1.9)

Next, steady state behaviour is analysed using the isocline concept, that is

obtaining the population sizes N and P at which the rate of change of these pop-

ulations are zero (see [16] for the detailed explanation), and we only explain the

facultative mutualism when both species survive. An assumption (i) in (1.9) that

states D1f(N,P ) < 0 implies that the N -isocline f(N,P ) = 0 can be written as

N = ϕ(P ) ≥ 0, with ϕ′(P ) ≥ 0 on some interval 0 ≤ P < ∞ and ϕ(0) = K > 0

such that f(K, 0) = 0. It is also assumed that ϕ(∞) = K∗ < ∞ so that the effect

of mutualism of species P cannot allow the species N to increase unbounded.

The notation K∗ may be viewed as an increased carrying capacity for species N

obtained by the effect of mutualism of species P , where K∗ ≥ K since ϕ is an

9

increasing function.

A similar argument can be used for assumption (ii). Equation P = ψ(N) ≥ 0

is obtained from the P -isocline g(N,P ) = 0 with ψ′(N) ≥ 0 on some interval

0 ≤ N < ∞ and ψ(0) = M > 0 such that g(0,M) = 0. We also assume ψ(∞) =

M∗ <∞

For an interior steady state, denoted by (N∞.P∞), which is the intersection

between N -isocline and P -isocline, trace and determinant calculations are ap-

plied to investigate the stability. First a community matrix of (1.8) at (N∞, P∞) is

established and expressed as

J∞ = J(N∞, K∞) =

N∞D1f(N∞, P∞) N∞D2f(N∞, P∞)

P∞D1g(N∞, P∞) P∞D2g(N∞, P∞)

.

This matrix is called the Jacobian of the nonlinear system (1.8) at (N∞, P∞). From

here, the trace can be obtained as

tr (J∞) = N∞D1f(N∞, P∞) + P∞D2g(N∞, P∞) < 0

by using the assumptions in (1.9). Meanwhile the determinant can be calculated

by

det(J∞) = N∞P∞(D1f(N∞, P∞)D2g(N∞, P∞)−D2f(N∞, P∞)D1g(N∞, P∞)

).

This determinant has a positive sign (which implies (N∞, P∞) to be asymptoti-

cally stable) if only and only if the intersection of the N -isocline and P -isocline is

such that the N -isocline is above the P -isocline to the right of N∞.

Since f(K, 0) = 0, g(0,M) = 0 and f(0, 0) > 0, the origin is an unstable node

and both (K, 0) and (0,M) are saddle points. As P -isocline is above theN -isocline

for small N , the interior steady state (N∞, P∞) is asymptotically stable. Figure 1.4

shows the direction field of four regions which are divided due to the isoclines

10

Figure 1.4: Facultative interactions for both species.

crossing. In this figure we can observe that all orbits starting at the interior of the

first quadrant tends to the interior node (N∞, P∞).

The following is an example of a mutualism interaction model of two species

refers to (1.8), whose per capita growth rates are specified by linear forms and

satisfy the assumptions (1.9), expressed as

dN

dt= N(λ− aN + bP ),

dP

dt= P (µ+ cN − dP ).

(1.10)

In this example we only consider facultative mutualism. Therefore the constants

λ and µ are assumed to be positive. There are two possibilities depending on

the relationship between the slope a/b of the N -isocline and the slope c/d of the

P -isocline as shown in Figure 1.5. When ad > bc, the slope of the N -isocline is

greater that the slope of the P -isocline so we find a steady state, namely (N∞, P∞),

which is the intersection between the two isoclines. On the other hand, if ad < bc

there is a region where solutions become unbounded. Thus, we need to restrict

the model in (1.10) so that only the case ad > bc occurs. As has been explained

earlier in the general Kolmogorov model, only the interior steady state (N∞, P∞)

has asymptotically stable behaviour, where N∞ and P∞ are obtained from the

11

(a) (b)

Figure 1.5: Isoclines of facultative interactions when (a) ad > bc and (b) ad < bc

intersection of the lines aN − bP = λ and cN −dP = −µ. Thus any orbit will tend

to this steady state.

Now we know the importance of population models of either one or two

species in analysing the qualitative behaviour, especially for Kolmogorov model

where that analysis is not only for a specific growth rate function but also for any

class of functions that satisfies certain assumptions. But it is more important to

assign the carrying capacity of the population as variable since it closely describes

real world population behaviour.

1.3 Variable carrying capacity

The assumption of carrying capacity as a variable is said to be more realistic be-

cause the maximum population size can change due to one or more factors, for

example the revolution from agriculture to industry-based society which caused

the population to reach an equilibrium for the first time and then increases again

until it finds the second equilibrium. This model is known as bi-logistic growth,

introduced by Meyer [82] and Meyer & Ausubel [83]. Another observation is

that the dynamics of carrying capacity can behave like population growth, for

instance, the carrying capacity which grows periodically [72, 101] or stochasti-

12

cally [5]. Since the carrying capacity is assigned as variable, the logistic model

in (1.1) now can be reexpressed as

dN

dt= rN

[1− N

K(t)

]. (1.11)

Some variable carrying capacity models have been proposed as an explicit

expression, for instance, by Ebert and Weisser [35] who established a population

model of viral parasites that propagate inside their host, where the parasites car-

rying capacity is defined as

K(t) =K0K1e

ct

K1 +K0(ect − 1).

The constant K0 denotes the host size at the time of infection, K1 is the maximum

of the carrying capacity, and c represents the maximum per capita growth rate

of the host. The carrying capacity formulation has also been done by Safuan et

al. [104] through a population model of microbial biomass, written as

K(t) = Ks

(1− be−ct

),

where Ks describes the bacterial saturation level, c represents the saturation con-

stant and b = 1 −K0/Ks, with K(0) = K0. Other carrying capacity models with

explicit expressions can also be found in [30, 56, 60, 103].

The logistic model with variable carrying capacity can also be constructed

with more than one differential equation. This occurs when there is interaction

between population N and carrying apacity K. For example, a coupled system

of population and carrying capacity differential equations as introduced in [105]

is

dN

dt= rN

(1− N

K

),

dK

dt= K(a− bN),

(1.12)

13

where a and b denote the carrying capacity rate of development and depletion,

respectively. With this model, we can investigate the steady state behaviour of the

population and carrying capacity, as well as the population size that can affect the

dynamics of carrying capacity growth, and not only the carrying capacity that af-

fects the population size as described by model (1.11). Thornley & France [112]

proposed the model of population size N with the final asymptotic level K, gov-

erned by

dN

dt= (1− flim)rN

(1− N

K

),

dK

dt= −Dflim(K −N).

(1.13)

In this model, parameter D is viewed as the rate of differentiation, development

or ageing, while flim is a fraction (0 ≤ flim ≤ 1), which reflects a possible growth

limitation.

Observing the two models in (1.12) and (1.13), we can subsume these models

into a general class of the form

dN

dt= Nf(N,K),

dK

dt= Kg(N,K).

(1.14)

Observe that (1.14) has the same form of the Kolmogorov model given in (1.8).

However, the interaction is not for two species but between a population and its

carrying capacity. Thus some assumptions for f and g in (1.14) can be assigned

to cover both models in (1.12) and (1.13).

1.4 Literature Survey

The logistic model for population growth has been utilised extensively to study

the cause and effect relationship between the so-called ‘carrying capacity’ (i.e. the

population size that available resources can support) and the population size [16,

14

46, 91, 115]. It is typically assumed that the carrying capacity is constant in time.

Consequently, for a certain intial population value the logistic model exhibits a

sigmoidal shape when the population is plotted as a function of time. It has also

been studied in the case when the population oscillates due to periodic seasonal-

ity [71], and many others.

Human Population

Human population growth exhibits more complex behaviour, unlike population

species grown in laboratory cultures which have a fixed amount of space and

resources. It is therefore more realistic to assume a time-varying carrying capacity

when using the logistic model for describing human population dynamics.

Meyer [82], as well as Meyer & Ausubel [83], considered a bi-logistic model,

which is essentially a logistic model but with a sigmoidal time-varying carrying

capacity. The bi-logistic model was applied to the English and Japanese popula-

tions, where a second growth occurred due to a shift from an agriculture based

society to an industrialised one [83]. Cohen [29] proposed a human population

growth model with a variable carrying capacity, which in turn changes as a func-

tion of the population itself. A conclusion from the above models is that the

inclusion of a variable carrying capacity is more reflective of the human condi-

tion.

Cohen [29] and Meyer & Ausubel [83] have attempted to illustrate human

carrying capacity for the purpose of presenting robust models to accurately es-

timate the human population size that can be supported. However, there is no

consensus with regards to appropriate models for human carrying capacity [29].

It is more or less accepted that human carrying capacity is influenced by several

factors such as changes in technology, culture, economics etc. Specific examples

of new technologies and resources are those that permitted the increase in crop

yields, as well as other innovations that have brought about the increase in hu-

15

man food availability. Hence food availability is deemed as an important factor

that affects human population growth [55]. Furthermore, Hopfenberg has de-

fined the population carrying capacity as a function of food availability [54]. He

postulated that food production data is the sole variable that influences human

carrying capacity and assumed a simple linear relationship between human car-

rying capacity and the food production index. He then modelled his postulate

in the form of a single ODE, where the carrying capacity is replaced by a linear

function of the food production index. The constant of proportionality is chosen

instead of calculated in order to fit the approximated world population number

with its actual data.

System of ODE Population Model

In addition to explicit function, the carrying capacity dynamics can also be rep-

resented as the ODE of the growth rate with interaction to the population dy-

namics. In other words, the population model is expressed as a coupled system

of two ODEs, and one of the examples is introdued by Safuan et al. [105]. Their

model eliminates the need for prior knowledge of the carrying capacity or con-

straints to be placed upon the initial conditions. Another model is proposed by

Thornley & France [112] as an ‘open-ended’ form of the logistic equation by con-

sidering a system of two ODEs representing the coupled processes of growth and

development. Their model is ‘open-ended’ in the sense that dynamic changes in

nutrition and environment can influence growth and development, which in turn

may affect the asymptotic carrying capacity value.

However the coupled system of ODEs can also be represented in general

as described by Kolmogorov in (1.8), but instead of prey-predator interaction,

this model now relates to the interaction between a population and its carrying

capacity as shown in (1.14).

Analytical Solution

16

It is undeniable that describing the qualitative behavior of a population model

with variable carrying capacity can be done through various approaches, and this

effort is not as complicated as generating analytical model solution. Neverthe-

less, some researchers have succeded in using a method to derive the analytical

solution.

Safuan et al. has found an exact solution of a non-autonomous logistic equa-

tion with a special form for the carrying capacity and expressed the solution as

a series [104]. Meanwhile Shepherd & Stojkov [108] studied the logistic equa-

tion with a slowly varying carrying capacity and used multiple scale analysis

to obtain an approximate closed form solution. Thornley et al. [113] found an

analytical solution of the Thornley-France model. The solution of the system of

ODEs is expressed in terms of the solution of a single ODE of power-law logistic

type, also referred to as the θ-logistic model, which frequently arises in ecology

and elsewhere (see Gilpin & Ayala [41] for instance). Von Bertalanffy [13] and

Richards [99] studied power-law logistic models. The principal (nonnegative)

parameter of these models is denoted by θ. The Gompertz model and the logis-

tic model are recovered when θ = 0 and θ = 1, respectively. Larger values of θ

behave like a logistic model but with an increasingly sharper cessation of growth

as the asymptote (i.e. the constant carrying capacity limit) is approached [113].

As a fraction of the asymptote, inflexion can occur over the range from 1/e (Gom-

pertz), through 1/2 (‘ordinary’ logistic) and then to 1 (for large θ). Determining

the point where inflexion takes place can be especially important when fitting the

model to actual data that exhibit a sigmoidal trend. See Banks [10] for a detailed

analysis of the θ-logistic model.

Optimal Harvesting

There are many results and explanations regarding population models in which

harvesting is performed. The most basic form of a population harvesting model is

a harvesting that is done at a constant rate or at a changing rate but with constant

17

effort as described in [16, 88]. This constant harvesting factor has been analysed

by Doust [34] through the steady state and solution behaviour by finding the first

integral, solution curve and phase diagram. The same analysis has also been done

with a non-constant harvesting factor. Liu et al. [74] investigated the interaction

between the Allee effect and harvesting behaviour. They used the population

growth rate model governed by a Smith growth function to inspect some bifur-

cations induced by the interaction, which results in a strong Allee efect with low

harvesting rate and can lead to bistability. Meanwhile Idlango et al. [59] consid-

ered that not only the harvesting factor but also the parameters of growth rate

and carrying capacity vary with time. Such variation may occur in the surround-

ing environment. They divided the time scales into two, the normal and the slow

time, and utilised a perturbation technique to approximate the solution of the

population model. In a specific area, such as in fisheries management, some re-

searchers have proposed some analysis with regards to the qualitative behaviour

of population dynamics [7, 58, 86] and provided several harvesting strategies ei-

ther for managing the renewable resources [3] or for economic reasons, e.g. to

obtain the maximum revenue [7, 58].

Harvesting for maximum profit, one needs to take into account not only

that economic purpose but also the ecological implications. On the one hand,

over-harvesting or unrestricted harvesting might lead to extinction of the species.

On the other hand, under-harvesting might cause economic losses. Therefore, a

harvesting strategy needs to be established such that profits can be earned op-

timally while sustainability is maintained. The fishing industry is an example

of an industry that faces a harvesting problem that must be resolved. Fisheries

management is needed as a way to balance the profits gained and the ecological

implications by regulating catches so that they earn maximum profit as well as

preserving fish species. One such strategy is measuring the intensity of the fish-

ing operation as well as controlling a fishing effort, even though it is difficult to

establish a policy which has more than one control variable [21, 66].

18

An optimal harvesting policy for renewable resources has also been dis-

cussed in [31, 121] where exploitation would lead to population extinction. Mean-

while, Kar [65] investigated a harvesting policy using Pontryagin’s Maximal Prin-

ciple [95] for a prey-predator system, and Mesterton-Gibbons [81] has described

a technique for finding an optimal harvesting policy in a Lotka-Voltra ecosystem.

On the other hand, Suri [111] used the calculus of variations and the Hamilto-

nian function method to gain the maximum revenue of harvesting in a fishing

industry.

Contribution

The human population model initiated by Hopfenberg [54] seems interesting and

provides a new perspective in defining variable carrying capacity governed by

food availability. However, the mathematical analysis regarding the carrying

capacity model is inaccurate and we think it really needs to be fixed to meet a

satisfied result. This model applies the ‘trial and error’ concept by choosing the

value of the proportionallity constant between the carrying capacity and the food

produciton index in the expectation of obtaining an approximation of popula-

tion number that is graphically ’close’ to the actual data. Although it is ‘close’,

it does not mean that the value is considered to give the best fit approximation.

Therefore, instead of guessing and trying many times, the constant value needs

to be estimated to minimise errors between population approximaton and actual

data. Furthermore, the corelation between the carrying capacity and the food

availability can be improved by a nonlinear relation. Another improvement or

modification can be done related to the model expression, that the carrying ca-

pacity is not only expressed as a solution but also as an ODE that forms a system

of two ODEs.

When the carrying capacity takes form of an ODE, the population model

now can be modelled in a general form containing two functions of per capita

growth rate. Although this model has a similar form to that of Kolmogorov given

19

in (1.8), the assumptions of the functions can be modified. By assuming some

suitable functions properties, we can generate analytical solutions in addition to

the qualitative behaviour as explained earlier in Section 1.2.

The qualitative behaviour of the Kolmogorov model can be analysed further

with regards to harvesting that affects populatoin dynamics. In other words, the

model can be modified by adding the harvesting term to the population growth

rate. The harvesting value is analysed to see whether it influences the number of

steady states that appear in the model as well as their stability.

In fisheries management, harvesting has an important role in gaining opti-

mal profit. Several methods can be applied to obtain the optimal solution, and

two of them are the Hamiltonian function and the calculus of variations that were

used by Suri [111]. However, the population model she proposed consists of the

carrying capacity which is considered as constant. Hence, we can replace the con-

stant to become a variable, then make the carrying capacity as a function of food

availability. This means that both fish population and its food, i.e. seaweed, are

harvested in such a way that optimal profit is obtained while maintaining the fish

population.

1.5 Thesis outline

Based on the literature survey description that gives rise to our research ideas,

the contents of this thesis are outlined as follows. The basic theory utilised to

support the thesis is provided in Chapter 2. Chapter 3 analyses logistic-type pop-

ulation models for a single ODE, where the carrying capacity is defined as an

explicit function and represented by several distinctive functions of food avail-

ability. Then the models are extended to a coupled system of population and car-

rying capacity differential equations. An integral-based method is implemented

to estimate parameters within both models, where actual data of human popula-

tion size and food availability are used. A model consisting of a coupled system

20

of two ODEs is also described in Chapter 4, but now it is constructed by two

general functions as in the Kolmogorov model. However, the model given in

Chapter 4 does not describe the interaction of two populations, but instead the

interaction between a single population and its carrying capacity. With some

assumptions assigned to these general functions, the procedure to find analytic

solutions is investigated and then compared to numerical solutions.

The presence of a constant harvesting term in a population model with vari-

able carrying capacity is then presented in Chapter 5. The changing number of

steady states that appear in the model, as well as their stability, are investigated

with regards to the varying harvesting values. Graphical illustrations are used

to help understand the steady state analysis. Like in the previous chapter, a Kol-

mogorov model is also used here but with the added harvesting term and with

distinctive assumptions of the class functions. This chapter is concluded by pre-

senting some examples that satisfy those assumptions so that the population dy-

namics can readily be observed.

The last main content of the thesis in Chapter 6 uses a logistic model to study

how to maximise profit from population harvesting, specifically fish harvesting,

that is maintained by controlling the harvesting effort with respect to time. Then

the model is modified by introducing another population as a food source for

fish which is also harvested, so that the model now comprises two ODEs with a

controllable variable in each. Two methods are applied to solve this optimisation

problem numerically. One method is used to solve the problem with a single

controllable variable, and the other method is for a problem with two controllable

variables.

Finally, all results from the main chapters are summarised in the conclusion

of the thesis, and they are compiled in Chapter 7. This chapter also provides a

brief statement of some possible future works and further directions.

21

Chapter 2

Basic Theory

In this chapter, basic theories that support the main chapter of this thesis are col-

lected here. Section 2.1 describes the correlation between food availability and

human carrying capacity in solving a population growth problem. In Section 2.2,

the integration-based parameter estimation used to estimate the parameters con-

figuring the human populaton is also discussed here. Then a harvesting popu-

lation model is provided in Section 2.3, where the harvesting rate is set either

as a constant or a variable. Finally two methods, calculus of variations and the

method of Hamiltonian functions, which are described in Section 2.4 and Sec-

tion 2.5, respectively, are implemented to solve the optimisation problem with

constant carrying capacity which is given in Section 2.6.

2.1 Human carrying capacity and food availability

The study of human population is essential for any country which tells us about

the age, composition, distribution, growth and so on. It points to many items

to measure such as the success in growing, expanding, migrating, civilising and

industrialising. Although human population has succeded to attain those mea-

sures, the increasing human population has been responsible for accelerating eco-

logical and environmental problems [18, 93]. Along with the rapid growth of the

22

human population, the level of resources consumption is higher than its avail-

ability. As a result more than three billion people are malnourished, as reported

by The World Health Organization [117]. With the current world population sur-

passing more than 7.8 billion [118], the issue of population size is a serious prob-

lem which primarily impacts the Earth’s ability to cope. In other words, human

population cannot continue to grow indefinitely, as there are limited resources

that can be provided. With such conditions, we call the maximum number of

human life (or any species in general) that the environment can support as the

carrying capacity.

Although a growing population is affected by some supporting factors such

as space for living, water consumption, ecological conditions and food necessi-

ties [40, 120], it is believed that food production is the most important factor that

must be taken into account to feed the rapidly increasing human population [96].

This implies that massive agricultural improvement is needed to support the

abundance of human population. The correlation between food production and

human population growth as a global problem has been predicted by Malthus

as early as 1798. Through the article ”An Essay on the Principle of Population”,

he stated that populations of nations would be restricted by the food availability

due to incapability to control the birth rate.

Knowing that the human population problem extends to all parts of the

world, an insight about the relationship between food availability and human

population size, as well as its carrying capacity, is highly essential, and this can

be illustrated through mathematical models. Some authors such as Cohen [29]

and Meyer & Ausubel [83] have provided robust models to illustrate human car-

rying capacity and attempted to estimate human carrying capacity. This estimate

relates to the changes in culture, economics and technology.

Although it is very difficult for ecologists and scientists to calculate human

carrying capacity, Hopfenberg [54] constructed a mathematical model that gives

23

a contribution to the understanding of evaluating human carrying capacity K(t)

and its relation to food production index I(t), governed by

K(t) = c1c2I(t), (2.1)

where c1 is a positive constant obtained by dividing the total population by the

food production index in the initial year, while a positive value for c2 is cho-

sen such that the estimated population size is close to the actual data. To quan-

tify food production data, a measure of global food availability must be estab-

lished. The Food and Agriculture Organisation (FAO) obtains data from official

and semi-official reports of crop yields, area under production and livestock num-

bers. The food production index covers food crops that are considered edible and

that contain nutrients (see [54] for more details on how the FAO determines this

food production index, as well as a related livestock production index).

The model starts with the dynamics of human population size N(t) and is

governed by the well-known logistic function, written as

dN

dt= rN

[1− N

K(t)

],

with population growth rate denoted by a constant r > 0. From this growth rate

model, the approximate solution is obtained and expressed as

N(t+∆t) =K(t)

1 +[K(t)N(t)

− 1]e−r∆t

. (2.2)

Here, ∆t defines the change in time and is set to one year, while r was selected

with a value of 0.02. The population size N(t) represents the world human popu-

lation number from year 1962 to 2010, collected from the World Bank [32]. Mean-

while, the carrying capacity K(t) is calculated from (2.1), where the food produc-

tion index I(t) data points are gathered from the Food and Agricultural Organi-

sation [39], the same year interval with population. Calculating (2.2), we obtain

24

the estimated human population size, and its values are shown graphically in

Figure 2.1.

1962 1 970 1 980 1 990 2 000 2 010

0

10

20

30

40

50

Figure 2.1: Human carrying capacity, determined from the food production in-dex data, gives a good population size approximation

The simulation uses the parameter value r = 0.02 and calculates c1 which

results 0.095. On the other hand, the constant c2 = 4.480 is chosen such that

the estimated carrying capacity gives the estimated human population close (ob-

tained by inspection) to the actual data points. As we can observe from Figure 2.1,

the carrying capacity grows far above the population as time goes on. This means

that the food supply should be more than enough to feed the world population.

Hence, from this data the issues of malnutrition and hunger that occur in some

parts of the world are not actually related to the food availability, but are closely

connected to the distribution complexities.

2.2 Integration-based parameter estimation method

Parameter estimation plays an important role in determining the unknown val-

ues of model parameters to provide an optimal fit between the simulation and

experimental data such that the system behaviour can be accurately described.

25

There are several methods that can be used to do parameter estimation, and the

most common one is the use of least squares [8, 11, 49, 73]. This involves fit-

ting a line such that the sum of the squared distances between data points and

the regression line is minimised. Meanwhile, the integration-based method [52]

is suitably used to solve inverse problems by estimating the parameter values

within a system of ODEs. The technique reduces to solving an algebraic system

of equations. This method can be widely used to systems of differential equa-

tions, and the number of parameters to be estimated is not restricted.

Now, suppose that an ODE or a system of ODEs with m unknown param-

eters is given. The first step in this technique is to multiply the left and the

right-hand side of the ODEs by a weight function containing a controllable pa-

rameter β, called an equation-generating parameter, then perform integration to

both sides of the ODEs over a finite interval. This method needs m different

values of β to estimate m parameters. Consequently, there will be m algebraic

equations after substituting m different values of β to the transformed equations.

These transformed equations, called algebraic equations, will have some integral

terms that are able to be calculated by numerical approximation using observa-

tion data.

To understand more about the method, here is a detailed explanation of how

to estimate the parameters. It should be noted that a system of ODEs can use

this method by breaking it into a single ODE sequence, then for each ODE the

parameter estimates are carried out progressively. The reason for breaking into

single ODEs is to avoid overdetermined parameters in a system for the case when

there is a parameter used in more than one ODE in the system so that estimation

cannot be done.

Now, let an ODE take the form

x′(t) = f(t, x(t);P ), x(0) = x0, (2.3)

26

where x : R → R, f : R2 → R, P ∈ Rm, x0 ∈ R. The aim here is to estimate the

parameter set of P . First we assume that x is observed over a time interval I , then

we consider a weight function φ : I × R → R which is integrable as well as its

derivative with respect to time. Multiplying the weight function to (2.3) and then

integrating both sides over the interval I yields

φ(t; β)x(t)|I −∫I

φ′(t; β)x(t) dt =

∫I

φ(t; β)f(x(t), t;P ) dt. (2.4)

For notational convenience, the left-hand side of the latter equation is denoted

by kβ and the right hand side by Fβ(P ). Thus (2.4) now becomes kβ = Fβ(P ).

Since there arem unknown parameters, m different values βi, i = 1, 2, ...,m, must

be chosen and substituted into the equation (2.4) so that it is transformed into a

system of algebraic equations expressed as

kβ1

...

kβm

=

Fβ1(P )

...

Fβm(P )

. (2.5)

Suppose that there is a function gi : R2 → R for i = 1, ...,m and parameter

estimate P = (p1, p2, ..., pm) such that an ODE defined in (2.5) can be rewritten

as

f(t, x(t);P ) = p1g1(t, x(t)) + p2g2(t, x(t)) + ...+ pmgm(t, x(t)).

As can be observed that f is a linear functional of P in the equation. In this case

the matrix in (2.5) takes the form of a linear system of algebraic equations, where

its solution can be obtained as long as the weight function φ is chosen such that

the resulting coefficient matrix of P in (2.5) is invertible. We repeat these steps

from the beginning to estimate other parameters that appear in each ODE within

a system of ODEs.

Next, two population growth models are given as examples to obtain the

27

parameter estimates using this method. One is a population model with a single

ODE, and the other one is a system of ODEs containing two equations.

Example 2.2.1. In this example the logistic population model is used to apply the

integration-based method so that the function f(t, x(t);P ) in (2.3) can be written

as

f(t, N(t); r) = rN(t)[1− N(t)

K

], (2.6)

where N(t) is the population size at time t, and r is the population growth rate.

Meanwhile K is a constant carrying capacity which we assume to be known. In

other words r is the only parameter that is going to be estimated with given K

and N(t) for 0 ≤ t ≤ T .

By following the steps explained before, the weight function of exponen-

tial form φ(t; β) = e−βt is chosen here for example. Thus the equation derived

from (2.4) can be generated as

e−βtN(t)|T0 + β

∫ T

0

e−βtN(t) dt = r

∫ T

0

e−βtN(t)[1− N(t)

K

]dt. (2.7)

Since r is the only parameter to be estimated, a value of β (say, β1) is the only

controllable parameter needed to calculate r.

Eventually we obtain the estimation as

r =e−β1tN(t)|T0 + β1

∫ T

0e−β1tN(t) dt∫ T

0e−β1tN(t)[1−N(t)/K] dt

,

provided the coefficent of r which is∫ t

0e−β1tN(t)

(1− N(t)

K

)dt is nonzero.

The procedure of parameter estimation from the logistic model (2.6) is then

extended to two parameters. Now, r is not the only unknown parameter to esti-

mate but also K. But in this case, K is not actually the parameter to be estimated

directly since it appears to be nonlinear, as seen on the right-hand side of the

equation (2.6). Thus u = r/K is defined to transform the nonlinearity of K into

28

linearity in r and u, and (2.6) is now rewritten as

f(t, N(t); r,K) = rN(t)− uN(t)2

The function in exponential form φ(t; β) = e−βt is also used here as the weight

function. Multiplying this weight function then integrating both sides of the lo-

gistic model yields

e−βtN(t)|T0 + β

∫ T

0

e−βtN(t) dt = r

∫ T

0

e−βtN(t) dt− u

∫ t

0

Te−βtN(t)2 dt. (2.8)

Two controllable parameters, say β1 and β2, are then used and substituted

into (2.8) to generate a system of two algebraic equations, which can be repre-

sented in a matrix form by

∫ T

0e−β1tN(t) dt −

∫ T

0e−β1tN(t)2 dt∫ T

0e−β2tN(t) dt −

∫ T

0e−β2tN(t)2 dt

ru

=

e−β1tN(t)|T0 + β1∫ T

0e−β1tN(t) dt

e−β2tN(t)|T0 + β2∫ T

0e−β2tN(t) dt

.Provided the coefficient matrix is invertible, the parameters r and u can be calcu-

lated which results in obtaining the parameter K = r/u.

Example 2.2.2. In this part, the Lotka-Volterra prey-predator model is used. This

system model consists of two ODEs and each ODE contain two parameters given

by

x′(t) = ax(t)− bx(t)y(t), (2.9)

y′(t) = cx(t)y(t)− dy(t). (2.10)

where a, b, c and d are positive parameters to be estimated.

The ODE in (2.9) describes the prey growth rate, while the second one in (2.10)

represents the predator growth rate. Since these two ODEs contain the parame-

ters which are linear and the parameters in both ODEs are independent to each

29

other, the two parameters of each ODE can be estimated simultanously. First ap-

ply the method to (2.9) to obtain a and b. After that, apply it to the ODE in (2.10)

to get c and d, or it can be done the other way around.

Each ODE requires two controllable parameters, β1 and β2, to establish a

system of algebraic equations, then finally the parameters can be obtained via

∫ T

0e−β1tx(t) dt −

∫ T

0e−β1tx(t)y(t) dt∫ T

0e−β2tx(t) dt −

∫ T

0e−β2tx(t)y(t) dt

ab

=

e−β1tx(t)|T0 + β1∫ T

0e−β1tx(t) dt

e−β2tx(t)|T0 + β2∫ T

0e−β2tx(t) dt

for the ODE in (2.9), and

∫ T

0e−β1tx(t)y(t) dt −

∫ T

0e−β1ty(t) dt∫ T

0e−β2tx(t)y(t) dt −

∫ T

0e−β2ty(t) dt

cd

=

e−β1ty(t)|T0 + β1∫ T

0e−β1ty(t) dt

e−β2ty(t)|T0 + β2∫ T

0e−β2ty(t) dt

.for the ODE in (2.10), provided the coefficient matrices are invertible.

2.3 Population models with harvesting

The natural resources and environmental systems have become an important con-

sideration over the last few years that we must face, and one of them is controlling

the population size in an ecosystem. This population may undergo exponential

growth, at least for short periods, which means the population grows without

bounds. In this case, undertaking harvesting strategies to control population

size could be used to keep the size in check and also to avoid extinction. But

of course the environment cannot support an unlimited population, instead its

growth rate will decrease along with the increase in population size due to a fac-

tor, namely carrying capacity, and this phenomenon has been modelled, called

logistic growth model.

In general, populations governed by logistic growth do not require harvest-

ing to sustain population equilibrium. In other words, the main purpose of har-

30

vesting is not to control the population but to gain a substantial harvest from the

population. A proper and efficient management strategy is very important for

harvesting, although it is difficult to be done since many conflicting factors arise

such as economic, technical, legislational as well as environmental protection and

resource conservation, so all of these factors need to be balanced. With a good

harvesting management, resources as well as population size can still be main-

tained at a maximum growth rate, known as maximum sustainable harvesting,

while minimum efforts are expected to avoid extinction.

Numerous mathematical models have emerged to help examine the envi-

ronmental and ecological impacts of harvesting as well as to establish and for-

mulate cost-effective management policies. A population model which consists

of a harvesting term has been widely investigated by many scientists in which

the harvesting rate is set as a constant or a variable. However, such populations

may grow in a randomly fluctuating environment [37, 76, 119] or in an environ-

ment where the Allee effect occurs [74]. Several other population models have

introduced harvesting strategies to two-population interaction models such as

predator-prey models [6, 7] or competition models [85]. Meanwhile, the eco-

nomic point of view as a factor influencing the optimal harvesting strategy has

been studied by Clark [24, 26, 27]. In this thesis we consider the harvesting rate

as either a constant or a variable with catch effort.

2.3.1 Constant harvesting rate

Suppose that a population in an environment is governed by logistic growth and

harvested by the constant rate H . Hence the population dynamics can be mod-

elled asdN

dt= rN

(1− N

K

)−H, (2.11)

where r denotes the constant population growth rate, while K represents the car-

rying capacity, which is also constant. For this model, steady states and their

31

stability are inspected. First, equating the right-hand side in (2.11) to zero, we ob-

tain two non-negative steady states, namely NL and NU , where NL < NU , written

as

NL =K −

√K2 − 4HK/r

2and NU =

K +√K2 − 4HK/r

2. (2.12)

These steady states exist provided K2 − 4HK/r ≥ 0 or 0 ≤ H ≤ rK/4, other-

wise the roots are complex valued. This causes the population hitting zero in

finite time, which means the system is considered to have collapsed. Observe

that when H = rK/4, the steady state is unique since NL and NU give the same

value. There are cases associated with a number of steady states that appear in

the model based on the H value.

• In the case when 0 ≤ H < rK/4, the two steady states exist as written

in (2.12). As H increases from 0 to rK/4, NL increases from 0 to K/2. On the

other hand, NU decreases from K to K/2 as H increases. For the stability,

we let (2.11) to be rewritten as N ′ = F (N) so that we can determine the

requirement of stability condition for a steady state (sayN∞) as F ′(N∞) < 0,

that is, the first derivative of F (N) = rN(1−N/K) at N∞ is negative which

implies

N∞ >K

2. (2.13)

Since 0 ≤ NL <K2

, which means NL does not meet the stability requirement

as given in (2.13), the steady state NL is said to be always unstable. On

the other hand, NU is always asymptotically stable since its value is in the

interval K2< NU ≤ K which satisfies (2.13). In conclusion, the population

size approaches the steady state K/2 as H tends to rK/4.

• When H reaches rK/4 the two steady states coalesce and cause the discon-

tinuity of the system behaviour. We call this value Hc = rK/4 as the critical

harvesting rate of the population model (2.11).

32

• IfH > rK/4, then the square roots in (2.12) are negative such that the steady

states are no longer available, which leads to the population size hitting zero

value in finite time.

Figure 2.2: Steady states shown by the intersection between the growth ratef(N) and the constant harvesting rate H

Figure 2.2 shows us the two graphs of a curve, say f1(N) = rN(1 − N/K) and

a line f2(N) = H , where their intersections describe the steady states. As we

can observe, the number of steady states may change as H varies. If we choose

H < Hc, then we find two points of intersection as the steady states, those are NL

on the left and NU on the right. But when H is increased to reach Hc, then the

line f2 moves upward to the peak of f1. This means both steady states coalesce

into a unique steady state. Furthermore, as H gets larger and passes Hc, the line

f2 moves away from f1 such that the steady state dissapears.

2.3.2 Variable harvesting rate

Now, let the population be harvested at a changing rate, where this rate depends

on the population size itself (usually called density dependent harvesting). This

33

type of problem arises in the fishery resources problem and is modelled by

dN

dt= rN

(1− N

K

)− qEN. (2.14)

This model is known as the Schaefer model [107] and has been utilised in many

other commercial fisheries management. The constant E in model (2.14) denotes

the fishing effort, a certain standardised measure of the number of vessels oper-

ating per unit time. Meanwhile, q denotes the coefficient of catchability or the

ability to harvest the resource. The model in (2.14) has two steady states, one at

N∞ = 0 and the other one is given by

N∞ = K(1− qE

r

). (2.15)

Observe that the condition 0 ≤ E < r/q must be satisfied to avoid a negative

steady state. With this condition, the steady state decreases from K to 0 when the

effort is made to increase from 0 to r/q.

Like the model with constant harvesting rate in (2.11), we can also rewrite the

model (2.14) as N ′ = F (N) with F (N) = rN(1 − N/K) − qEN and then inspect

the sign of F ′(N∞) to verify that the steady state N∞ = 0 is unstable and N∞ is

asymptotically stable provided E is in the interval 0 ≤ E < r/q. From this stable

condition for N∞, we can define and calculate a yield, namely the sustainable

yield Y , for a given effort 0 ≤ E < r/q as

Y (E) = qEN∞, (2.16)

where N∞ is given in (2.15). Using this definition, we want to know how much

effort is needed to harvest to obtain the maximum sustainable yield. First of all,

substituting N∞ from (2.15) to (2.16), the sustainable yield becomes

Y (E) = qKE(1− qE

r

)= qKE − q2K

rE2. (2.17)

34

As we can see, Y (E) is now a quadratic function of E, where in a graphical rep-

resentation, this function takes the form of a concave downward curve. Thus

from (2.17), we have

Y ′(E) = qK − 2q2K

rE

to obtain E∗ = r/(2q) as the optimal effort which maximises the sustainable yield

and Y (E∗) = rK/4 is the maximal sustainable yield. Figure (2.3) depicts the sus-

tainable yield with respect to the effort. With the effort given to attain r/(2q), the

Figure 2.3: maximum sustainable yield for Schaefer model

sustainable yield increases to the value rK/4. But as more effort is made beyond

E∗ = r/(2q), it becomes counterproductive since this decreases the sustainable

yield. This situation occurs because the increased effort leads to a reduced num-

ber of fish population, which implies the reduction in the harvest received.

2.4 Calculus of variations

One technique that can solve the dynamic optimisation problem is the calculus

of variations. It uses variation or small changes of functions or functionals to

find the maxima or minima of an objective function. This method is explained in

detail by Chiang [22] and also can be found in [14, 50].

35

Suppose that we have an optimisation problem, where the objective is to

maximise (or minimise) the functional

J(x(t)) =

∫ T

0

f(t, x(t), x′(t)) dt (2.18)

with respect to x = x(t), subject to

x(0) = x0, x(T ) = xT ,

where x0, xT are given. The aim of this method is to obtain a function x∗ = x∗(t),

called extremal, that maximises (minimises) the value of J(x(t)) by perturbing

x∗(t) with the plausible paths p(t). Therefore, x(t) is expressed as

x(t) = x∗(t) + ϵp(t), (2.19)

where ϵ is small. Since x∗(t) must connect the two given endpoints (0, x0) and

(T, xT ), these endpoints of the paths should satisfy

p(0) = p(T ) = 0. (2.20)

As x(t) depends on ϵ, J now can be expressed in terms of ϵ, and it is important to

note that each ϵ value gives particular paths x(t) for given x∗(t) and p(t). Further-

more, the necessary condition for optimum value of J(ϵ) must satisfy

dJ

dϵ= 0.

By the Leibniz Rule, we may write

dJ

dϵ=

∫ T

0

∂f

∂ϵdt =

∫ T

0

(∂f

∂x

dx

dϵ+∂f

∂x′dx′

dϵ

)dt = 0 (2.21)

36

Notice that the first derivative of x(t) in (2.19) can be expressed as

x′(t) =dx

dt=

dx∗

dt+ ϵ

dp

dt.

Using the first derivative of the equation in (2.19) and the latter equation with

respect to ϵ, equation (2.21) becomes

∫ T

0

∂f

∂xp dt+

∫ T

0

∂f

∂x′dp

dtdt = 0. (2.22)

This equation is already free from the arbitrary ϵ, but another arbitrary p and its

derivative still appear in the equation. In order to make the necessary condi-

tion of optimality work properly, the functions p and dp/dt should also be elimi-

nated.

Now, integrating the second term of (2.22) by parts, we have

∫ T

0

p

[∂f

∂x− d

dt

( ∂f∂x′

)]dt+

∂f

∂x′p

∣∣∣∣T0

= 0. (2.23)

Then, by evaluating the endpoints of p(t) given in (2.20), we now have another

form of the necessary condition as

∫ T

0

p

[∂f

∂x− d

dt

( ∂f∂x′

)]dt = 0, (2.24)

where dp/dt has vanished in this equation, while p still exists. But since p is

arbitrary function, which means it can be any path, including the nonzero path,

then the multiplier of p must be zero in (2.24). Thus we can infer that for all

t ∈ [0, T ],∂f

∂x− d

dt

( ∂f∂x′

)= 0. (2.25)

This equation, which is called the Euler–Lagrange equation, is now completely

free from arbitrary variables so that this equation can be solved to find the optimal

value of the objective function. This Euler–Lagrange function can also be written

37

as∂2f

∂x′2x′′(t) +

∂2f

∂x∂x′x′(t) +

∂2f

∂t∂x′− ∂f

∂x= 0 (2.26)

by expanding the total derivative of the second term on the left-hand side of

equation (2.25).

Example 2.4.1. Suppose that we want to maximise the objective function

J(x(t)) =

∫ 2

0

12tx(t) +

[x′(t)

]2dt,

with two boundary conditions given as x(0) = 0 and x(2) = 8. By referring to

(2.18), we have f(t, x, x′) = 12tx+ (x′)2, which has derivatives

∂f

∂x= 12t,

∂f

∂x′= 2x′,

∂2f

∂x′2= 2,

∂2f

∂x∂x′=

∂2f

∂t∂x′= 0.

Referring to the Euler–Lagrange equation in (2.26), we obtain

2x′′ − 12t = 0,

which upon integrating gives

x′ = 3t2 + c1.

Therefore

x(t) = x∗(t) = t3 + c1t+ c2.

Since two boundary conditions are given, we eventually have the particular so-

lution of x(t) that maximises J(x(t)), expressed by

x∗(t) = t3.

38

Transversality condition

For problems with fixed initial point and endpoint, the two given boundary con-

ditions provide sufficient information to determine the two arbtrary constants.

On the other hand, if the initial point or endpoint is variable, then a boundary

condition will no longer be available, and there may be many paths satisfying the

Euler–Lagrange equation. Thus a transversality condition needs to be applied to

solve this problem. With a transversality condition, one can select the optimal

path among those of the satisfied Euler–Lagrange equation, or at least rule out

the non-optimal paths.

Recall that the Euler–Lagrange equation is a necessary condition. In (2.26),

two arbitrary constants appear when solving the ordinary differential equation,

as shown clearly in the previous example. These two constants are able to be

determined as long as two boundary conditions are defined and fixed. On the

other hand, when either initial or terminal point is defined as a variable, then a

boundary condition is no longer available. In this case, the transversality condi-

tion needs to be applied to fill that void.

Here, we choose the terminal boundary point to be variable while the initial

point is fixed. Suppose that we have an optimisation problem

Maximise J(x(t)) =

∫ T

0

f(t, x(t), x′(t)) dt

subject to x(0) = x0, x(T ) = xT ,

(2.27)

where x0 is given (fixed) and xT , as well as the terminal time T , are free (vari-

able). In addition to generating the neighbouring paths of x(t) as was explained

previously, the terminal time T is also perturbed by

T = T ∗ + ϵ∆t.

The arbitrary chosen small changes of T is represented by ϵ∆t, where ϵ is small.

39

Meanwhile T ∗ denotes the optimal terminal time that it is suppose to be known.

Since T is considered as the function of ϵ, hence its derivative can be written as

dT

dϵ= ∆T. (2.28)

Next, the first-order necessary condition for the maximum of J(x(t)) can be ob-

tained by equating its first derivative with respect to ϵ to zero. Starting with the

derivative and employing the Leibniz Rule [4], we may write

dJ

dϵ=

∫ T (ϵ)

0

∂f

∂ϵdt+ f

(T, x(T ), x′(T )

)dTdϵ. (2.29)

The first term of the right-hand side of (2.29) can be expanded similarly to that of

the right-hand side expressed previously in (2.23), but the terminal value p(T ) =

0 since T is free. As a result, the first term in equation (2.29) can be rewritten

as ∫ T

0

p

[∂f

∂x− d

dt

( ∂f∂x′

)]dt+

∂f

∂x′

∣∣∣∣t=T

p(T )

and the second term as

f(T, x(T ), x′(T )

)∆T

by referring to (2.28). Then, applying dJ/dϵ = 0 from (2.29), the necessary condi-

tion can be obtained as

∫ T

0

p

[∂f

∂x− d

dt

( ∂f∂x′

)]dt+

∂f

∂x′

∣∣∣∣t=T

p(T ) + f(T, x(T ), x′(T )

)∆T = 0. (2.30)

As can be seen from this equation, each term has a free arbitrary element; those

are the neighbouring curve p, the terminal value p(T ) and the arbitrary change

∆T . Consequently, each term must be equal to zero.

When the first term is set to zero, we find that the Euler–Lagrange equation

remains as a necessary condition for this endpoint problem. On the other hand,

equating the second or the third term to zero, the transversality conditions must

40

be applied since these two terms relate to the terminal time T . Therefore the next

step to be done is eliminating p(T ) in (2.30) by transforming this element into ∆T

and ∆xT . It is easier to explain how to perform this transformation by a graphical

point of view as illustrated in Figure 2.4 (reproduced from [22]).

Figure 2.4: Transforming p(T ) in terms of ∆T and ∆xT .

Suppose we have a curve AB in the interval [0, T ]. Then we can draw its

neighbouring path AB′ by perturbing it with ϵp(t). Note that we set ϵ = 1 for

convenience. Now, we can measure p(T ) as the distance between B and B′, while

p(0) = 0 since x(0) is fixed. The distance BB′ also measures the direct change

in xT caused by perturbation. Then, as we change the terminal time at value

∆T , the curve AB′ is stretched out to AB′′ which implies the terminal path xT

is pushed up vertically with the distance B′B′′. This distance is calculated at

x′(T )∆T . Consequently, ∆xT can be approximated by

∆xT = p(T ) + x′(T )∆T.

From this equation, we now have the transformed p(T ) as

p(T ) = ∆xT − x′(T )∆T. (2.31)

41

Substituting (2.31) into the second term of the equation in (2.30) and eliminating

the first term, we have the general transversality condition as

(f − x′∂f

∂x′)∣∣∣t=T

∆T +∂f

∂x′

∣∣∣t=T

∆xT = 0. (2.32)

With this equation, there are several types of variable terminal points that are

considered to give rise to the transversality condition, three of which are as fol-

lows:

• Vertical terminal line (Fixed-time-horizon problem).

This case means the terminal time T is fixed which implies ∆T vanishes.

Thus, the first term of (2.32) drops out. Meanwhile, since the second term

has ∆xT which is arbitrary, in order to have the zero value for the second

term, we must set∂f

∂x′

∣∣∣t=T

= 0. (2.33)

Thus this equation is the transversality condition for the fixed-time-horizon

problem.

• Horizontal terminal line (Fixed-end-point problem).

Another case is when ∆xT is fixed. Thus the second term in (2.32) is elim-

inated. Since ∆T is free that means it is not always zero, we have the

transverslity condition from the first term as

(f − x′

∂f

∂x′

)∣∣∣∣t=T

= 0.

• Terminal curve.

With this case, neither xT nor T is fixed. Therefore either ∆xT or ∆T is not

equal to zero which means no term in (2.32) is ruled out. Now let xT = ψ(T )

is defined as the terminal curve. For a small arbitrary ∆T , the deviation of

the terminal curve can be written as ∆xT = ψ′(T )∆T . This equation is then

42

substituted into (2.32) to obtain

(f − x′

∂f

∂x′+∂f

∂x′ψ′)∣∣∣∣

t=T

∆T = 0.

Since ∆T is arbitrary, this gives rise to the transversality condition

[f + (ψ′ − x′)

∂f

∂x′

]∣∣∣∣t=T

= 0.

2.5 Optimal control

The calculus of variations has been extended by the work of Bellman and Pon-

tryagin as the fundamental ideas of optimal control theory which can solve prob-

lems that the calculus of variations cannot. Bellman introduced dynamic pro-

gramming and the associated optimality principle, whereas Pontryagin and his

associates introduced the maximum principle which is used only for determin-

istic problems and produces a similar solution to that of dynamic programming.

There are several references that explain optimal control theories such as Bade-

scu [9], Chiang [22], Kamien et al. [63] and Kirk [68].

The optimal control deals with handling a control variable for a dynamical

system over the optimised objective function. In other words, the purpose of

optimal control theory is to decide the control signals from a problem that leads

to a process to fulfil constraints as well as to maximise (or minimise) the objective

criterion. Once the optimal control has been found, the corresponding optimal

state can be obtained.

There are three aspects required to formulate the optimal control problem:

a mathematical model, physical constraints and a performance criterion. Let us

consider that

x1(t), x2(t), ..., xn(t)

43

be the state variables at the time process t, with t0 ≤ t ≤ t1, and

u1(t), u2(t), ..., um(t)

be the control inputs at time t, with t0 ≤ t ≤ t1. Therefore the mathematical model

may be described by the following n first-order differential equations

dx1dt

= g1(x1, x2, ..., xn, u1, u2, ..., um),

dx2dt

= g2(x1, x2, ..., xn, u1, u2, ..., um),

...

dxndt

= gn(x1, x2, ..., xn, u1, u2, ..., um).

(2.34)

After describing the mathematical model, the physical constraints have to be es-

tablished, for instance the state constraints defined at t0 is defined as

xi(t0) = xi0, i = 1, 2, ..., n.

The last aspect is the performance criterion, which can be given by the following

objective function

J =

∫ t1

t0

f(t, x1(t), ..., xn(t), u1(t), ..., um(t)) dt.

For convenience, we first consider the problem formulation consisting of a

single state variable x(t) and a single control variable u(t). Hence, the optimal

control problem is now reformulated as

Maximise∫ t1

t0

f(t, x(t), u(t)) dt,

subject todx

dt= g(t, x(t), u(t)),

x(t0) = x0, x(t1) = x1.

(2.35)

44

The functions f and g are assumed to be continuously differentiable, u is a piece-

wise continuous function and x changes over time based on the mathematical

model of the differential equation.

Notice that the variational calculus mathematical problem can also be solved

with optimal control theory by transforming

Maximise∫ t1

t0

f(t, x(t), x′(t)) dt

subject to x(t0) = x0,

(2.36)

into the optimal control formulation as follows

Maximise∫ t1

t0

f(t, x(t), u(t)) dt

subject todx

dt= u(t), x(t0) = x0.

(2.37)

The simplest optimal control problem has a free value of the state variable at the

terminal point, which is unlike in the calculus of variations that involves fixed

endpoints of the state variables. Now consider the following simple optimal con-

trol problem with the given endpoints at [0, T ] as

Maximise J =

∫ T

0

f(t, x(t), u(t)) dt (2.38a)

subject todx

dt= g(t, x(t), u(t)), (2.38b)

x(0) = x0 fixed, x(T ) = xT free. (2.38c)

A very important issue in optimal control theory is to find the first-order

necessary condition that must obey the maximum principle. This theory was for-

mulated by Pontryagin et al. [95] with the aim of finding the best possible control

from the state variable in the presence of constraints. This involves the concepts

of the costate variable, called Lagrange multiplier λ(t), and the Hamiltonian func-

45

tion which is defined as

H(t, x(t), u(t), λ(t)) = f(t, x(t), u(t)) + λ(t)g(t, x(t), u(t)). (2.39)

With this function, (2.38b) can be re-expressed in the form of the Hamiltonian

function as∂H

∂λ= g(t, x(t), u(t)) =

dx

dt. (2.40)

Unlike the Euler–Lagrange equation, which is a single second-order differential

equation of the state variable x(t), the maximum principle involves two first-

order differential equations, those are in terms of the state variable x(t) and the

costate variable λ(t).

Now let us redefine the objective functional in (2.38a) into the new expression

J =

∫ T

0

[f(t, x(t), u(t)) + λ(t)g(t, x(t), u(t))− λ(t)x′(t)

]dt. (2.41)

This equation has the same value as the previous objective functional in (2.38a)

when the last two terms of the integrand are equal to zero, which satisfies (2.38b).

Introducing the Hamiltonian function in (2.39) into (2.41), now the objective func-

tional becomes

J =

∫ T

0

[H(t, x(t), u(t), λ(t))− λ(t)x′(t)

]dt.

Then, integrating the last term by parts, the functional can be expanded to be-

come

J =

∫ T

0

[H(t, x(t), u(t), λ(t)) + x(t)λ′(t)

]dt− λ(T )xT + λ(0)x0. (2.42)

Next, we turn to the state variable x(t) that corresponds to the control variable

u(t). The necessary condition to solve the optimisation problem is utilising the

calculus of variations concept by finding the first derivative of the family curves

46

x(t) = x∗(t) + ϵp(t), written as

dx

dt=

dx∗

dt+ ϵ

dp

dt. (2.43)

Since the state variable corresponds to the control variable and the optimisation

problem can be solved by the calculus of variations as explained earlier in (2.37)

such that dx/dt = u(t), thus the family of curves in (2.43) can be redefined as

u(t) = u∗(t) + ϵq(t), (2.44)

where q(t) = dp/dt. Furthermore, if endpoint and the state variable at that point

are considered free, then we can also write

T = T ∗ + ϵ∆T and xT = x∗T + ϵ∆xT ,

which implies thatdT

dϵ= ∆T and

dxTdϵ

= ∆xT . (2.45)

Next, we want to provide the first-order condition for the objective function. First,

(2.43) and (2.44) are substituted into (2.42) so that J now depends on ϵ, specified

by

J(ϵ) =

∫ T

0

H(t, x∗(t) + ϵp(t), u∗(t) + ϵq(t), λ(t)

)+[x∗(t) + ϵp(t)

]λ′(t)

dt

− λ(T (ϵ))xT (ϵ) + λ(0)x0.

(2.46)

Then the first derivative with respect to ϵ is taken. For the integral term in (2.46)

we can write its first derivative as

∫ T (ϵ)

0

[(∂H

∂xp+

∂H

∂uq

)+ λ′p

]dt+

(H + λ′x

)∣∣t=T

dT

dϵ, (2.47)

whereas for the middle term, the two equations in (2.45) are employed to yield

47

−λ(T )dxTdϵ

− xTdλ(T )

dT

dT

dϵ= −λ(T )∆xT − xTλ

′(T )∆T. (2.48)

The last term of equation (2.46) vanishes after differentiation. Therefore the first-

order condition is obtained when the sum of (2.47) and (2.48) is equal to zero,

expressed as

dJ

dϵ=

∫ T (ϵ)

0

[(∂H

∂x+ λ′

)p+

∂H

∂uq

]dt

+H∣∣t=T

∆T − λ(T )∆xT = 0.

(2.49)

There are three different components on each terms in (2.49) which are arbitrary,

those are p(t) and q(t) on the integral term, ∆T at the second term, and ∆xT

at the last term. As a consequence, each term has to be zero to satisfy (2.49).

Furthermore, when the integral term equals zero, so do its two components since

p and q are arbitrary. Hence two conditions emerge from the integral term, i.e. the

multiplier equation and the optimality condition, written respectively as

dλ

dt= −∂H

∂xand

∂H

∂u= 0. (2.50)

Recall that the simple optimal control problem that we set earlier stating that the

terminal point T is fixed but xT is free. Therefore the second term in (2.49) drops

out due to ∆T = 0, while

λ(T ) = 0, (2.51)

is gained from the last term in (2.49) as ∆xT can be any value.

Based on the explanation, the maximum principle conditions for problem

(2.38) with the Hamiltonian function defined in (2.39) are provided by (2.40),

48

(2.50) and (2.51), or written collectively as

dx

dt=∂H

∂λ, state variable dynamics,

dλ

dt= −∂H

∂x, Lagrange multiplier dynamics,

∂H

∂u= 0, maximizing H,

λ(T ) = 0, transversality condition.

(2.52)

Current-value Hamiltonian

For many problems in economics, discounted future values are applied. That

is, the integrand objective function contains the additional factor e−ρt, where

ρ > 0. Thus, instead of f(t, x(t), u(t)

), now the integrand in (2.41) is replaced

by e−ρtf(t, x(t), u(t)

). This also applies to the Hamiltonian function, where by the

standard definition it now takes the form

H(t, x(t), u(t), λ(t)

)= e−ρtf

(t, x(t), u(t)

)+ λ(t)g

(t, x(t), u(t)

). (2.53)

Since the maximum principle involves the differentiation of H with respect to x

and u, and the additional discount factor gives more terms from the derivatives,

it is convenient to define a new Hamiltonian function which is free of the dis-

counted factor, namely the current-value Hamiltonian. This concept corresponds

to the Lagrange multiplier such that it defines the new (current-value) Lagrange

multiplier as

m(t) = eρtλ(t),

or

λ(t) = e−ρtm(t). (2.54)

Therefore the current-value Hamiltonian of (2.53), denoted by Hc, can be ex-

49

pressed by

Hc(t, x(t), u(t),m(t)) = eρtH(t, x(t), u(t), λ(t))

= f(t, x(t), u(t)

)+m(t)g

(t, x(t), u(t)

).

(2.55)

Seeing that we utilise the new definition of the Hamiltonian function Hc, all

conditions of the maximum principle given in (2.52) have to be inspected to see

whether those conditions need revision.

As we know that the first derivative of Hc in (2.55) with respect to the La-

grange multiplier m(t) is equal to function g(t, x(t), u(t)

), hence the first condi-

tion stated in (2.52) can be redefined in terms of the Hamiltonian current value

as∂H

∂λ= g

(t, x(t), u(t)) =

∂Hc

∂m.

Thus we can revise the first maximum principle condition in (2.52) as

dx

dt=∂Hc

∂m.

Now for the second equation in (2.52), we transform the left-hand side the equa-

tion using (2.54) to obtain

dλ

dt= e−ρtdm

dt− ρe−ρtm,

and the right-hand side using (2.55) to become

−∂H∂x

= −e−ρt∂Hc

∂x

from (2.55). As a result, the second condition of the maximum principle in (2.52)

is deduced asdm

dt= −∂Hc

∂x+ ρm. (2.56)

50

The condition remains unchanged for the third condition, which is finding the

maximum H with the corresponding value u. This is due to the discount factor

e−ρt in the equation (2.55) being independent of u. Therefore the equation now

becomes∂Hc

∂u= 0. (2.57)

The last equation, when λ(T ) = 0, can be revised using (2.54) as

e−ρTm(T ) = 0, (2.58)

Since e−ρT is nonzero for finite T , the transversality condition in the last equation

of (2.52) changes to

m(T ) = 0.

In conclusion, the maximum principle conditions based on the current-value Ha-

miltonian function, defined in (2.55), can be written as

dx

dt=∂Hc

∂m, state variable dynamics,

dm

dt= −∂Hc

∂x+ ρm, Lagrange multiplier dynamics,

∂Hc

∂u= 0, maximizing Hc,

m(T ) = 0, transversality condition.

(2.59)

The optimal control problem with two variables

Next the problem is extended to two state variables x1 and x2, while the number

of control variables is assigned the same, which is u. Thus the optimal control

51

problem in (2.38) now becomes

Maximise J =

∫ T

0

f(t, x1(t), x2(t), u(t)) dt

subject todx1dt

= g1(t, x1(t), x2(t), u(t)),

dx2dt

= g2(t, x1(t), x2(t), u(t)),

x1(0) = x10, x2(0) = x20 fixed,

x1(T ) = x1T x2(T ) = x2T free.

(2.60)

Meanwhile, the Hamiltonian function now includes two Lagrange multipli-

ers, say λ1(t) and λ2(t), defined as

H(t, x1(t), x2(t), u(t), λ1(t), λ2(t))

=f(t, x1(t), x2(t), u(t)) + λ1(t)g1(t, x1(t), x2(t), u(t))+

λ2(t)g2(t, x1(t), x2(t), u(t)).

(2.61)

To solve the optimisation control problem (2.60), we just follow the same way

with that of the maximum principle of one variable as given in the previous prob-

lem. Thus with this new problem the maximum principle condition for one state

variable given in (2.52) ends up with the following conditions.

dxidt

=∂H

∂λi, i = 1, 2,

dλidt

= −∂H∂xi

, i = 1, 2,

∂H

∂u= 0,

λi(T ) = 0, i = 1, 2.

(2.62)

Observe that these conditions are very similar to those of the one variable prob-

lem given in (2.52).

Next, when the current value Hamiltonian is applied, it has two Lagrange

multipliers, and we can denote those as mi(t) = e−ρtλi(t) for i = 1, 2, such that

52

the conditions in (2.62) eventually change, specified as

dxidt

=∂Hc

∂mi

, i = 1, 2,

dmi

dt= −∂Hc

∂xi+ ρmi, i = 1, 2,

∂Hc

∂u= 0,

mi(T ) = 0, i = 1, 2.

(2.63)

2.6 Optimal control in harvesting

The purpose of harvesting is unlikely to gather as many resources as possible

which may lead to the destruction or extinction of resources. Instead, its aim is to

obtain the maximum of the objective with a certain level of harvesting so that the

resources can still be sustained. In fisheries management, one of the objectives

is to earn the maximum long-term profit which is represented by present value

of discounted net economic revenue. Present value (PV) is the current value of a

future sum of money or cash flows given a specific rate of return. It states whether

the amount of money today is worth more than the same amount in the future.

Meanwhile, a discounted rate is an interest rate which relates a present value of

money and its growth in the future, i.e. a discount rate is deducted from a future

value of money to provide its present value.

Even though earning maximum net profit is a very common objective, this

does not seem to be the only policy objective that motivates most fisheries man-

agement [20]. The management of the sustainability issue in an ecosystem is also

important. When sustainability is implemented in the objective, the standard

model used by economics is no longer applicable. This issue has been inves-

tigated by Clark [23] that under extreme circumstances, the “optimal” fisheries

management leads to resource extinction. However, the policy objective of eco-

nomic net profit is used in this thesis, omitting the sustainability issue.

53

Now let the general discounted present value [25] be modelled as

PV =

∫ ∞

0

α(t)π(t, N(t), E(t)) dt, (2.64)

where α represents the discount rate, π is the net revenue that the harvester earns

with the harvesting effort E(t) that changes over time. Here, we use SFU (Stan-

dardized Fishing Unit) as the unit of effort, while the fish population N(t) is cal-

culated as a biomass in tonnes.

Next, we specify the discount rate in the following way. The profit is dis-

counted at the rate ρ as a unit of money today is worth more than a unit amount

of money tomorrow. Furthermore, a unit of money after t units of time in the fu-

ture is worth (1−ρ)t today. If the discount is compounded n times per unit of time

such that the discount periods are nt, then the present value obtained after t units

of time in the future is (1− ρ/n)nt. Thus, allowing for continuous compounding,

the present value of a unit of money is calculated by

limn→∞

(1− ρ

n

)nt

= e−ρt.

Gordon [45] established the economic model in fisheries, where the net revenues

obtained from fishing are defined as a function of total sustainable revenue de-

ducted by total cost, or we can write it as

Net Revenue = ph− cost, (2.65)

where p is the constant price per unit harvest, h is the sustainable yield.

Harvesting with linear cost

Following the proposed optimal harvesting strategy by Clark & Munro [28], as

well as Rodin [100], the harvesting rate and the harvesting cost are assumed to be

linearly proportional with respect to E(t), formed by h(E(t), N(t)

)= qE(t)N(t)

54

and cost = cE(t), respectively. Hence, from (2.65), the net revenue can be calcu-

lated as

π(t, N(t), E(t)) = ph(E(t), N(t))− cE(t)

= pqE(t)N(t)− cE(t).

As a result, recalling (2.64), the discounted present value with linear cost can be

expressed as

PV =

∫ ∞

0

e−ρt[pqN(t)− c]E(t) dt.

Harvesting with quadratic cost

Next, the cost of harvesting is set to be quadratic with respect to the effort, given

by c1E(t)+ c22E(t)2. Meanwhile, the harvesting rate remains unchanged in a linear

function of the effort. This quadratic form of cost harvesting appear to be more

acceptable than the linear one, as has been pointed out by Sancho & Mitchel [106]

and Holt et al. [53]. Furthermore, Hanson [47] stated that the quadratic form can

occur due to the use of unspecialised boats and vessels for additional effort when

the best ones have already been employed. Thus based on (2.65), the net revenue

can be specified by

π(t, E(t), N(t)) =pqE(t)N(t)−[c1E(t) +

c22E(t)2

]=[pqN(t)− c1 −

c22E(t)

]E(t).

(2.66)

Therefore the discounted present value with quadratic cost is written as

PV =

∫ ∞

0

e−ρt[pqN(t)− c1 −

c22E(t)

]E(t) dt.

Next, we provide a problem with the goal is to maximise the present value

of discounted profit with quadratic cost by optimising the control variable effort

55

E(t), where the initial time is zero and the terminal time is T . Thus we wish to

determine the objective function of the optimal control problem as

Maximise J =

∫ T

0

e−ρt[pqN(t)− c1 −

c22E(t)

]E(t) dt (2.67)

subject todN

dt= rN(t)

[1− N(t)

K

]− qE(t)N(t), (2.68)

N(0) = N0. (2.69)

Another condition also needs to be considered, that is the range of harvesting

effort such that the negative value of population does not happen. This can be

done by examining the population size steady state, taking the equation in (2.68)

to be zero to obtain

N∞ = K

[1− qE

r

].

This steady state value has the maximum number at N∞ = K when E = 0.

However, when the effort is made increasing until reaches E = r/q, then the

steady state reduces towards zero. Beyond this value the steady state will tend

to negative value. Thus, to make the steady state plausible, the effort must stay

in the range [0, r/q]. In other words for a biologically realistic situation, we must

assume Emax = r/q.

This optimisation problem and its formulation is derived from [111], where

the two approaches, the calculus of variations and the Hamiltonian function me-

thod, are applied to solve this problem as well as some numerical simulations.

2.6.1 Variational approach

Let E∗(t) denote the effort value that optimises the objective functional with the

corresponding biomass of fish population N∗(t). Employing variation to both

variables, we can write

E(t) = E∗(t) + s(t),

56

N(t) = N∗(t) + x(t),

where s(t) and x(t) are arbitrary and small. Substituting these into (2.67), we have

the variation of J as

∆J =

∫ T

0

e−ρt[pq(N∗ + x)− c1 −

c22(E∗ + s)

](E∗ + s) dt

−∫ T

0

e−ρt(pqN∗ − c1 −

c22E∗

)E∗ dt.

(2.70)

Since we want to maximise J , (2.70) should be set to be zero. Therefore, by sim-

plifying (2.70), we obtain

∆J =

∫ T

0

e−ρts(pqN∗ − c1 − c2E∗) dt+

∫ T

0

e−ρtx(pqE∗) dt = 0. (2.71)

by discarding the second order terms in s and x. Similarly, from (2.68) we have

d

dt(N∗ + x) = r(N∗ + x)

(1− N∗ + x

K

)− q(E∗ + s)(N∗ + x).

Expanding the latter equation, then applying the first line (2.68) at E∗ with corre-

sponding N∗ and setting the second order of s and x equal to zero yields

dx

dt= r

(1− 2N∗

K

)x− qE∗x− qN∗s.

From this equation, we can express s as

s =r(1− 2N∗/K

)x− dx/dt− qE∗x

)qN∗ ,

then substituting it into (2.71), we can write the latter as

∫ T

0

e−ρt

(pqN∗ − c1 − c2E

∗)[r(1− 2N∗/K)− qE∗

qN∗

]+ pqE∗

x dt

−∫ T

0

e−ρt(pqN∗ − c1 − c2E

∗

qN∗

)dxdt

dt = 0.

(2.72)

57

Integrating the second integral in (2.72), the equation (2.72) now becomes

∫ T

0

e−ρt

(pqN∗ − c1 − c2E

∗)[r(1− 2N∗/K)− qE∗

qN∗

]+ pqE∗

−ρ(pqN∗ − c1 − c2E

∗

qN∗

)− c1

q

( 1

N∗

)′− c2

q

(E∗

N∗

)′x dt

−e−ρT

[p− c1 − c2E

∗(T )

qN∗(T )

]x(T ) +

[p− c1 − c2E

∗(0)

qN∗(0)

]x(0) = 0

(2.73)

Recall that x(t) is arbitrary, hence each of the three terms in (2.73) must be equal

to zero. As a result, we can now break up the expression (2.73) into three parts.

For the first part or integral term, since x(t) can be any function, we have

e−ρt

(pqN∗ − c1 − c2E

∗)[r(1− 2N∗/K)− qE∗

qN∗

]+ pqE∗

−ρ(pqN∗ − c1 − c2E

∗

qN∗

)− c1

q

( 1

N∗

)′− c2

q

(E∗

N∗

)′

= 0.

(2.74)

Then, for the second part in (2.73) we obtain

[p− c1 − c2E

∗(T )

qN∗(T )

]= 0 (2.75)

since the terminal state N∗(T ) is free which implies x(T ) = 0. Meanwhile, at

the initial state, x(0) = 0 since N0 is given that makes the last term in (2.73) van-

ish. Rearranging equation (2.74) and (2.75), we now have the boundary condition

E∗(T ) =pqN∗(T )− c1

c2(2.76)

and the dynamical system of the control variable

dE∗

dt=rE∗N∗

K+ ρE∗ +

pq

c2(r − ρ)N∗ − 2rpq

c2KN∗2 +

c1c2

(ρ+

rN∗

K

),

respectively

In conclusion, to obtain the effort that maximises (2.67), we need to numer-

ically solve the following dynamical system for the optimal control E∗ with the

58

corresponding N∗ as follows

dE∗

dt=rE∗N∗

K+ ρE∗ +

pq

c2(r − ρ)N∗ − 2rpq

c2KN∗2 +

c1c2

(ρ+

rN∗

K

),

dN∗

dt= rN∗

(1− N∗

K

)− qE∗N∗.

(2.77)

Since the initial valueN0 is known and terminal value is now defined in (2.76), the

initial value E(0) can be approximated using a numerical method that relates to

two-point boundary value problems, for instance, the finite difference or shooting

method which can be found in [36, 57, 61]. This simulation uses parameter values

which are utilised by Suri [111] and provided in Table 2.1. Also, the initial fish

population biomass is chosen to be N0 = K = 106 tonnes. Figure 2.5 shows the

graph of the optimal effort over the time 0 ≤ t ≤ 1 as the numerical solution

of (2.77).

Parameter Description Value Unit

ρ Discount rate 0.1 year−1

r Intrinsic growth rate 0.71 year−1

p Unit harvest price 0.5 $ tonnes−1

q Catchability coefficient 0.0001 SFU−1 year−1

c1 Unit effort cost coefficient 1 0.01 $ SFU−1 year−1

c2 Unit effort cost coefficient 2 0.01 $ SFU−2 year−1

K Carrying capacity 106 tonnes

Table 2.1: Parameters used to perform simulation of the model in (2.77).

As can be observed, the optimal effort stays at ranges between approximately

3640 to 3860 SFU over the time period [0, 1], which is plausible since these values

do not exceed the maximum effort Emax = r/q = 7100 SFU. Also, notice that

initially the effort does not need to be increased to obtain the optimal net profit,

but decreasing the value until it is half way keeps the net profit optimum. Then

the effort needs to rise again to maintain the optimal net profit.

59

0.0 0.2 0.4 0.6 0.8 1.0

3640

3740

3840

Figure 2.5: The effort values that maintain the optimal net profit in (2.67) usingcalculus of variations method.

2.6.2 Hamiltonian method

In this section, the Hamiltonian method based on the Pontryagin maximum prin-

ciple is implemented to find the maximum of the objective function, which is

the same previously defined in (2.67). For convenience, we rewrite the problem

here:

Maximise J =

∫ T

0

e−ρt[pqN(t)− c1 −

c22E(t)

]E(t) dt (2.78)

subject todN

dt= rN(t)

(1− N(t)

K

)− qE(t)N(t). (2.79)

Therefore, as defined in (2.55), the current-value Hamiltonian function (denoted

by H instead of Hc for convenience) can be expressed here as

H(E,N,m) =(pqN(t)− c1−

c22E(t)

)E(t)+m(t)

[rN(t)

(1− N(t)

K

)− qE(t)N(t)

],

(2.80)

where m(t) denotes the current-value Lagrange multiplier.

As we wish to find the control variable E(t) that maximises the objective

60

function, we need to solve the system of Euler–Lagrange equations given in (2.56)

and (2.57). From condition (2.56), we expand its right-hand side of the equation

using the first derivative of (2.80) with respect to N to obtain

dm

dt= ρm(t)− rm(t)− q

c2

[p−m(t)

][p−m(t)qN(t)− c1

]+

2rm(t)N(t)

K,

or we may write

dm

dt= (ρ− r)m(t) +

2rm(t)N(t)

K−q2[p−m(t)

]2N(t)

c2+c1q

[p−m(t)

]c2

. (2.81)

Meanwhile from condition (2.57), we can write

∂H

∂E=

[pqN(t)− c1 − c2E(t)

]−m(t)qN(t) = 0

to obtain

E(t) =

[p−m(t)

]qN(t)− c1

c2. (2.82)

The transversality condition is also taken into account that referred to (2.58) such

that we can deduce

m(T ) = 0 (2.83)

since the endpoint T is finite. Subtituting (2.82) into (2.79) we can reformulate

the population dynamical system. By this dynamical system and (2.81) with the

terminal value (2.83), we establish the following system of ODEs to solve the

control variable that maximises the objective function as

dN

dt= rN(t)

[1− N(t)

K

]− q2

c2

[p−m(t)

]N(t)2 +

c1qN(t)

c2,

dm

dt= (ρ− r)m(t) +

2rm(t)N(t)

K−q2[p−m(t)

]2N(t)

c2+c1q

[p−m(t)

]c2

,

(2.84)

where

N(0) = N0 and m(T ) = 0. (2.85)

61

Implementing numerical solution to this two-point boundary problem, we can

0.0 0.2 0.4 0.6 0.8 1.0

3640

3740

3840

Figure 2.6: The effort values that maintain the optimal net profit in (2.78) usingHamiltonian method

solve N(t) and λ(t) that implies the solution E(t) given by (2.82). As in the pre-

vious simulation, we also use the parameter values given in Table 2.1, and the

numerical simulation is given in Figure 2.6. .

62

Chapter 3

Modelling Carrying Capacity as Food

Availability

In this chapter we propose more sophisticated human population growth models

that relate human carrying capacity with food production data. The first popula-

tion model is proposed as a single ODE with a variable carrying capacity, where

three classes of carrying capacity models that depend on food availability are im-

plemented. An integration-based method for model fitting is used to obtain ex-

plicit formulas for the parameters. Numerical simulations for the three carrying

capacity models are presented to examine the population dynamics and to find

out which carrying capacity model gives the best fit compared to the actual popu-

lation data. For information, this model has been published in 2020 by ANZIAM

Journal, Vol. 62, pp. 318–333 with title “Modelling human carrying capacity as a

function of food availability” [122].

The population model is then modified to a coupled system of two ODEs

by adding a carrying capacity growth rate derived from [105] and [112] into the

model. Like the previous model, we use three classes of carrying capacity models

that depend on food availability and we utilise an integration-based method for

model fitting to obtain explicit formula for the parameters. Lastly, numerical sim-

63

ulations are also performed to see population behaviours, as well as comparison

of theoretical and actual results.

3.1 Population model with one ODE

As a first step in modelling human population growth with a variable carrying ca-

pacity, we follow Hopfenberg [54] and postulate that food production data is the

sole variable that influences human carrying capacity. In fact, Hopfenberg [54]

assumed a simple linear relationship between human carrying capacity and food

production index. Using the available FAO food production data and despite a

crude fitting procedure, model parameters were estimated that yielded popula-

tion estimates that closely approximate actual population numbers. However,

population forecasting was not discussed in [54].

3.1.1 Mathematical formulation

Let us consider the classical logistic equation and assume a variable carrying ca-

pacity, namelydN

dt= rN

[1− N

K(t)

], (3.1)

where N(t) is the human population number at time t, r is the (constant and

positive) intrinsic growth rate and K(t) is the carrying capacity at time t. As

stated in the previous section, we suppose for simplicity that food production

data is the only variable that influences human carrying capacity. More precisely,

we assume that K(t) = f(I(t)) (where I(t) is the food production index at time t)

for some suitable smooth function f of I such that f(0) = 0 and f(I) > 0 for I > 0.

The former says that there is zero carrying capacity if no food is available (this is

a mathematical idealisation since I(t) > 0 in practice; hence K(t) > 0), while the

latter is due to the fact that carrying capacity is a positive quantity.

Moreover, we will consider three models depending on the properties of the

64

function f .

(a) f ′(I) > 0 for I > 0 and f(∞) = ∞.

This model assumes that human carrying capacity increases indefinitely

with increasing food production. A family of examples is f(I) = αIp, where

α > 0 and p ≥ 0. When p = 1, we recover Hopfenberg’s model [54], while

p = 0 reduces to the classical logistic equation with a constant carrying ca-

pacity.

(b) f ′(I) > 0 for I > 0 and 0 < f(∞) <∞.

This is similar to the previous model in that human carrying capacity in-

creases with increasing food production but it does not do so indefinitely

and tends to some finite positive limiting value. We say that the human

carrying capacity is ‘self-limiting’. Some examples are f(I) = αI/(1 + I) or

f(I) = α(1− e−I), where α > 0. It is easy to see that 0 < f(∞) = α <∞.

(c) There exists I∗ > 0 such that f ′(I) > 0 for 0 < I < I∗ and f ′(I) < 0 for I >

I∗, i.e. f has a unique global maximum at I∗. Furthermore, 0 < f(∞) <∞.

Here we assume that there is a critical threshold value for the food pro-

duction index. If the food production index is below the threshold, then

the carrying capacity increases with the food supply, like in models (a) and

(b). However, too much food production (and hence a food production in-

dex greater than the threshold) leads to a lowering of the carrying capacity.

Some examples are f(I) = αI(1 + I)/(1 + I2) or f(I) = α(I − 1)e−I + α,

where α > 0.

The above examples for f can be expressed in the form f(I) = αg(I), where

α > 0 and g(I) > 0 for I > 0. The parameter α is to be estimated by fitting

the model to the population data, while the functional form for g is specified

according to the behaviour desired for the human carrying capacity. Of course,

in principle, g may also depend on one or more parameters that will also have

65

to be estimated. However, as an initial attempt at modelling and to keep the

parameter estimation tractable, we will assume that g does not depend on any

unknown parameters.

3.1.2 Integration-based parameter estimation method

If we set K(t) = αg(I(t)), then (3.1) becomes

N ′(t) = rN(t)− r

α

N(t)2

g(I(t)). (3.2)

Suppose for the moment that N(t) and I(t) are known for all 0 ≤ t ≤ T for some

positive T . Our goal here is to find explicit formulas for α and/or r using the

integration-based method of Holder and Rodrigo [52].

Let w = w(t; s) be a suitable positive weight function parametrised by s ≥

0. Some possible weight functions are, for example, w(t; s) = e−st or w(t; s) =

1/(1 + t)s. If we choose the exponential function, then∫ T

0w(t; s)N(t) dt can be

viewed as a finite Laplace transform. Multiplying both sides of (3.2) by w(t; s)

and integrating by parts, we obtain

w(T ; s)N(T )− w(0; s)N(0)−∫ T

0

w′(t; s)N(t) dt

= r

∫ T

0

w(t; s)N(t) dt− r

α

∫ T

0

w(t; s)N(t)2

g(I(t))dt.

(3.3)

To simplify the notation, define

a(s) =

∫ T

0

w(t; s)N(t) dt,

b(s) = −∫ T

0

w(t; s)N(t)2

g(I(t))dt,

c(s) = w(T ; s)N(T )− w(0; s)N(0)−∫ T

0

w′(t; s)N(t) dt,

(3.4)

so that (3.3) becomes

ra(s) +r

αb(s) = c(s). (3.5)

66

Note that in (3.5), a(s), b(s) and c(s) are known quantities for a fixed s. We can

think of (3.5) as a ‘generating equation’ that is used to generate algebraic equa-

tions for α and/or r by assigning specific values to s. In principle, the same values

for α and/or r should be obtained for any value of s ≥ 0 provided the logistic

equation (3.1) were an exact model of human population growth. In practice, of

course, this may not be the case. However, if we believe in the validity of the

logistic model, the parameter values thus obtained should be robust with respect

to the choice of s although only a heuristic justification of this was given in [52].

Rate r is known

If the intrinsic growth rate r is assumed to be known as in [54], then choosing

s = s0 ≥ 0 in (3.5) yields the explicit formula

α =rb(s0)

c(s0)− ra(s0), (3.6)

provided that it is positive.

Remark 3.1.1. From (3.4) we see that b(s0) < 0. Suppose that there exists M >

0 such that 0 < N(t) < M for all 0 ≤ t ≤ T . For definiteness assume that

w′(t; s0) < 0 for all 0 ≤ t ≤ T . This is the case, for example, when w(t; s) = e−st or

w(t; s) = 1/(1 + t)s. The case when w′(t; s0) > 0 for all 0 ≤ t ≤ T can be treated

similarly. Then

w(T ; s0)N(T )− w(0; s0)N(0)−∫ T

0

w′(t; s0)N(t) dt ≥ w(T ; s0)N(T )− w(0; s0)N(0)

−M [w(T ; s0)− w(0; s0)].

This gives

c(s0)− ra(s0) ≥ −[M −N(T )]w(T ; s0) + [M −N(0)]w(0; s0)− rM

∫ T

0

w(t; s0) dt.

67

If s0 is such that

[M −N(T )]w(T ; s0)− [M −N(0)]w(0; s0) + rM

∫ T

0

w(t; s0) dt < 0, (3.7)

then c(s0) − ra(s0) > 0 and therefore α < 0, a contradiction. For example, if

w(t; s) = e−st, then (3.7) simplifies to

[M −N(T )]e−s0T − [M −N(0)] +rM

s0(1− e−s0T ) < 0. (3.8)

Since the limit of the left-hand side of (3.8) as s0 → ∞ is −[M −N(0)] < 0, we de-

duce that s0 cannot be taken too large. For a more general weight functionw(·; s0),

s0 should not be chosen so that (3.7) holds.

Remark 3.1.2. Here we investigate the robustness of α in (3.6) with respect to s0.

One way is to sketch α vs. s0 and determine subintervals of s0 where α is “almost

constant” and positive. We then choose any s0 in such subintervals. Another way

is to consider dα/ds0. Differentiating (3.6) with respect to s0 yields

dα

ds0= r

b′(s0)c(s0)− b(s0)c′(s0)− ra(s0)b

′(s0) + ra′(s0)b(s0)

[c(s0)− ra(s0)]2. (3.9)

In particular, if w(t; s) = e−st, then (3.4) gives

a′(s0) = −∫ T

0

te−s0tN(t) dt,

b′(s0) =

∫ T

0

te−s0tN(t)2

g(I(t))dt,

c′(s0) = −T e−s0TN(T ) +

∫ T

0

e−s0tN(t) dt− s0

∫ T

0

te−s0tN(t) dt.

If N = N(t) is an exact solution of (3.2), then of course dα/ds0 = 0 for any s0.

Otherwise, sketching dα/ds0 vs. s0 would indicate subintervals of s0 where the

graph is close the s0-axis. We then choose s0 in one of these subintervals.

Rate r is unknown

68

If the intrinsic growth rate r is not assumed to be known, then we need to de-

termine α and r simultaneously. For this we choose two convenient nonnegative

values of s, e.g. s1 and s2 with s1 = s2, in (3.5) to produce the linear algebraic

system a(s1) b(s1)

a(s2) b(s2)

r

r/α

=

c(s1)c(s2)

for r and r/α. More specifically,

α =c(s1)b(s2)− c(s2)b(s1)

a(s1)c(s2)− a(s2)c(s1), r =

c(s1)b(s2)− c(s2)b(s1)

a(s1)b(s2)− a(s2)b(s1), (3.10)

assuming that both quantities are positive. Conditions analogous to (3.7) to en-

sure the positivity of α and r in (3.10) can also be derived that give restrictions

on s1 and s2. Similarly, the robustness of α and r with respect to s1 and s2 can be

investigated by looking at regions in the s1s2-plane where either (i) the surfaces

α and r given in (3.10) are “almost constant” or (ii) the surfaces ∂α/∂s1, ∂α/∂s2,

∂r/∂s1 and ∂r/∂s2 are “close” to the s1s2-plane.

Remark 3.1.3. In practice, N(t) and I(t) are not known for all 0 ≤ t ≤ T . Rather,

discrete values Nj and Ij , where j = 0, 1, . . . , n, are given at corresponding time

values tj such that t0 = 0 and tn = T . Thus the integrals appearing in (3.4) will be

evaluated using numerical quadrature.

Remark 3.1.4. Eq. (3.1) is a Bernoulli equation, whose exact solution is

N(t; r, α) =1

e−rt/N(0) + (r/α)∫ t

0e−r(t−u)/g(I(u)) du

. (3.11)

A nonlinear least squares approach to estimate r and α involves the minimisation

of the squared error

E(r, α) =n∑

j=0

[N(tj; r, α)−Nj]2.

In the case of a constant carrying capacity, the integral appearing in (3.11) can be

evaluated explicitly and partial derivatives of E with respect to r and α can be

69

calculated in principle. Here, however, this is not straightforward since one of

the unknown parameters r appears inside the integral, which cannot be evalu-

ated explicitly since it depends on g(I(u)). The integration-based method we use

in this article is easy to implement as we have explicit formulas for α and/or r

involving integrals that can be evaluated numerically.

3.1.3 Numerical simulations

We now present the results of the model fitting and population forecasting. The

world population data [32] and world food production index data [39] can be

downloaded from the World Bank website. Both data sets are visualised in Fig-

ure 3.1.

1962 1970 1980 1990 2000 2014

4

6

3

5

7

3.5

4.5

5.5

6.5

7.5

Year

World Population

(in billion)

(a)

1962 1970 1980 1990 2000 2014

100

40

60

80

120

Year

World Food Index

(b)

Figure 3.1: Data sets for (a) world population (in billion) and (b) world foodindex, the net food production of the agricultural sector in the world per person,from years 1962 to 2014.

The food production index is a measure of the net food production of a coun-

try’s agricultural sector per person. This covers all edible agricultural products

that contain nutrients. The FAO determines these numbers relative to the aver-

age food production for three years and sets the average for these three years

equal to 100. Hopfenberg [54] used the three-year period from 1989-1991 while

we use the current three-year period from 2004-2006. In Figure 3.1b, an index

70

value greater than 100 means food production is increasing with respect to the

base years 2004-2006; otherwise it is decreasing.

There are three steps to be implemented in the procedure. The ‘parameter

estimation’ step applies the integration-based technique from the previous sec-

tion and makes use of the data from 1962 (t = 0) to 1991 (t = 29); thus T = 30.

We use three different values of s0 or (s1, s2) to find out which one gives the best

estimates for α and/or r, respectively. The choice of three values is guided by

the heuristic arguments given in Remark 3.2. The ‘error estimation’ step uses the

data from 1992 (t = 30) to 2014 (t = 53). Here we use the estimated parameters

from the parameter estimation step and solve the logistic equation (3.2) numeri-

cally to approximate the population from 1962 (t = 0) to 2014 (t = 53). Note that

the last available population data are for 2014. Then we calculate the root mean

square (RMS) error between the numerically obtained population number and

the actual population data from 1992 (t = 30) to 2014 (t = 53). The magnitude of

the errors will give an indication of which of the models (a), (b) or (c) with corre-

sponding s0 or (s1, s2) gives the best fit to the given data. Finally, the ‘population

forecasting’ step is to solve (3.2) numerically from 2015 (t = 54) to 2120 (t = 158),

thus predicting the population trend after 2014.

As we can observe in Figure 3.1b, the food production index exhibits an ex-

ponential trend. Therefore it is reasonable to implement a linear least squares

technique to obtain the approximate curve I(t) ≃ 32.86e0.025t using the data from

1962 (t = 0) to 1991 (t = 29).

For the weight function we take w(t; s) = e−st. Note that numerical simu-

lations were also performed with the weight function w(t; s) = 1/(1 + t)s and

similar results were obtained.

Rate r is known

Here we estimate the parameter α only and fix r = 0.03, as in [54]. By trying out

three different values for s0, the model fitting step using (3.6) and then the error

71

estimation step are implemented. Table 3.1a shows the results for the three car-

rying capacity models. It shows that for model (a) with f(I) = αI , the value α =

0.23 gives the best estimate since the RMS error has the lowest value. Meanwhile,

for models (b) with f(I) = αI/(1 + I) and (c) with f(I) = αI(1 + I)/(1 + I2), the

values α = 11.38 and α = 10.97 are the respective best approximations.

0 10.2 0.4 0.6 0.80.1 0.3 0.5 0.7 0.9

0

2

4

6

Model(a)

Model(b)

Model(c)

Figure 3.2: Root mean square error between numerical and actual populationnumbers as a function of s0 for three carrying capacity models.

To justify the choices of s0 in Table 3.1a, Figure 3.2 depicts the RMS error on

the interval [0, 1], where the larger s0 yields the larger value of the error. The value

of s0 that results in the smallest RMS error is then used to approximate the popu-

lation number as provided in Figure 3.3a. On the other hand, Figure 3.4a shows

the graph of α vs. s0 from (3.6) and Figure 3.4b shows dα/ds0 vs. s0 from (3.9).

We see that any s0 in the subinterval [0.00, 0.5] gives an “almost constant” and

positive α and an “almost zero” dα/ds0; hence the choices of s0 = 0.00, 0.01, 0.10

in Table 3.1a.

From these estimations we can now implement the population forecasting

step to find out if the population will grow without bounds, which is arguably

unrealistic, or if it tends to some limiting population value for large times. Fig-

ure 3.3b shows that the models (b) and (c) give a reasonable result as the pop-

ulation number reaches a ‘limiting carrying capacity’ α. This is because when t

is large, then I(t) is also large as it is approximated by an exponentially increas-

ing function. Since K = f(I), when I is large, then the carrying capacity will

72

1962 1980 1990 2000 2014

4

6

8

3

5

7

3.5

4.5

5.5

6.5

7.5

Year

World population

(in billion)

Model(a)

Model(b)

Model(c)

Actual data

(a)

2015 2040 2060 2080 2100 2120

0

20

40

60

10

30

50

Year

World population

(in billion)

Model(a)

Model(b)

Model(c)

(b)

Figure 3.3: Comparison of the best three carrying capacity models (when r isknown) with respect to (a) approximation with the actual data from 1962-2014and (b) forecasting from 2015-2120.

tend towards α. On the other hand, when model (a) is applied, then the carrying

capacity will approach infinity since I becomes large, therefore the population

number increases without bound.

In summary, since models (b) and (c) give smaller errors than model (a) af-

ter fitting actual population data, the numerical simulations indicate that around

100 years from now, the projected world population is about 11 billion (the ap-

proximate value of α in models (b) and (c)). This is in stark contrast to the pro-

jected population from model (a), which is around 92 billion. Note that model (a)

with f(I) = αI is identical to the model proposed by Hopfenberg [54].

Rate r is unknown

Next we estimate both parameters α and r using (3.10). This time we choose three

pairs of values for (s1, s2). We again choose f(I) = αI/(1 + I) for model (b) and

f(I) = αI(1 + I)/(1 + I2) for model (c) as before when r is assumed to be given.

However, if we choose f(I) = αIp, with p = 1 for model (a), then (3.10) yields

negative values for r as (s1, s2) is made to vary. Also, when p > 1 is taken, the neg-

ative sign of α is obtained even though r is positive. Thus we choose f(I) = αI1/4

73

0 10.2 0.4 0.6 0.80.1 0.3 0.5 0.7 0.9

0

20

10

5

15

Model(a)

Model(b)

Model(c)

(a)

0 10.2 0.4 0.6 0.80.1 0.3 0.5 0.7 0.9

0

20

40

60

80

Model(a)

Model(b)

Model(c)

(b)

Figure 3.4: Graphs showing (a) the parameter α estimated as a function of s0 (cal-culated by (3.6)) and (b) the gradient of α with respect to s0 (obtained from (3.9)).

instead (other forms, where 0 < p < 1 may be taken as well). Table 3.1b sum-

marises the results for the α and r estimates together with the corresponding

RMS errors.

We see that when (s1, s2) = (0.05, 0.10) all three models give very good es-

timates, and again models (b) and (c) are (marginally) better than model (a). To

justify the choices of (s1, s2) in Table 3.1b, Figure 3.5 presents the RMS error of all

three models, where the subinterval [0.0, 0.15] × [0.0, 0.15] gives consistant val-

ues. Therefore, by this subinterval, any (s1, s2) yields “almost constant” and

positive α and r as shown by Figure 3.6. This is why we chose (s1, s2) = (0.00,

0.01), (0.00,0.10), (0.05,0.10) in Table 3.1b. Population approximations for the best

value (s1, s2) for each model are provided in Figure 3.7a. Meanwhile, popula-

74

Model (a): f(I) = αI

s0 α RMS0.00 0.23 0.4150.01 0.23 0.461

0.10 0.26 0.970

Model (b): f(I) = αI/(1 + I)

s0 α RMS0.00 11.41 0.0004

0.01 11.38 0.00030.10 11.16 0.0017

Model (c): f(I) = αI(1 + I)/(1 + I2)

s0 α RMS0.00 10.97 0.00010.01 10.94 0.0002

0.10 10.66 0.0065

(a)

Model (a) : f(I) = αI1/4

(s1, s2) r α RMS(0.00, 0.01) 0.032 3.86 0.021

(0.00, 0.10) 0.032 3.78 0.018

(0.05, 0.10) 0.032 3.75 0.016Model (b) : f(I) = αI/(1 + I)

(s1, s2) r α RMS(0.00, 0.01) 0.027 13.36 0.011

(0.00, 0.10) 0.028 12.93 0.008

(0.05, 0.10) 0.028 12.71 0.006Model(c) : f(I) = αI(1 + I)/(1 + I2)

(s1, s2) r α RMS(0.00, 0.01) 0.027 13.32 0.011

(0.00, 0.10) 0.027 12.88 0.008

(0.05, 0.10) 0.028 12.65 0.006(b)

Table 3.1: Parameter estimation for three carrying capacity models when (a) r isknown at r = 0.03 and (b) r is unknown.

tion projections using the best approximation for each of the models are shown

in Figure 3.7b.

We summarise as follows. Similar to the case when r was assumed known,

from the numerical simulations we infer that around 100 years from now, the

projected world population is about 13 billion (the approximate value of α in

models (b) and (c)). However, this time model (a) predicts a population of around

19 billion when α and r are simultaneously estimated, compared to around 92 bil-

lion when r was fixed and only α was estimated. It should be noted, of course,

that the functional forms for f are different although they both belong to the class

of functions in model (a).

3.1.4 Discussion

In this part, we assumed that the human carrying capacity is a function of food

availability. Extending the classical logistic equation, we proposed three different

classes of models (a), (b) and (c) that describe how the carrying capacity varies

with the food production index. Model (a) assumed that the human carrying ca-

75

0

0.20.4

0.10.3

0.5

0

0.2

0.4

0.6

0

0.1

(a)

0

0.20.4

0.10.3

0.5

0

0.2

0.4

0.6

0

0.2

(b)

0

0.20.4

0.10.3

0.5

0

0.2

0.4

0.6

0

0.2

(c)

Figure 3.5: Root mean square error between numerical and actual populationnumbers as a function of s1 and s2 for three carrying capacity models.

pacity increases without bound as the food production increases, whereas mod-

els (b) and (c) assumed that there is a limit to the carrying capacity even as food

production is increased indefinitely.

We also proposed an integration-based method to estimate the parameters

and gave explicit formulas for them. The method provides an alternative to a

76

00.1

0.20.3

0.40.5

0

0.3

0.6

4

3

5

00.2

0.4

0.10.3

0.5

0

0.3

0.6

0.04

0.03

0.035

(a)

00.2

0.4

0.10.3

0.5

0

0.3

0.610

15

00.2

0.4

0.10.3

0.5

0

0.3

0.6

0.03

0.025

0.035

(b)

00.2

0.4

0.10.3

0.5

0

0.3

0.610

15

00.2

0.4

0.10.3

0.5

0

0.3

0.6

0.03

0.025

(c)

Figure 3.6: The estimated values of α and r which both depend on s1 and s2,calculated by (3.10).

nonlinear least squares approach when an explicit analytical formula of the solu-

tion to the differential equation is not available or is not easy to implement. In

essence, instead of minimising the squared error, the integration-based method

‘averages out the potential errors’ by taking the integrals of associated functions.

This statement was not proved in [52] but can be heuristically motivated here as

77

1962 1980 1990 2000 2014

4

6

3

5

7

3.5

4.5

5.5

6.5

Year

World population

(in billion)

Model(a)

Model(b)

Model(c)

Actual data

(a)

2015 2040 2060 2080 2100 2120

10

2

4

6

8

12

14

16

18

Year

World population

(in billion)

Model(a)

Model(b)

Model(c)

(b)

Figure 3.7: Comparison of the best three carrying capacity models (when r isunknown) with respect to (a) approximation with the actual data from 1962-2014and (b) forecasting from 2015-2120.

follows. A naive discretisation of (3.2) is

N(tj+1)−N(tj)

tj+1 − tj− rN(tj) +

r

α

N(tj)2

g(I(tj))= 0.

Suppose that r is given and we wish to estimate α. For a fixed j0, we substitute

tj0 , I(tj0) and N(tj0) into the above equation and solve for α (which is basically a

collocation method). However, for each j = j0, the left-hand side will introduce a

residual term which may be positive or negative. By multiplying (3.2) by a weight

function and integrating over [0, T ], in effect we are ‘averaging out the potential

errors’.

From the integration-based model fitting using actual world population and

food production index data, our results suggest that models (b) and (c) give the

best fit. This implies that although an increase in food availability implies an in-

crease in carrying capacity, there is an upper limit to the carrying capacity, which

is not unreasonable to expect. In fact, looking at Figures 3.3b and 3.7b, our mod-

els (b) and (c) predict human population in 2050 to be roughly 10 billion, which

is comparable to the Population Reference Bureau prediction of 9.8 billion.

78

Potential extensions of this work would be to include other factors that in-

fluence human carrying capacity, e.g. water supply, living space and environ-

mental conditions (for example, see [40, 94, 120]). Agriculture requires water

for food production that accounts for almost 70% of all water withdrawals, and

up to 95% in some developing countries [84]. Thus water supply can be con-

sidered as a factor that influences human carrying capacity since greater food

production leads to a decrease in water supply, which in turn could potentially

decrease human carrying capacity. Food production may also reduce the popula-

tion number due to deaths from diseases caused by food plant infection. This is

one mechanism that explains model (c), for example. One well-known example

of dieback (see [17] as well as [79] for more information) occurred in Ireland after

a fungus infection destroyed the potato crop in 1845. This was called the Irish

Potato Famine or Famine of 1845-49. It was reported that as a result of the potato

famine, approximately one million people died and three million more emigrated

to other countries. The challenge of mathematical modelling is of course how to

quantify such factors. In this chapter, although we elected to model human car-

rying capacity explicitly as a function of the food production index only, it is not

unreasonable to expect that the effects of other factors (e.g. water supply) is im-

plicitly reflected in the observed population and food production data, and such

effects are encapsulated in the parameters estimated via the model fitting proce-

dure.

3.2 Population model with two ODEs

The population model given in (3.1) in Section 3.1 is now extended to a system of

two differential equations by adding the carrying capacity growth rate. As it is

assumed that carrying capacity is a function of food availability, then the carry-

ing capacity growth rate is converted into a growth rate for the food production

index. In this section we use the same definition of carrying capacity as a func-

79

tion of the food production index as previously defined in Section 3.1, namely

K(t) = f(I(t)) = αg(I(t)), as well as function g(I(t)) which is broken down into

three types of models based on the assumptions given in Section 3.1.

As in the population model with one ODE, this section will also carry out

parameter estimation and numerical simulations. However, there are two addi-

tional parameters to be estimated which are contained in the growth rate of the

food production index. This implies that there are a total of four parameters that

will be estimated using the integration-based method

3.2.1 Mathematical formulation

Here we employ the carrying capacity growth rate differential equation, pro-

posed by Safuan et al. [105], which is written as

dK

dt= βK

(1− N

γ

), (3.12)

where β represents the carrying capacity development rate, while γ = β/c with c

denoting the interaction rate between the population and carrying capacity. But,

unlike Safuan model that all parameters apear in the model are assigned positive,

here we asssume that β and γ can be either positive or negative. The negative

value β means the reduction rate of carrying capacity. On the other hand, the sign

of γ is affected by the sign of β and c. Thus when the interaction rate c < 0 occurs,

this means the increase of population number will raise the development effect

of carrying capacity. By substituting K(t) = αg(I(t)) on the left-hand and right-

hand sides of (3.12), the rate of change in the food production index is obtained

as follows,dI

dt= β

g(I)

g′(I)

(1− N

γ

). (3.13)

Another form of a population model with variable carrying capacity was also

investigated by Thornley & France [112], where the carrying capacity growth rate

80

is expressed bydK

dt= −β(K −N), (3.14)

where β > 0 in this equation represents the process of development or progress.

Thus the latter equation can be converted in terms of the food production index

by the following differential equation

dI

dt=

β

αg′(I)

[N − αg(I)

]. (3.15)

3.2.2 Integration-based parameter estimation method

Now we can form two population models by combining the differential equations

of the population (3.2) and the carrying capacity, either (3.13) or (3.15). First, we

use (3.2) and (3.13), so that the model can be written as

dN

dt= rN − r

α

N2

g(I),

dI

dt= β

g(I)

g′(I)− β

γ

g(I)

g′(I)N.

(3.16)

As seen in (3.16), there are four parameters contained in the system of differen-

tial equations, namely r, α, β and γ. We assume that all of these parameters are

unknown, therefore we need to estimate them all.

Estimation is carried out separately between the first ODE and the second

ODE. That is, the parameters r and α are obtained from the first ODE, whereas

β and γ are acquired using the second ODE. Hence the r and α estimates can be

computed using (3.10). Meanwhile, β and γ can be generated in the same way as

estimating r and α based on the following integration-based method.

Let w(t; s) be a positive weight function, with parameter s ≥ 0. Multiplying

w(t; s) to both sides of the second ODE in (3.16) and then integrating them, we

81

have

w(T ; s)I(T )− w(0; s)I(0)−∫ T

0

w′(t; s)I(t) dt

= β

∫ T

0

w(t; s)g(I(t))

g′(I(t))dt− β

γ

∫ T

0

w(t; s)g(I(t))

g′(I(t))N(t) dt,

or written compactly as

β a(s) +β

γb(s) = c(s),

where

a(s) =

∫ T

0

w(t; s)g(I(t))

g′(I(t))dt,

b(s) = −∫ T

0

w(t; s)g(I(t))

g′(I(t))N(t) dt,

c(s) = w(T ; s)I(T )− w(0; s)I(0)−∫ T

0

w′(t; s)I(t) dt.

(3.17)

Since we want to determine β and γ simultaneously, we only need to select two

different non negative values for s in (3.17), i.e. the (s1, s2)-pairs, where each pair

corresponds to the pair used to estimate r and α. As a result, the linear algebraic

system for solving β and γ can be constructed as

a(s1) b(s1)

a(s2) b(s2)

β

β/γ

=

c(s1)c(s2)

,or more specifically as

γ =c(s1)b(s2)− c(s2)b(s1)

a(s1)c(s2)− a(s2)c(s1), β =

c(s1)b(s2)− c(s2)b(s1)

a(s1)b(s2)− a(s2)b(s1). (3.18)

Note that even though the solution in (3.18) looks similar to the solution of α and

β in (3.10), they both have different solutions due to the different definitions of

a(s), b(s) and c(s).

Another thing that needs to be observed is that since the estimated parame-

82

ters do not overlap between the two ODEs (see (3.16)), we can also first estimate

the parameters of the second ODE and then the first one, which will give the same

result.

Next, the second model, which is established by (3.2) and (3.15), is given by

dN

dt= rN − r

α

N2

g(I),

dI

dt=

β

αg′(I)

[N − αg(I)

].

(3.19)

Using a similar way, we obtain the formulas to calculate α and β within the sec-

ond ODE of (3.19) as

α =c(s1)b(s2)− c(s2)b(s1)

a(s1)c(s2)− a(s2)c(s1), β =

c(s1)b(s2)− c(s2)b(s1)

a(s1)b(s2)− a(s2)b(s1),

where, for s = s1, s2,

a(s) = −∫ T

0

w(t; s)g(I(t))

g′(I(t))dt,

b(s) =

∫ T

0

w(t; s)N(t)

g′(I(t))dt,

c(s) = w(T ; s)I(T )− w(0; s)I(0)−∫ T

0

w′(t; s)I(t) dt.

In this model, the overlapping parameter α from the first and second ODE

occurs (see (3.19)). One can estimate the parameters from the first ODE to obtain

r and α, then use this value of α to the second ODE to obtain another parameter

which is β. Conversely, from the second ODE, parameters β and α are yielded,

then α is substituted to the first ODE to estimate parameter r. Thus, these two

ways of estimating parameters are applied to not only the model (3.19) but also

model (3.16), and their numerical results are discussed in the next section.

83

3.2.3 Numerical simulations

Here we use the same data points from the world population and food produc-

tion index as employed in the one-ODE model given in (3.1) and presented in

Figure 3.1. The procedure for performing the simulation is the same, starting

with the parameter estimation in two consecutive estimations, r and α are ob-

tained from the first ODE, while β and γ from the second ODE. Furthermore, we

use the same pairs (s1, s2) in performing the two estimates. In other words, the

four parameters are generated based on the corresponding pairs (s1, s2).

The next step is validating the estimations by investigating the error between

the approximate and the actual data for the remaining data points. Note that here

we are using the RMS error not only inN(t) but also in I(t) since we approximate

the solution for both N(t) and I(t) as given by system (3.16). Thus the RMS error

is defined as

E =

√√√√ 1

n

n∑i=1

[(Ni −Ni)2 + (Ii − Ii)2

], (3.20)

where Ni and Ii represent the approximate values, while Ni and Ii denote the

actual values. Note that unlike the one-ODE model, we do not need to perform

model fitting for I(t) since the solution I(t) can be obtained numerically from the

second ODE of (3.16).

Finally, the population, size as well as the carrying capacity, are forecasted.

We do forecasts for carrying capacity instead of the food production index to see

the interactions between the population and its maximum limits. Meanwhile the

food production can be regulated by the government to protect the people’s need

rather than making predictions.

We use the same carrying capacity models as provided in Section 3.1, with

each model of the form of f(I) = αg(I), where α > 0 and g(I) > 0 for I > 0, using

the following examples:

• Model (a) : f(I) = αI1/4.

84

• Model (b) : f(I) = αI/(1 + I).

• Model (c) : f(I) = αI(1 + I)/(1 + I2).

The reason we chose f(I) = αI1/4 in model (a) instead of f(I) = αI is because

once again, the parameter r has negative values as (s1, s2) varies.

Regarding the number of data points used for estimation, we not only used

the first 30 data points like the previous one ODE model, but also tried more val-

ues expecting better results, which is 50 data points, then compared those results.

We therefore present two scenarios for performing the simulation based on the

number of data points used for the estimation.

Scenario one : Estimation with 30 data points

First of all, 30 data points are employed to estimate the four parameters with cor-

responding (s1, s2). With all parameters values for each carrying capacity model

and for each (s1, s2), we choose the one that has the smallest RMS error, specified

in (3.20), presented in Table 3.2.

Model (s1, s2) r α β γ RMS ErrorModel (a) (0.01, 0.05) 0.032 3.80 0.0003 22.36 6.89

Model (b) (0.00, 0.01) 0.027 13.36 0.0002 6.99 20.84

Model (c) (0.00, 0.01) 0.027 13.32 −0.0002 7.10 19.78

Table 3.2: Parameter estimation of the model given in (3.16) for three carryingcapacity models using 30 data points.

As can be seen in the table, the model (a) has the smallest RMS error com-

pared with the other two models, which is three times larger.

For the Thornley-France model, inconsistent results occur regarding the es-

timated values when the other way of estimation is applied, that is, β and γ, are

obtained first from the second ODE, then r and α from the first ODE. The results

are shown in Table 3.3. The first row for each model in Table 3.3 informs us the

estimated values when the population ODE was first employed followed by the

food production index ODE, while the second row for each model tells us that

the estimations are performed in reverse. As we can see, they have very different

85

parameter values, including the RMS error. Hence we will not go further with

the Thornley-France model but focus on the Safuan et al. model instead.

Model (s1, s2) r α β RMS ErrorModel (a) (0.30, 0.046) 0.028 4.89 −0.098 4.360

0.060 2.05 −0.021 32.237

Model (b) (0.15, 0.01) 0.028 12.82 −0.001 8.6140.051 6.84 −0.001 16.999

Model (c) (0.15, 0.01) 0.027 12.76 0.001 7.9700.048 6.81 0.001 16.257

Table 3.3: Parameter estimation of Thornley-France model for three carryingcapacity models using 30 data points, where two ways of estimation are per-formed.

A comparison between the approximated and the actual data for the popu-

lation and food production index are also shown graphically in Figure 3.8. Al-

though the graphs of the world population for models (b) and (c) “look better”

than model (a), they are actually not when looking at the comparison to the food

production index. In Figure 3.8b, it can be seen that models (b) and (c), both are

declining too soon, while the actual data shows it is still climbing up, the same

trend with model (a).

1962 1980 2000 20143

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Year

World p

opula

tion (

in b

illio

n)

Model(a)

Model(b)

Model(c)

Actual data

(a)

1962 1980 2000 2014

40

60

80

100

120

Year

World food index

Model(a)

Model(b)

Model(c)

Actual data

(b)

Figure 3.8: Comparison of the best three carrying capacity models with the ac-tual data from 1962-2014 using 30 data points for (a) world population and (b)food production index.

86

The next step is to forecast both the world population and the carrying ca-

pacity to compare the three models and to investigate whether the population has

reached its maximum value by using the obtained parameters given in Table 3.2.

Let us start with population growth forecasting as depicted in Figure 3.9. The

2015 2050 2 100 2 160

10

6

8

12

14

16

18

Year

Wo

rld

po

pu

latio

n (

in b

illio

n)

Model(a)

Model(b)

Model(c)

Figure 3.9: Forecasting comparison of world population number for each of thebest three carrying capacity models from 2015-2160 using 30 data points.

three models give quite different values, the discrepancies between them emerge

more as the years go by, where model (a) shows us that the population is growing

faster than the others. On the other hand, like the prediction of 9.8 billion by 2050

mentioned by the Population Reference Bureau [64], models (b) and (c) appear

to be closer to the prediction with the respective values of 10.2 and 10.3 billion,

while model (a) looks to have increased slightly to 10.9 billion.

To see whether the population has reached or is far from its maximum value,

Figure 3.10 shows the approximation between population number and carrying

capacity up to 2160. The graphs for model (a) and model (c) show that the popu-

lation and carrying capacity keep growing, but the population itself does not gain

the maximum number until 2160. Whereas for model (b), the population has at-

tained its maximum starting around 2130 and at that time it has also reached the

87

2015 2050 2100 2160

8

12

16

20

Year

World p

opula

tion (

in b

illio

n)

(a)

2015 2050 2100 2160

6

8

10

12

Year

World p

opula

tion (

in b

illio

n)

(b)

2015 2050 2100 2160

4

8

12

16

Year

World p

opula

tion (

in b

illio

n)

(c)

Figure 3.10: Forecasting the population and carrying capacity for each carryngcapacity models from 2015 - 2160 using 30 data points.

equilibrium value. Based on the overall simulations, model (a) seems to be better

in terms of RMS error especially for the food production index approximation,

but when we talk about population forecasts, especially prediction in 2050, mod-

els (b) and (c) seem to be better, even though the food production index approx-

imation starts to move away from the actual data in around 1995. Therefore, in

88

order to make the food production index approximation of the three carrying ca-

pacity models to be closer to the actual data until the end of the interval, another

simulation has been conducted, where the number of data points for estimation

is now added to 50, while 3 data points for validating.

Scenario two: Estimation with 50 data points

In this scenario, we increase the data points to 50 for estimation and 3 data points

remaining for validation. This scenario is purposed to examine whether the mod-

els (b) and (c) give better fit than model (a) like in the single ODE model. Table 3.4

shows the parameter estimates for each carrying capacity model resulting in the

lowest RMS error with the corresponding (s1, s2). From this table, we find a dif-

Model (s1, s2) r α β γ RMSModel(a) (0.05, 0.21) 0.034 3.410 0.0001 55.24 0.908

Model(b) (0.05, 0.11) 0.029 11.74 0.0001 8.06 0.888

Model(c) (0.05, 0.11) 0.029 12.69 −0.0001 8.18 0.864

Table 3.4: Parameter estimation of the model given in (3.16) for three carryingcapacity models using 50 data points.

ferent conclusion from scenario one, where model (b), particularly model (c) is

better than model (a) with respect to the RMS error. In addition, we can see that

γ in model (a) is very large, which is seven times larger than the other models.

Even when we compare with the 30 data point estimation, the γ for the three

models increases. This means that the carrying capacity can develop more and

affect population growth. On the other hand, the rest of the parameters do not

look much different with the first scenario.

Figure 3.11 validates the world population and food production index by

showing a comparison between the approximated and the actual data. As ex-

pected, the more information used for estimation, the better validation result is,

where the forecast graph is closer to the actual one.

Now when the forecast of the three carrying capacity models is performed

as depicted in Figure 3.11, we can say that there is no trend change in popula-

89

1962 1980 2000 2014

4

6

3

5

7

3.5

4.5

5.5

6.5

7.5

Year

World p

opula

tion (

in b

illio

n)

Model(a)

Model(b)

Model(c)

Actual data

(a)

1962 1980 2000 2014

100

40

60

80

120

Year

World food index

Model(a)

Model(b)

Model(c)

Actual data

(b)

Figure 3.11: Comparison of the best three carrying capacity models with theactual data from 1962-2014 using 50 data points for (a) world population and (b)food production index.

2015 2050 2100 2160

20

10

8

12

14

16

18

Year

Wo

rld

po

pu

latio

n

(in

bill

ion

)

Model(a)

Model(b)

Model(c)

Figure 3.12: Forecasting comparison of world population number for each thebest three carrying capacity model from 2015-2160 using 50 data points.

tion growth when compared to the first scenario. On the other hand, model (a)

shows a faster population growth, while models (b) and (c) tend to grow more

90

slowly. The approximated population in 2050 gives us a smaller but closer value

to the PRB’s prediction for models (b) and (c) of 9.6 billion, with 10.6 billion for

model (a). Regarding the carrying capacity, Figure 3.13 shows model (a) has not

shown that it has reached or is close to its maximum value until 2160, but this is

not the case for models (b) and (c).

3.2.4 Summary

In this part, the carrying capacity differential equation was added to the popula-

tion model such that it forms a coupled system. Then the carrying capacity K(t)

differential equation is converted into the food production index I(t) differen-

tial equation using the definition of K(t) = αg(I(t)). We used the same classes

and functions of carrying capacity models as proposed in the one-ODE popula-

tion model to compare and find out which model gives the best fit to the actual

data.

An integration-based method was also applied here to estimate the four pa-

rameters (two for each ODE) using the first 30 data points, and the results are

consistent with that for the one-ODE population model. However, unlike the

one-ODE model, model (a) in the two-ODE model gives the best fit with regards

to the mean error, whereas models (b) and (c) result a very large discrepancy

between the actual and the approximation in I(t). An alternative population

model with variable carrying capacity has also been investigated by referring to

the model proposed by Thornley & France [112]. We found different and incon-

sistent parameter values when two orders of estimation are performed; when the

parameters from the population ODE are first estimated and when the parameter

estimations from the food production index ODE are done first. Therefore we

only used the Safuan et al. model to do further simulations.

Another scenario is performed by adding the number of data points used for

estimation to 50 to see what changes will occur in the population number if the

91

2015 2050 2100 2160

20

10

5

15

25

Year

Wo

rld

po

pu

latio

n (

in b

illio

n)

(a)

2015 2050 2100 2160

10

4

6

8

12

Year

Wo

rld

po

pu

latio

n (

in b

illio

n)

(b)

2015 2050 2100 2160

10

4

6

8

12

Year

Wo

rld

po

pu

latio

n (

in b

illio

n)

(c)

Figure 3.13: Forecasting population and carrying capacity for each best threecarrying capacity model from 2015-2160 using 50 data points.

three model estimates are close to all the actual data points I(t). We suggest that

models (b) and (c) give the best fit for this scenario, not only from the RMS error

92

but also the prediction in 2050 that the population will reach 9.6 billion or close to

the prediction mentioned by the Population Reference Bureau [64] of 9.8 billion,

and even closer than that of the one ODE model given in (3.1), which reaches out

10.2 billion. On the other hand, model (a) predicts 10.2 billion for this scenario,

while for the one-ODE model, it predicts to reach 11.3 billion.

Based on these comparisons, we deduce that the more data points for esti-

mation, which is 50 data points, the better result we obtain where models (b) and

(c) are the better fit compared with model (a).

93

Chapter 4

Analytical Solution of a General

Population Model with Variable

Carrying Capacity

In this chapter we consider a general population model with a variable carry-

ing capacity consisting of two coupled ODEs, and which includes the Thornley-

France [112], Safuan-Jovanoski-Towers-Sidhu [105] and Meyer-Ausubel [83] mod-

els as special cases. Moreover, when the carrying capacity is kept constant, the

system reduces to a single-ODE population model and recovers the Gompertz,

‘ordinary’ logistic and θ-logistic models, amongst others. The idea is to extract

the essential properties of such models without getting ‘bogged down’ by partic-

ular cases. We provide a procedure for obtaining, when possible, the analytical

solution of this general population model. Different models can then be cho-

sen depending on the particular phenomenon being modelled. An important

tractable special case is when the per capita growth rates of the population and

carrying capacity are proportional to each other. We also give a criterion for when

inflexion may occur. Several examples are provided. The results are based on

the article “Population models with variable carrying capacities: analytical solu-

tions”, currently under review.

94

4.1 Population model formulation

Consider the initial value problem (IVP)

dN

dt= Nf(N,K),

dK

dt= Kg(N,K), N(0) = N0, K(0) = K0, (4.1)

where N(t) and K(t) are the population and carrying capacity at time t, re-

spectively. The functions f = f(x, y) and g = g(x, y) are assumed to be C1 in the

region (x, y) : x, y > 0, and satisfy

D1f(x, y) < 0, D2f(x, y) > 0,

where Dj denotes the partial derivative with respect to the independent variable

in the jth position. The initial values N0 and K0 are positive and given.

The assumptions above means that the population per capita growth rate

1

N

dN

dt

decreases with increasing population and increases with increasing carrying ca-

pacity. As for the signs of D1g(x, y) and D2g(x, y), it is not obvious what these

should be since the behaviour of the carrying capacity per capita growth rate

1

K

dK

dt

may depend on the particular population species.

When g(x, y) = 0 for all x, y > 0, then K(t) = K0, a constant function, and

the IVP (4.1) reduces to

dN

dt= Nf(N,K0), N(0) = N0.

A well-known example is the θ-logistic (or power-law logistic) model, expressed

95

as (see Banks [10])

f(N,K0) =r

θ

[1−

( NK0

)θ],

where r > 0 is the intrinsic growth rate and θ ≥ 0 is a parameter related to the

point of inflexion of the solution. The choice θ = 1 gives the ‘ordinary’ logis-

tic model. In the limit as θ → 0+, the power-law logistic model reduces to the

Gompertz model [44]

f(N,K0) = r log(K0

N

).

In the case of a variable carrying capacity, the Thornley-France model [112]

takes the form

f(N,K) = a(1− N

K

), g(N,K) = −b

(1− N

K

), (4.2)

where a, b > 0. We observe that D1g(x, y) > 0 and D2g(x, y) < 0.

Safuan et al. [105] proposed the model

f(N,K) = a(1− N

K

), g(N,K) = b− cN, (4.3)

where a, b, c > 0. Here we see that D1g(x, y) < 0 and D2g(x, y) = 0.

Meyer [82] and Meyer & Ausubel [83] assumed that

f(N,K) = a(1− N

K

), g(N,K) = b− cK,

where a, b, c > 0. This time D1g(x, y) = 0 and D2g(x, y) < 0.

96

4.2 Analytical solution procedure

Suppose that there exists a positive C1-function F such that y = F (x) solves the

IVPdy

dx=yg(x, y)

xf(x, y), y = K0 when x = N0. (4.4)

Define

G(x) =

∫ x

N0

1

zh(z)dz, h(z) = f(z, F (z)). (4.5)

We claim that G(N(t)) = t implicitly defines the solution of the IVP

dN

dt= Nh(N), N(0) = N0. (4.6)

It is clear that G(N(0)) = G(N0) = 0. Implicitly differentiating G(N(t)) = t with

respect to t yields

1 = G′(N(t))N ′(t) =N ′(t)

N(t)h(N(t))

or dN/dt = Nh(N). This proves the claim. The formal solution of (4.1) is therefore

given implicitly by

G(N(t)) = t, K(t) = F (N(t)). (4.7)

Rather than attempt to determine general sufficient conditions on f and g

which would imply the solution (4.7), here we outline the procedure to arrive

at (4.7), assuming that certain hypotheses are satisfied, and then provide illustra-

tive examples.

Before we consider some examples, let us first investigate where inflexion

may occur for the function N . This step is important since we expect that, for a

certain condition, the population dynamic shows the logistic-type behaviour that

forms an S-shaped curve. Differentiating (4.6) with respect to t, we have

d2N

dt2= [h(N) +Nh′(N)]

dN

dt= Nh(N)[h(N) +Nh′(N)].

97

Therefore inflexion for the functionN may occur at some t∗ > 0 such thatN(t∗) =

N∗ and h(N∗) +N∗h′(N∗) = 0. But (4.5) implies that

h′(z) = D1f(z, F (z)) +D2f(z, F (z))F′(z).

If the function

H(z) = f(z, F (z)) + zD1f(z, F (z)) + zF ′(z)D2f(z, F (z)) (4.8)

has a positive root z = N∗, then inflexion may occur for the function N when it

reaches N∗, and this will happen when t = t∗, where

t∗ = G(N(t∗)) = G(N∗) =

∫ N∗

N0

1

zh(z)dz. (4.9)

Example 4.2.1. Consider the model with per capita growth rates

f(N,K) = a(1− N

K

), g(N,K) = a− cN

K, (4.10)

where 0 < a < c; compare with the model (4.3) proposed by Safuan et al. [104].

Note that f and g are not proportional. Moreover, D1g(x, y) < 0 and D2g(x, y) >

0. Then the ODE in (4.4) becomes

dy

dx=y(a− cx/y)

ax(1− x/y)=

(y/x)(ay/x− c)

a(y/x− 1).

This is a homogeneous ODE which can be transformed via v = y/x to the separa-

ble ODE

xdv

dx=a− c

a

v

v − 1, v =

K0

N0

when x = N0.

The solution is

v − log(v) =a− c

alog(x) + log(α), (4.11)

98

where

log(α) =K0

N0

− log(K0

N0

)− a− c

alog(N0).

The latter implies that

α =N

c/a0 eK0/N0

K0

> 0

and (4.11) can be rewritten as

−ve−v = − 1

αxc/a−1.

Let us recall here some properties of the Lambert W -function. The equa-

tion zez = u, where u, z ∈ R, can be solved for z if u ≥ −1/e, and z = W (u). In

addition, W (u) < 0 for −1/e ≤ u < 0 and

W ′(u) =W (u)

u[1 +W (u)], u = −1

e, 0. (4.12)

Identifying

u = − 1

αxc/a−1, z = −v

and assuming that

0 < x ≤(αe

)a/(c−a)

(to ensure that u ≥ −1/e), we obtain v = −W (−xc/a−1/α) = y/x. Note that v > 0

since theW -function is negative here because α, x > 0. Hence the solution of (4.4)

is

y = F (x) = −xW(− 1

αxc/a−1

).

It follows from the definition that W (zez) = W (u) = z if u = zez ≥ −1. Thus

F (N0) = −N0W(−K0

N0

e−K0/N0

)= K0

99

if K0 ≤ N0. This verifies the initial condition in (4.4).

Equation (4.5) yields

h(z) = f(z, F (z)) = a1 +W (−zc/a−1/α)

W (−zc/a−1/α)

and therefore

G(x) =

∫ x

N0

W (−zc/a−1/α)

az[1 +W (−zc/a−1/α)]dz.

The substitutions

u = − 1

αzc/a−1, du =

c− a

a

u

zdz

allows us to write

G(x) =1

c− a

∫ −xc/a−1/α

−Nc/a−10 /α

W (u)

u[1 +W (u)]du

=1

c− a

[W

(− 1

αxc/a−1

)−W

(− 1

αN

c/a−10

)],

where we used (4.12) in the last step. Finally, we see from (4.7) that the analytical

solution for this model is

W(− 1

α[N(t)]c/a−1

)= −K0

N0

+ (c− a)t, K(t) = −N(t)W(− 1

α[N(t)]c/a−1

),

provided

0 < N(t) ≤(αe

)a/(c−a)

, K0 ≤ N0. (4.13)

The functions N and K are depicted in Figure 4.1, which also includes for com-

parison the numerical solution of the system (4.1) with per capita growth rates

given in (4.10) (Note that all numerical simulations in this chapter are generated

by the built-in ’ode’ scilab syntax code). Here we choose a = 0.1, c = 0.4 and

initial condition (N0, K0) = (5, 4.5) to satisfy the second requirement in (4.13).

With regards to the dynamics of population and carrying capacity, Figure 4.2 tells

us that with increasing time, the population experiences a decline as carrying

capacity decreases. Observe that the first condition in (4.13) is satisfied since

100

0 21 30.5 1.5 2.5

0

2

4

1

3

5

0.5

1.5

2.5

3.5

4.5

Figure 4.1: Comparison of exact and numerical solutions for population andcarrying capacity with per capita growth rates (4.10).

N(t) < (α/e)a/(c−a) = 5.0089422. Furthermore, when the initial conditions are

0 1 2 3 4 5

0

1

2

3

4

5

K

N

Figure 4.2: Phase plane for (4.1) with per capita growth rates (4.10).

varied, the population undergoes extinction in finite time as the carrying capac-

ity can no longer support what the population needs to survive.

Example 4.2.2. Now suppose that the per capita growth rates are

f(N,K) = a log(KN

), g(N,K) = a log

(KN

)+ b, (4.14)

101

where a, b > 0. We see that f and g are not proportional. Then the ODE in (4.4) is

dy

dx=y[a log(y/x) + b]

ax log(y/x)=

(y/x)[a log(y/x) + b]

a log(y/x),

another homogeneous ODE which can be transformed via v = y/x to the separa-

ble ODE

xdv

dx=b

a

v

log(v).

The solution of this separable ODE is

[log(v)]2 =2b

alog(x) + log(α) = log(αx2b/a),

where

log(α) =[log

(K0

N0

)]2− 2b

alog(N0).

Note that we have to assume x2b/a > 1/α to get a real-valued solution. Therefore,

if log(v) > 0, then log(v) = [log(αx2b/a)]1/2 and

y = F (x) = xe[log(αx2b/a)]1/2 , x < y.

From (4.5) we obtain

h(z) = f(z, F (z)) = a[log(αz2b/a)]1/2

and therefore

G(x) =

∫ x

N0

1

az[log(αz2b/a)]1/2dz.

Letting

u = log(αz2b/a), du =2b

a

1

zdz,

102

we see that

G(x) =1

2b

∫ log(αx2b/a)

log(αN2b/a0 )

u−1/2 du =1

b[log(αx2b/a)]1/2 − 1

b[log(αN

2b/a0 )]1/2.

Hence, using (4.7), we deduce the analytical solution

N(t) = N0

(K0

N0

)at

eabt2/2, K(t) = K0

(K0

N0

)at

eabt2/2+bt. (4.15)

This solution is valid provided

[N(t)]2b/a >1

α, N(t) < K(t). (4.16)

Some simulations are performed to show the population and carrying capacity

0 20 4010 305 15 25 35

0

100

20

40

60

80

120

140

Figure 4.3: Comparison of exact and numerical solutions for population andcarrying capacity with per capita growth rates (4.14).

behaviour based on per capita growth rates (4.14). The parameter values a = 0.1,

b = 0.01 and initial condition (N0, K0) = (3, 5) are chosen to calculate the exact

solution (4.15). As in the previous example, in Figure 4.3 we also make a compar-

ison between the exact and numerical solutions of the system (4.1). Also, we can

see that the population is increasing, and thereforeN(t) ≥ N0 > (1/α)a/2b = 0.814.

103

Inspecting the dynamics in Figure 4.4, both population and carrying capacity in-

0 20 4010 30 505 15 25 35 45 55

0

100

20

40

60

80

120

140

Figure 4.4: Increasing carrying capacity and population size. Symbol ‘x’ denotesthe initial point and ‘o’ is the endpoint.

crease. Note that the second condition in (4.16) is verified since

K(t)

N(t)=K0

N0

ebt > 1.

Furthermore, Figure 4.5 shows that when N0 < K0 (respectively, N0 > K0), both

0 10 20 30 40 50

10

20

30

40

50

K

N

Figure 4.5: Phase plane for (4.1) with per capita growth rates (4.14).

N(t) and K(t) increase (respectively, decrease) when t increases.

104

4.3 Special case: proportional per capita growth rates

There is a nontrivial special case when the calculations in Section 4.1 can be sim-

plified. Suppose that the per capita growth rates for the population and carrying

capacity are proportional to each other, i.e. there exists α ∈ R such that effectively

we have

g(N,K) = αf(N,K).

Then the solution of (4.4) becomes

y = F (x) =K0

Nα0

xα. (4.17)

We have from (4.5) that

G(x) =

∫ x

N0

1

zf(z,K0zα/Nα0 )

dz (4.18)

and the solution of (4.1) using (4.7) is

∫ N(t)

N0

1

zf(z,K0zα/Nα0 )

dz = t, K(t) =K0

Nα0

[N(t)]α.

Eqs. (4.8) and (4.9) simplify to

H(z) = f(z,K0

Nα0

zα)+zD1f

(z,K0

Nα0

zα)+αK0

Nα0

zαD2f(z,K0

Nα0

zα)

(4.19)

and

t∗ = G(N∗) =

∫ N∗

N0

1

zf(z,K0zα/Nα0 )

dz, (4.20)

respectively.

Example 4.3.1. We propose a model with per capita growth rates

f(N,K) =a

θ

[1−

(NK

)θ], g(N,K) = − b

θ

[1−

(NK

)θ], (4.21)

105

where a, b > 0 and θ ≥ 0. The ‘value’ when θ = 0 is meant to be understood as

the limit when θ → 0+, and this is related to the Gompertz model. The Thornley-

France model (4.2) is a special case if we take θ = 1.

If we denote by

N∞ = limt→∞

N(t), K∞ = limt→∞

K(t),

then any equilibrium state of the above model will have K∞ = N∞. Combining

this with K∞ = F (N∞) from (4.17), we obtain

K∞ =(K0

Nα0

)1/(1−α)

(4.22)

and therefore

y = F (x) = K1−α∞ xα = K∞

( x

K∞

)α

.

We have

f(x, y) =a

θ− a

θ

(xy

)θ

, g(x, y) = − bθ+b

θ

(xy

)θ

and α = −b/a < 0, so that

D1f(x, y) = −ax

(xy

)θ

, D2f(x, y) =a

y

(xy

)θ

.

Eq. (4.18) yields

G(x) =θ

a

∫ x

N0

1

z[1− (z/K∞)θ(1−α)]dz.

With the substitutions

u =zθ(1−α)

Kθ(1−α)∞

,1

udu = θ(1− α)

1

zdz,

106

we can express the integral as

G(x) =1

a(1− α)

∫ (x/K∞)θ(1−α)

(N0/K∞)θ(1−α)

1

u(1− u)du.

Recall the formula

I(u) =

∫1

u(1− u)du = log

( u

1− u

), 0 < u < 1.

If I(b0) − I(a0) = c0, where 0 < a0 < b0 < 1 and c0 > 0, then it is not difficult to

show that

b0 =1

1 + [(1− a0)/a0]e−c0.

Taking a0 = (N0/K∞)θ(1−α), b0 = [N(t)/K∞]θ(1−α) and c0 = a(1− α)t, the first

equation in (4.7) gives

[N(t)

K∞

]θ(1−α)

=1

1 + [(K∞/N0)θ(1−α) − 1]e−a(1−α)t, (4.23)

while from the second equation in (4.7) we have

K(t) = K1−α∞ N(t)α = K∞

[N(t)

K∞

]α. (4.24)

As we expect, N(t) and K(t) both tend to N∞ = K∞ as t→ ∞.

Equating H(z) in (4.19) to zero, we obtain

( z

K∞

)θ(1−α)

=1

1 + θ(1− α),

which gives the positive root z = N∗ satisfying

N∗

K∞=

[1

1 + θ(1− α)

]1/[θ(1−α)]

, (4.25)

with K∞ given by (4.22). Note that N∗ is positive since α < 0 and θ ≥ 0. Finally,

107

we see from (4.9) that

t∗ = G(N∗) =1

a(1− α)log

( (N∗/K∞)θ(1−α)

1− (N∗/K∞)θ(1−α)

)− 1

a(1− α)log

( (N0/K∞)θ(1−α)

1− (N0/K∞)θ(1−α)

).

(4.26)

As I is an increasing function of u (since I ′(u) > 0) and assuming that N0 < N∗,

it follows that (N0/K∞)θ(1−α) < (N∗/K∞)θ(1−α) and therefore t∗ > 0.

It was already noted that θ = 1 reduces to the Thornley-France model (4.2).

Then (4.23) and (4.24) recover the analytical solution found in Thornley et al. [113].

Eq. (4.25) indicates where inflexion occurs as a fraction of the asymptotic carrying

capacity, while (4.26) gives the time of inflexion (compare with Eqs. (17) and (18),

respectively, in [113] with an appropriate renaming of parameters). When θ > 0

and we let θ → ∞, then (4.25) shows that N∗/K∞ tends to unity so that, simi-

lar to the Thornley-France model when θ = 1, exponential growth is sustained

for longer and the inflexion value N∗ moves closer to the asymptotic carrying

capacity value K∞.

If θ > 0 and α = 0 (corresponding to b = 0 in (4.21)), thenK(t) = K0 and (4.1)

reduces to a single ODE

dN

dt=a

θN

[1−

( NK0

)θ],

which is the θ-logistic model. In particular, if θ = 1 and α = 0 (i.e. the ‘ordinary’

logistic model), then (4.25) shows that the inflexion value is one half the asymp-

totic carrying capacity value, which is well known. On the other hand, if θ → 0+,

then we deduce from (4.25) thatN∗/K∞ tends to 1/e for any α = −b/a < 0. This is

similar in behaviour to the case when α = 0 (i.e. b = 0 in (4.21)), so thatK(t) = K0

and therefore (4.1) simplifies to the Gompertz model

dN

dt= aN log

(K0

N

).

108

Thus N∗/K∞ tends to 1/e as θ → 0+ even for the coupled system with nonlinear-

ities given in (4.21).

0 30 60 90 120 150

10

4

6

8

12

14

16

18

inflx point

Figure 4.6: Population dynamics of (4.21) for four different values of θ =0.01, 1, 5, 20 with the inflexion points calculated from (4.25) and (4.26).

Here we include some results of numerical simulations of (4.1) with per

capita growth rates in (4.21). We choose the parameter values a = 0.2, b = 0.01

and initial condition (N0, K0) = (5, 20). We take four different values of θ to see

and compare the population dynamics as shown in Figure 4.6, as well as the car-

rying capacity behaviour in Figure 4.7. As can be seen in both figures, the larger

the value of θ, the longer it takes for the population and carrying capacity to reach

equilibrium.

Figure 4.8 depicts the trajectory in theNK-plane and tells us that the carrying

capacity experiences a decline as the population grows larger. For readability, we

only show the case θ = 0.01, which approximates the Gompertz case, since the

other three values of θ give similar looking curves. Moreover, when the initial

conditions are varied, the trajectories approach the equilibrium line K∞ = N∞ as

shown in Figure 4.9.

109

0 30 60 90 120 150

20

19

18.6

18.8

19.2

19.4

19.6

19.8

Figure 4.7: Carrying capacity dynamics for four different values of θ =0.01, 1, 5, 20.

4 6 8 10 12 14 16 18 20

20

19

18.6

18.8

19.2

19.4

19.6

19.8

20.2

Figure 4.8: Declining carrying capacity for increasing population size. Symbol‘x’ denotes the initial point and ‘o’ is the endpoint.

4.4 Conclusion

In this chapter we focused on finding analytical solutions of a general popula-

tion model with a variable carrying capacity (4.1) modelled by a coupled system

of two nonlinear ODEs. While it was clear that the assumptions D1f(x, y) < 0

110

0 5 10 15 20

0

5

10

15

20

K

N

Figure 4.9: Phase plane for (4.1) with per capita growth rates in (4.21) for θ =0.01.

and D2f(x, y) > 0 are reasonable since they describe the behaviour of the popu-

lation per capita growth rate, we showed through several explicit examples that

corresponding assumptions for D1g(x, y) and D2g(x, y) that describe the carrying

capacity per capita growth rate are not obvious and may be model-dependent.

One possible reason for this is that carrying capacity is not directly observable,

unlike the population size.

If the per capita growth rates are proportional, then in addition to the an-

alytical solution we also found a criterion for the occurrence of inflexion in the

population profile as a fraction of the asymptotic carrying capacity. This crite-

rion does not apply to the models in (4.10) and (4.14) these models do not have a

nontrivial equilibrium state.

In addition to analytical solution, some numerical simulations are also pre-

sented here with purpose to not only see the population dynamics but also com-

pare them with the exact solutions. From these figures, we conclude that the

numerical simulations are in agreement with the exact solution.

111

Chapter 5

Qualitative Behaviour of a General

Harvesting Population Model

The harvesting model used in this chapter consists of a coupled system of two dif-

ferential equations for the population and carrying capacity growth rates, where

either population or carrying capacity growth is governed by a general function

with some given properties. Meanwhile, a constant rate is chosen as the harvest-

ing factor. The model is analysed qualitatively by varying the value of the har-

vesting rate that influences the number of steady states that arise in the model,

or even when there is no steady state in the model due to a specific condition.

The condition for the appearance of a steady state is also used to investigate its

stability.

The model is then broken down into two special cases based on the the char-

acter of the carrying capacity function class, i.e functions with separable forms for

the population and carrying capacity, and functions with no population-dependence.

Examples are given for each special case, as well as to the original model by em-

ploying a specific function that is derived from some articles. But aside from

referring to some articles, an example of a carrying capacity function class is also

introduced in one of two special cases. Therefore from these examples we can

112

observe the population dynamics, including the inspection of the steady state

behaviour through graphical simulations. However, before the harvesting pop-

ulation model with variable carrying capacity is introduced, a preliminary result

for a harvesting model with constant a carrying capacity is provided to ease us to

establish the ideas for assigning the function class properties and formulating the

theorem with regards to the harvesting rate which determines how many steady

states the models have as well as their stabilities.

5.1 Preliminary result

Before we begin with a harvesting population model with variable carrying ca-

pacity, we first analyse a model where the carrying capacity is set as a constant.

This model uses a function class for the population per capita growth rate with

some properties given, and harvesting at a constant rate is taken into account.

Now, we let the population per capita growth rate be governed by the func-

tion f . This function is assumed to be the class C1(R+), where R+ = (0,∞), and

it includes a constant carrying capacity K. The population N(t) is then harvested

by a constant rate H > 0 as shown by

dN

dt= Nf(N)−H. (5.1)

Here, we assume that f = f(x) satisfies the following properties.

(i) There exists K > 0 such that f(K) = 0 and

f(x) > 0 for 0 < x < K, f(x) < 0 for x > K.

(ii) x 7→ xf(x) is strictly concave for 0 < x < K.

(iii) limx→0+

xf(x) = 0.

(5.2)

Observe that these three assumptions are satisfied by the common popula-

113

tion models, such as the Logistic and Gompertz models [44], as well as the models

proposed by Smith [110] and Gilpin & Ayala [41]. The strict concavity for xf(x)

in property (ii) is explained in Appendix A.

From the model given in (5.1), a maximum harvesting yield that can still

maintain the population size and meets the equilibrium, called a critical har-

vesting rate Hc, is inspected and calculated. Furthermore, the number or steady

states, as well as their stability, can be determined with regards to the varying

harvesting yield values.

Now, suppose that there exists the population number N∗ which maximises

the population growth rate in the absence of harvesting term, expressed as

N∗ = argmax(xf(x)

), 0 ≤ x ≤ K. (5.3)

Notice that when x close to zero and when x = K, we have f(x) = 0 (see the

properties (i) and (iii)). Thus the strict concavity of x 7→ xf(x), as mentioned in

property (ii), implies thatN∗ to be in the open interval (0, K). The strict concavity

also implies that any local maximum of x 7→ xf(x) is also a global maximum with

at most one point [15]. Hence x 7→ xf(x) has unique global maximum atN∗. With

this statement, we can establish a lemma as follows.

Lemma 5.1.1. Let f ∈ C1(R+) satisfy the properties given in (5.2). Then x 7→ xf(x)

has unique global maximum x = N∗ given in (5.3).

Next, harvesting is taken into account in the population dynamics as stated

in (5.1). As can be observed from the model, the larger the value ofH is, the lesser

population growth occurs until there is no growth at all and finally a growth

reduction occurs. In other words, there is a critical value of the harvesting rate

such that the population size can still be maintained, and we define that critical

value as

Hc = N∗f(N∗). (5.4)

114

We now define a function L to reexpress the population growth rate specified

in (5.1), which depends on population size x and parameter harvesting rate H ,

given by

L(x;H) = xf(x)−H. (5.5)

Notice that the steady state, say N∞, can be solved from L = 0. From a

geometric point of view, we can see that L and x 7→ xf(x) have similar graphs

since the constant H causes the graph of L to move in a vertical direction without

changing the graph shape. HenceL is also strictly concave. SinceN∗ is the unique

maximiser of x 7→ xf(x) as stated in Lemma 5.1.1, then L also has a unique max-

imiser at the same value N∗. Now we may state a lemma with regards to the

maximiser of L in (5.5) as follows.

Lemma 5.1.2. Let f ∈ C1(R+) satisfy the properties given in (5.2). Then L(x;H) =

xf(x)−H has unique global maximum N∗ given in (5.3).

With N∗ as the maximiser of L, and by referring to (5.4), the maximum value

of L can be written as

L(N∗;H) = N∗f(N∗)−H = Hc −H. (5.6)

Based on the explanation and the lemmas we have, we can now show that there

is saddle-node bifurcation in the parameter harvesting rate parameter from the

following theorem.

Theorem 5.1.3. Let f ∈ C1(R+) that satisfy the properties in (5.2). A saddle-node

bifurcation occurs for the population model (5.1) based on the varying parameter values

of H . Let Hc be defined in (5.4)

(i) If H = Hc, then there is only one steady state in the model.

(ii) If H > Hc, then there are no steady states in the model.

(iii) If H < Hc, then there are exactly two steady states in the model.

115

Proof. Case (i). By referring to (5.6) we have L(N∗;H) = 0 due to H = Hc. Hence

N∗ is also a steady state N∞. Since Lemma 5.1.2 says that N∗ is the unique max-

imiser for L, the steady state N∞ in model (5.1) is also unique and N∞ = N∗.

Case (ii). In this case we can write L(x;H) ≤ L(N∗;H) for 0 < x < K

since L(N∗;H) is the global maximum. Meanwhile, when H > Hc, we have

L(N∗;H) < 0 from (5.6). This implies that L(x;H) ≤ L(N∗;H) < 0 for 0 < x < K.

On the other hand, for x > K, we refer to the property (i) that xf(x) < 0 since x is

positive. Therefore L(x;H) = xf(x)−H < 0. From the results in the two regions

of x, we can infer that L(x;H) < 0 for all x > 0. In other words L never reaches

zero, which means there are no steady states that appear in the model.

Case (iii). We start with inspecting the value L for small x, for large x and

at x = N∗. First, property (iii) which says limx→0 xf(x) = 0 is used to obtain

limx→0 L(x;H) < 0. For large x, we use property (i) to have xf(x) < 0, then

xf(x) −H < 0, which means L(x;H) < 0 as defined in (5.5). Meanwhile, on the

interval 0 < x < K when x = N∗, equation (5.6) is used to show thatL(N∗;H) > 0

since H < Hc.

Next, we divide the interval x into two regions: the left region with 0 <

x ≤ N∗ and the right region with x ≥ N∗. On the left region, we can infer the

value L at small x and at x = N∗ as L(0+;H) < 0 < L(N∗;H). Then Bolzano’s

Intermediate Value Theorem is applied, so that there exists N∞,1 with 0 < N∞,1 <

N∗ where L(N∞,1;H) = 0. This means that N∞,1 is a steady state, and since L is

concave in this region with unique global maximiser, N∞,1 must be unique.

Similarly, for large x > K and x = N∗ we haveL(x;H) < 0 < L(N∗;H). Therefore

Bolzano’s Intermediate Value Theorem can be invoked, so that there exists N∞,2

with N∞,2 > N∗ that results in L(N∞,2;H) = 0. This means N∞,2 is a steady state

and it is unique since L is concave in this region with a unique maximiser. In

conclusion, there are exactly two steady states in this case.

The steady states of model (5.1) as described in Theorem 5.1.3 can be investi-

116

gated for their behaviour as presented in the following theorem.

Theorem 5.1.4. The stability of the steady states of (5.1) as referred to Theorem 5.1.3

can be infered upon the value H as follows:

(i) When H = Hc, the unique steady state is unstable.

(ii) When H < Hc, one of the two steady states which has greater value is

stable, and the other steady state is unstable.

Proof. A stable steady state of (5.1) occurs when the first derivative of the popula-

tion growth rate, which is represented by L in (5.5), at N∞ is negative. Before we

begin to prove each case, we first note that the first derivative at a critical point is

zero. Thus we have L′(N∗;H) = 0, where N∗ is the critical point for L as already

defined in (5.3).

Case (i). The value H = Hc leads to L(N∗;H) = 0, which means that N∗ =

N∞. Thus we have L′(N∞;H) = L′(N∗;H) = 0, from which we cannot conclude

anything about the stability. Thus, a further analysis needs to be done and here

we check the direction of the phase line of L(x;H) = dx/dt in the left and right

regions about N∗. Since H = Hc, we may write

L(x;H) = xf(x)−Hc = xf(x)−N∗f(N∗).

Since N∗f(N∗) is the global maximum, then L(x;H) < 0 either for 0 < x < N∗

or x > N∗, such that the phase line direction for L comes in to N∗ from the right

region and then goes out from N∗ to the left, which means the steady state is

unstable.

Case (ii). When H < Hc, there are two steady states arise in the model. We

start with L as a concave function, which implies that L′(x;H) > 0 for 0 < x < N∗

and L′(x;H) < 0 for x > N∗. Since 0 < N∞,1 < N∗ < N∞,2, then L′(N∞,1;H) >

0 and L′(N∞,2;H) < 0 which we can infer that only N∞,2 is the stable steady

state.

117

5.2 Steady state analysis

In this section we improve the harvesting model given in (5.1) by setting the car-

rying capacity as a variable denoted by K = K(t), instead of a constant, so that

the model is now composed of a coupled system of two ODEs and each ODE

consists of a general function of per capita growth rate f(N,K) for population

and g(N,K) for carrying capacity. Thus the harvesting population model is now

expressed as

dN

dt= Nf(N,K)−H,

dK

dt= Kg(N,K),

N(0) = N0, and K(0) = K0.

(5.7)

The functions f = f(x, y) and g = g(x, y) are assumed to be the classC1(R+×R+).

In this section we also assume that there is a one-to-one function y = q(x) which is

a continuously differentiable, obtained from the solution of g = 0, and a constant

x1 > 0 such that f satisfies the following properties:

(i) f(x, q(x)) > 0 for 0 < x < x1, and f(x, q(x)) < 0 for x > x1.

(ii) x 7→ xf(x, q(x)) is strictly concave for 0 < x < x1.

(iii) limx→0+

xf(x, q(x)) = 0.

(iv) D1f(x, y) < 0 and D2f(x, y) > 0 for all x, y > 0.

(5.8)

Notice that we use the similar assumptions with the previous one in (5.2), but

we add the derivative conditions in (iv) to support the stability analysis, but are

still satisfied by the common population models. Thus, like the previous model,

these assumptions are also fit the Logistic model and the models introduced by

Gompertz, Smith and Gilpin & Ayala. Since we have another general function g,

118

we also make some assumptions, more specifically

D1g(x, y) < 0, and D2g(x, y) < 0 for all x, y > 0. (5.9)

Next, the steady states (N∞, K∞) of (5.7) are examined. Since N and K are

positive, the steady states are obtained from each ODE in (5.7) by solving the

following two equations,

g(x, y) = 0 and xf(x, y) = H. (5.10)

As previously assumed, the first equation of (5.10) has the solution y = q(x).

Then, this function is substituted to the second one, so that we obtain

xf(x, q(x)) = H. (5.11)

By solving this equation, we obtain the steady state x = N∞ and a corresponding

y = K∞ = q(N∞) since y is a one-to-one function.

Now we suppose there exists N∗, the value of x which maximises the popu-

lation growth rate with the absence of harvesting in the interval [0, x1]. Thus we

can express it as

N∗ = argmax(xf(x, q(x))

), 0 ≤ x ≤ x1. (5.12)

Since x 7→ xf(x, q(x)) is strictly concave (see property (ii) in (5.8)), this implies

any local maximum of x 7→ xf(x, q(x)) is global maximum, and the optimal set

contains at most one point. With that reason, N∗ is said to be the unique global

maximum for x 7→ xf(x, q(x)). Then through this statement, we can state a lemma

regarding the maximiser N∗ as follows.

Lemma 5.2.1. Let f ∈ C1(R+ × R+) satisfy the properties given in (5.8), and let y =

q(x) be the unique solution g(x, y) = 0. Thus N∗ given in (5.12) is the unique global

119

maximiser for x 7→ xf(x, q(x)).

Now, harvesting is taken into account in the population dynamics of (5.7).

As we can notice from the model, the higher value of H gives a lower population

growth until no growth occurs. Moreover, a growth decay occurs. Therefore there

is a critical value of the harvesting rate such that the population size can still be

sustained, and we define it as

Hc = N∗f(N∗, q(N∗)). (5.13)

Next, let us redefine the population growth rate as a function that depends

only on x and the parameter H , expressed as

L(x;H) = xf(x, q(x))−H, (5.14)

where L = 0 is the condition to obtain the steady state solution x = N∞. From

a geometric point of view, as depicted in Figure 5.1, we observe that L and x 7→

xf(x, q(x)) have a similar graphs because the constant H causes the graph L to

move downwards, but the graph shape remains unchanged. For this reason, L

is also strictly concave and we can see the derivation to show that L is strictly

concave in Appendix A.

SinceN∗ is the unique maximiser of x 7→ xf(x, q(x)), as stated in Lemma 5.2.1,

then L also has a unique maximiser at the same value N∗. Thus we may state a

lemma that relates to the maximiser N∗ of L as follows.

Lemma 5.2.2. Let f ∈ C1(R+ × R+) satisfy the properties given in (5.8), and let y =

q(x) be the unique solution g(x, y) = 0. Thus L(x;H) = xf(x, q(x)) − H has unique

global maximum N∗ given in (5.12).

With N∗ as the maximiser of L, the global maximum of L in (5.14) can be

120

00

Figure 5.1: The curve that changes its position to move downward as H increasesfrom H1, H2, then H3

specified as

L(N∗;H) = N∗f(N∗, q(N∗))−H = Hc −H, (5.15)

whereHc is given in (5.13). Based on the explanation and the lemmas we have, we

now can present the relation between a saddle-node bifurcation and parameter

harvesting rate with the following theorem.

Theorem 5.2.3. Let f, g ∈ C1(R+×R+) satisfy the properties given in (5.8) and in (5.9),

respectively. Let y = q(x) be the unique solution of g(x, y) = 0. With Hc is given

by (5.13), a saddle-node bifurcation exists in model (5.7) based on the harvesting value as

follows:

(i) If H = Hc, then there is only one steady state of (5.7).

(ii) If H > Hc, then there are no steady states of (5.7).

(iii) If H < Hc, then there are exactly two steady states of (5.7).

Case (i). From (5.15) we have L(N∗;H) = 0 since H = Hc, which means N∗

is not only a global maximiser but also a steady state of L. Hence N∗ can also be

written as N∞. Recall Lemma 5.2.2, which says that N∗ is the unique maximiser

121

for L. Thus with the corresponding K∞ = q(N∗), the steady state (N∞, K∞) in

this case is also unique.

Case (ii). In this case we can write L(x;H) ≤ L(N∗;H) for 0 ≤ x ≤ x1 since

L(N∗;H) is the global maximum and implies thatL(x;H) ≤ L(N∗;H) = Hc−H <

0.

On the other hand, for x > x1, we use property (i) in (5.8) to obtain xf(x, q(x)) < 0

since x is positive. Thus L(x;H) = xf(x, q(x)) − H < 0. From these two results,

we can conclude that L(x;H) < 0 for all x > 0. This means L will never reach

zero and there are no steady states for the model.

Case (iii). We investigate the value L(x;H) by dividing x into two regions,

but first we need to know the value for small and large x as well as at x = N∗.

For small x, property (iii) in (5.8) that states limx→0+ xf(x, q(x)) = 0 is applied so

we deduce that limx→0+ L(x;H) < 0.

When x is large, i.e. x > x1, we use property (i) of f to see that xf(x, q(x)) < 0.

Since H is positive constant, we have L(x;H) = xf(x, q(x))−H < 0.

Meanwhile, at x = N∗, where 0 < x < x1, we have L(N∗;H) = Hc − H > 0.

With these results, we divide the interval x into two regions: the left region with

0 < x ≤ N∗, and the right region with x > N∗. To make it easier to understand

the explanation of this case, we illustrate this in the graph shown in Figure 5.2.

Starting from the left region, the value of L for small x and x = N∗ can be

used here to obtain L(0+;H) < 0 < L(N∗;H). Bolzano’s Intermediate Value

Theorem is then applied so that there exists N∞,1 with 0 < N∞,1 < N∗, where

L(N∞,1;H) = 0. This implies that N∞ is the steady state with the correspoding

K∞,1 = q(N∞,1). Since L is concave in this region and has a unique global max-

imiser N∗, then N∞,1 must be unique.

Similarly, for large x > x1 and x = N∗ we have L(x;H) < 0 < L(N∗;H).

Thus Bolzano’s Intermediate Value Theorem can be used here so that there exists

122

00

Figure 5.2: Two steady states obtained in case (iii)

N∞,2, with N∞,2 > N∗, where L(N∞,2;H) = 0. This means that N∞,2 is also a

steady state. Since L is concave in this region and N∗ is unique, then N∞,2 must

be unique with the correspoding K∞,2 = q(N∞,2). In conclusion, there are exactly

two steady states in this case.

The stability of the steady states of model (5.7) as described in Theorem 5.2.3

can be investigated through the following theorem.

Theorem 5.2.4. Let Hc is defined in (5.13). The stability for the steady states of (5.7)

can be determined by harvesting value as follows:

(i) The unique steady state is unstable when H = Hc.

(ii) The two steady states, say (N∞,1, K∞,1) and (N∞,2, K∞,2), where N∞,1 < N∞,2,

have distinct stability. (N∞,1, K∞,1) is unstable, while (N∞,2, K∞,2) is stable when

H < min(Hs, Hc),

where Hs = −N2∞,2D1f(N∞,2, K∞,2)−N∞,2K∞,2D2g(N∞,2, K∞,2).

Proof. The trace and determinant of the linearised system for (5.7) are used here

to investigate the steady states’ behaviour. We begin with the Jacobian matrix of

123

the linearised model for (5.7), written as

J(x,y) =

f(x, y) + xD1f(x, y) xD2f(x, y)

yD1g(x, y) g(x, y) + yD2g(x, y)

.Now, we calculate the Jacobian matrix at a steady state (N∞, K∞). But first recall

the two equations that fulfil the steady state solution given in (5.10). With these

equations, we can write g(N∞, K∞) = 0 and f(N∞, K∞) = H/N∞. Thus J at the

steady state can be re-expressed as

J∞ = J(N∞,K∞) =

HN∞

+N∞D1f(N∞, K∞) N∞D2f(N∞, K∞)

K∞D1g(N∞, K∞) K∞D2g(N∞, K∞)

.

Next, the trace and determinant for J∞ can be calculated, respectively, as

tr (J∞) =H

N∞+N∞D1f(N∞, K∞) +K∞D2g(N∞, K∞) (5.16)

and

det(J∞) =Hq(N∞)

N∞D2g(N∞, K∞) +N∞K∞

[D1f(N∞, K∞)D2g(N∞, K∞)

−D2f(N∞, K∞)D1g(N∞, K∞)],

(5.17)

where K∞ = q(N∞).

As we expect, tr (J∞) < 0 and det(J∞) > 0 have a stable steady state. Hence

two conditions need to be fulfilled, which are

H < −N2∞D1f(N∞, K∞)−N∞K∞D2g(N∞, K∞) (5.18)

from (5.16), and

H < −N2∞D1f(N∞, K∞) +N2

∞D2f(N∞, K∞)

D2g(N∞, K∞)D1g(N∞, K∞) (5.19)

124

from (5.17).

Case (i). Recall the proof of the Theorem 5.2.3 in part (i), which says that

whenH = Hc, thenN∗ = N∞ and the steady state (N∞, K∞) is unique. Therefore,

through the definition Hc in (5.13), we may write

H = Hc = N∗f(N∗, q(N∗)) = N∞f(N∞, K∞). (5.20)

Also, recall that N∗ is the global maximiser of L, which implies that L′(N∞, H) =

L′(N∗, H) = 0. Meanwhile, for all x > 0, we can expand the first derivative of L

in (5.14) to write

L′(x;H) = f(x, q(x)) + xD1f(x, q(x)) + xq′(x)D2f(x, q(x)). (5.21)

Since y = q(x) is a solution of g(x, y) = 0, then we can write g(x, q(x)) = 0 and its

derivative with respect to x can be written as

D1g(x, q(x)) +D2g(x, q(x)) q′(x) = 0,

and we obtain

q′(x) = −D1g(x, q(x))

D2g(x, q(x)).

Substituting q′(x) into (5.21), the first derivative of L can be transformed into

L′(x;H) = f(x, q(x)) + xD1f(x, q(x))− xD2f(x, q(x))

D2g(x, q(x))D1g(x, q(x)). (5.22)

Since L′(N∞;H) = 0, (5.22) yields

f(N∞, K∞) = −N∞D1f(N∞, K∞) +N∞D2f(N∞, K∞)

D2g(N∞, K∞)D1g(N∞, K∞).

Multiplying N∞ to both sides of this equation and then subsituting it into (5.20),

125

we obtain

H = −N2∞D1f(N∞, K∞) +N2

∞D2f(N∞, K∞)

D2g(N∞, K∞)D1g(N∞, K∞),

which obviously does not satisfy (5.19). Hence the steady state (N∞, K∞) is un-

stable.

Case (ii). We begin first withN∞,1. AsL is concave function, and as explained

in the proof in Theorem 5.2.3 that L(0+;H) < 0 < L(N∗;H), we know that L is

increasing from 0 to N∗. If 0 < N∞,1 < N∗, we have L′(N∞,1;H) > 0. From (5.22)

we may write

f(N∞,1, K∞,1) > −N∞,1D1f(N∞,1, K∞,1)+N∞,1D2f(N∞,1, K∞,1)

D2g(N∞,1, K∞,1)D1g((N∞,1, K∞,1).

Then we have

−N2∞,1D1f(N∞,1, K∞,1) <N∞,1f(N∞,1, K∞,1)

−N2∞,1

D2f(N∞,1, K∞,1)

D2g(N∞,1, K∞,1)D1g(N∞,1, K∞,1)

by multiplying the inequality with N∞,1. Recall the steady state solution in (5.10),

from the second equation we haveN∞,1f(N∞,1, K∞,1) = H . This implies the latter

inequality now becomes

−N2∞,1D1f(N∞,1, K∞,1)−H < −N2

∞,1

D2f(N∞,1, K∞,1)

D1g(N∞,1, K∞,1)D2g(N∞,1, K∞,1)

that results in

H > −N2∞,1D1f(N∞,1, K∞,1) +N2

∞,1

D2f(N∞,1, K∞,1)

D1g(N∞,1, K∞,1)D2g(N∞,1, K∞,1)

which does not satisfy (5.19). So steady state (N∞,1, K∞,1) is unstable.

Conducting in a similar way forN∞,2, we have L′(N∞,2;H) < 0 and we even-

126

tually find the condition

H < −N2∞,2D1f(N∞,2, K∞,2) +N2

∞,2

D2f(N∞,2, K∞,2)

D1g(N∞,2, K∞,2)D1g(N∞,2, K∞,2).

As we can see, this condition fulfils (5.19). Thus the other condition that is stated

in (5.18) needs be satisfied for stability. Since N∞,2 exists when H < Hc, we can

infer that the steady state (N∞,2, K∞,2) exists and is stable when

H < min(−N2∞,2D1f(N∞,2, K∞,2)−N∞,2K∞,2D2g(N∞,2, K∞,2), Hc).

Next, an example of the model in (5.7) with two specific functions f and g

is presented here to see the population dynamics as the harvesting yield value

varies.

Example 5.2.1. We use the well-known logistic growth rate for the population

dynamics, where f(N,K) = r(1−N/K), and we introduce the carrying capacity

per capita growth rate g(N,K) = b− c(K − e−N

), where b, c > 0. Thus the model

can be expressed as

dN

dt= rN

(1− N

K

)−H,

dK

dt= K

[b− c(K − e−N)

].

(5.23)

First of all, we need to inspect whether these functions f and g satisfy the proper-

ties in (5.8) and (5.9), respectively. Now, since y = q(x) = b/c+ e−x is the solution

of g(x, y) = 0, we can write xf(x, q(x)) as

xf(x, q(x)) = rx(1− x

b/c+ e−x

). (5.24)

Then the second derivative needs to be examined to find out if xf(x, q(x)) is

strictly concave (see part (ii) of Theorem A.0.4 in Appendix A). Differentiating

127

(5.24) two times with respect to x we obtain

d2

dx2(xf(x, q(x))) = − r

b/c+ e−x

[2 +

x

b/c+ e−x+

2e−x

b/c+ e−x+

xe−2x

(b/c+ e−x)2

].

As we can observe, the second derivative has negative value everywhere, and we

can conclude that x 7→ xf(x) is strictly concave for all x > 0, which also means it

is also strictly concave for 0 < x < x1 for any positive value x1. Thus property (ii)

of f in (5.8) is satisfied, while the remaining properties are easy to check as well

as the property of g in (5.9).

0 20 4010 30 505 15 25 35 45

10

2

4

6

8

3

5

7

9

11

Figure 5.3: Stable steady state in the model (5.23) occurs when H < Hc is chosen

In this example we perform some graphical simulations for population and

carrying capacity dynamics as illustrated in Figure 5.3. Here we use the param-

eter values r = 0.2, b = 1, c = 0.1 and the critical harvesting is calculated as

Hc = 0.5. Then H = 0.1 is chosen so that two steady states are obtained, with val-

ues (0.5, 10.6) and (9.5, 10). As can be seen from the figure, the population grows

toward the larger population steady state, which verifies Theorem 5.2.4. On the

other hand when H is chosen at a value greater than Hc for instance H = 0.8,

either population or carrying capacity is not showing that it moves to a limiting

value as depicted in Figure 5.4. Moreover, the population declines to negative

128

0 2 41 3 50.5 1.5 2.5 3.5 4.5

0

10

−2

2

4

6

8

12

14

Figure 5.4: No stable steady state appears in the model (5.23) when H > Hc ischosen

values which is unrealistic.

Figure 5.5 shows the bifurcation diagram for model (5.23) that summarizes

the value H that affects the number of steady state the model has. As we can see

that when H is chosen less than Hc = 0.5, two steady states are obtained which

are symbolised by the red and blue asteric. As value H is larger then reaching to

H = Hc, the two steady states collide which results N∞,1 = N∞,2, that is only one

steady state occurs. But when H is taken even larger such that H > Hc, no steady

state will occur.

Next, we break down the carrying capacity per capita growth rate model

into two special cases. The First model is when it is specified as a function of

population size only, and the second model is when the carrying capacity is the

only one that governs its per capita growth rate.

129

0 0.2 0.4 0.6 0.80.1 0.3 0.5 0.7

0

10

2

4

6

8

12

Figure 5.5: Bifurcation diagram N∞,1 and N∞,2 versus H for model (5.23).

5.2.1 Carrying capacity per capita growth rate depends on popu-

lation size only

Here we use a special case where the carrying capacity per capita growth rate

depends only on population size. Thus, we can rewrite the model given in (5.7)

as

dN

dt= Nf(N,K)−H,

dK

dt= Kg(N).

(5.25)

In this model, f is of C1(R+ ×R+) class and we assume it has some properties as

follows:

(i) For any x > 0, there exists y1 = y1(x) such that

f(x, y) < 0 for 0 < y < y1 and f(x, y) > 0 for y > y1.

(ii) limy→∞

f(x, y) exists and is positive for x > 0.

(iii) D1f(x, y) < 0 and D2f(x, y) > 0 for all x, y > 0.

(5.26)

130

Compared with the main carrying capacity per capita growth rate model in (5.7),

the concavity condition is removed from the assumption, whereas the first deriva-

tive criteria remain unchanged. Nevertheless, these modified assumptions still

encompass the logistic, Gompertz, Smith and Gilpin & Ayala population mod-

els.

On the other hand, we let g be a strictly decreasing function that belongs

to C1(R+) class, and we assume there exists a unique constant N∞ such that

g(N∞) = 0, which means N∞ is the population steady state obtained from the

second ODE in (5.7).

SubstitutingN∞ into the first ODE, we can obtain the carrying capacity steady

states by solving

N∞f(N∞, y) = H.

In other words, if we define

L(y;H) = N∞f(N∞, y)−H, (5.27)

then the carrying capacity steady state y = K∞ with the corresponding popula-

tion steady state N∞ can be determined when L equals zero.

Notice that H is the parameter that controls the number of steady states.

But first we need to formulate the critical harvestng value Hc which is defined

differently from the previous model in (5.13) due to the different assumptions on

f . Based on the assumptions in (5.26), we can say from a geometric perspective

that the graph f(N∞, y) increases to the limiting value limy→∞ f(N∞, y). For this

reason, the critical harvesting rate in this section is defined as

Hc = N∞ limy→∞

f(N∞, y), (5.28)

131

which implies that L in (5.27) for very large y becomes

limy→∞

L(y;H) = N∞ limy→∞

f(N∞, y)−H = Hc −H. (5.29)

Now we have a theorem regarding the existence of steady states which is influ-

enced by the harvesting value as follows.

Theorem 5.2.5. Let f ∈ C1(R+ ×R+) and satisfy the assumptions given in (5.26). Let

g be a strictly decreasing function in the C1(R+). WithHc defined in (5.28), the existence

of a steady state in the model (5.25) can be determined by the harvesting value as follows:

(i) If H < Hc, then there exists a unique steady state in the model.

(ii) If H ≥ Hc, then there are no steady states in the model.

Proof. First we investigate the value L in (5.27) for small y, and we have

limy→0+

L(y;H) = N∞ limy→0+

f(N∞, y)−H < 0 (5.30)

as f is negative for small y based on assumption (i).

Next we inspect the first derivative of L, and using assumption (iii) we obtain

L′(y;H) = N∞D2f(N∞, y) > 0 for all y > 0, (5.31)

which gives that L is an increasing function. Meanwhile, for large y, we can

observe from (5.29) that the sign L may change as H passes Hc.

Case (i). When H < Hc, we have positive L for large y. Since we have

negative value of L(y;H) for small y, we can apply Bolzano’s Intermediate Value

Theorem which says that there exists K∞ such that L(K∞;H) = 0, which means

K∞ is the carrying capacity steady state. As we have found out that L is an

increasing function as shown in (5.31), then K∞ is unique.

Case (ii). When H > Hc, we can see in (5.29) that L now has a negative value

for large y. On the other hand when H = Hc, the value L tends to but never

132

reaches zero as y large. Recall that for small y the value L is negative, and L is an

increasing function for all y. Since L never reaches positive or zero value for large

y, we can conclude that there are no steady states in the model (5.25). Figure 5.6

illustrates the value of L that leads the steady state to exist as in Case (i) and

disappears as in Case (ii).

00

Figure 5.6: The existence of the steady states in model (5.25) when H < Hc.

Furthermore, to investigate the steady state behaviour, the trace and determi-

nant calculations are utilised here to find out whether the steady state that exists

in the model is stable or not. Also, trajectories for population and carrying ca-

pacity are also traced whether they oscillate or focus straight to the steady state.

A theorem with regards to the steady state stability, as well as how the trajectory

moves for the model (5.25) is proposed as follows.

Theorem 5.2.6. Let f ∈ C1(R+ ×R+) and satisfy then assumptions given in (5.8). Let

g be a strictly decreasing function in the C1(R+). With Hs = −N2∞D1f(N∞, K∞) and

Hc defined in (5.28), the unique steady state (N∞, K∞) of the model (5.25) is stable when

H < min(Hs, Hc) (5.32)

with spiral-shape trajectory.

133

Proof. It is easy to check that the Jacobian matrix of the model (5.25) at steady

state J∞ has trace

tr (J∞) =H

N∞+N∞D1f(N∞, K∞). (5.33)

Also, the determinant can be expressed as

det(J∞) = −N∞K∞D2f(N∞, K∞)g′(N∞).

Since the determinant is always positive (see assumption (iii) and g is a strictly

decreasing function), a condition to have a stable steady state must be fulfilled

when the trace is negative, that is from (5.33) we obtain the condition

H < −N2∞D1f(N∞, K∞). (5.34)

This condition is possible since D1f(N∞, K∞) < 0 from assumption (iii) in (5.26).

As has been shown in Theorem 5.2.5 that H < Hc is the requirement for the

existence of steady state, and the condition in (5.34) is the necessity for the steady

state to be stable. Therefore the intersection of these two conditions, from which

the steady state of the model (5.25) is stable can be written as

H < min(−N2∞D1f(N∞, K∞), Hc). (5.35)

Now to investigate whether the trajectory moves to the steady state spirally, we

calculate

tr (J∞)2 − 4 det(J∞) =H2

N2∞

+D1f(N∞, K∞)(N2∞D1f(N∞, K∞) + 2H)

+ 4N∞K∞D2f(N∞, K∞)g′(N∞),

(5.36)

and the value must be negative. Thus the condition to have the negative sign, we

134

can write

H2

N2∞

+ 4N∞K∞D2f(N∞, K∞)g′(N∞) < −D1f(N∞, K∞)(N2D1f(N∞, K∞) + 2H).

Recall the stable condition given by (5.34). This implies

H2

N2∞

+ 4N∞K∞D2f(N∞, K∞)g′(N∞) < −D1f(N∞, K∞)H.

or we can also write this equation as

H2 +N2∞D1f(N∞, K∞)H + 4N3

∞K∞D2f(N∞, K∞)g′(N∞) < 0.

Solving this inequality we obtain

H <−N2

∞D1f(N∞, K∞) +N2∞√D1f(N∞, K∞)2 − 16N∞K∞D2f(N∞, K∞)g′(N∞)

2.

(5.37)

Notice that the inequality in (5.37) involves the steady state value so that the

condition H < Hc has to be met to make the steady state exist. Since g strictly

decreases and D2f(N∞, K∞) is positive, the square root value is positive. In con-

clusion, we have the condition (5.35) as the intersection region between (5.35)

and (5.37) that means the trajectory of population and carrying capacity moves

toward the steady state spirally with the condition given in (5.35).

Example 5.2.2. In this example we refer to the propulation growth rate model

from Smith [110] and the model from Safuan et al. [105] for the carrying capacity

growth rate, as shown by

dN

dt= Nf(N,K)−H =

rN(K −N)

K + aN−H,

dK

dt= Kg(N) = K(b− cN).

(5.38)

Note that the function f satisfies the assumption in (5.26) and g is a decreasing

135

function. In this example, some simulations are performed using the parameter

values r = 0.2, a = 0.3, b = 0.5, c = 0.1 and H = 0.3. By these values we obtain the

steady state (N∞, K∞) = (5, 7.78) and Hc = 1. Notice that our chosen H implies

the appearance of the steady state, but to know the stability, another calculation

needs to be done as mentioned in (5.32), from which we obtain Hs = 0.58. Since

H = 0.3 which is less than min(0.58, 1), the population and the carrying capacity

move toward the steady state (5, 7.78) in an oscillatory motion as depicted in

Figure 5.7. Notice that the calculations, as well as the population and carrying

0 10020 40 60 8010 30 50 70 90

10

4

6

8

12

5

7

9

11

Figure 5.7: Stable steady state emerges in the model (5.38) with oscillatory mo-tion by the chosen H < min(Hs, Hc)

capacity dynamics, verify Theorem 5.2.6.

Another simulation is performed when we use the same parameter values

except for H . As we can observe from (5.38) that the value of H influences the

value of K∞, which then affects the value of Hs. Now, when we choose H =

0.5, we obtain Hs = 0.44 so that H > min(0.44, 1). Although the steady state,

calculated at (N∞, K∞) = (5, 11.5), is still obtained since H is less than Hc = 1,

but since H > 0.44, the steady state is unstable. The population dynamics using

this value of H is shown in Figure 5.8.

136

0 10020 40 60 8010 30 50 70 90

0

20

40

10

30

50

5

15

25

35

45

Figure 5.8: No steady state emerges in the model (5.38) when H is chosen suchthat H > min(Hs, Hc)

5.2.2 Carrying capacity per capita growth rate depends on carry-

ing capacity only

Another special case for the harvesting population model presented in (5.7) is

when the function g now depends only on K. Thus the model is now established

by the general functions of f(N,K) and g(K), expressed as

dN

dt= Nf(N,K)−H,

dK

dt= Kg(K).

(5.39)

Here, f is in the class C1(R+ × R+) and we assume that there is a constant y1 > 0

such that f satisfies the following assumptions.

(i) f(x, y1) > 0 for 0 < x < y1 and f(x, y1) < 0 for x > y1.

(ii) x 7→ xf(x, y1) is strictly concave for 0 < x < y1.

(iii) limx→0+

xf(x, y1) > 0.

(iv) D1f(x, y1) < 0 and D2f(x, y1) > 0 for all x > 0.

(5.40)

137

Observe that these assumptions are similar to those of the main carrying capacity

growth rate model in (5.8). The difference lies in the type of carrying capacity

form used in the assumption, which is constant, whereas the main model uses a

function of x for the carrying capacity.

On the other hand, we let g a strictly decreasing function that belongs to

C1(R+), and we assume that there is a unique constant K∞ such that g(K∞) = 0.

In other words,K∞ is the carrying capacity steady state obtained from the second

ODE in (5.39).

Substituting K∞ into the first ODE, we can have a population steady state

solution by solving

xf(x,K∞) = H.

Now let us define F (x) = xf(x,K∞), where F (0) = 0. Notice that F (x) is strictly

concave on 0 < x < K∞ by applying assumption (ii). Thus there exists a max-

imiser N∗ of F (x), written as

N∗ = argmax(F (x)

), 0 ≤ x ≤ K∞.

With F (x) strictly concave, this implies that N∗ is the unique global maximiser,

as mentioned in Lemma 5.2.1 in the previous section.

Now let

L(x;H) = xf(x,K∞)−H. (5.41)

This function is also strictly concave since the presence of the constant H does

not affect the shape of L or its concavity. Therefore, x = N∗ is also the unique

maximiser of L.

Notice thatL = 0 in (5.41) is the way to obtain the population steady stateN∞

with the corresponding K∞. Now, as L is strictly concave, there is a harvesting

critical value, denoted byHc, such that the steady state can still be achieved. Since

the assumption f in this section is similar to that of the main harvesting model,

138

stated in (5.8), the definition for Hc can be proposed here in the similar form, and

certainly by referring to (5.41), as

Hc = N∗f(N∗, K∞). (5.42)

Thus, the global maximum L in (5.41), that is at N∗, can be expressed using (5.42)

as

L(N∗;H) = N∗f(N∗, K∞)−H = Hc −H. (5.43)

For this purpose, the number of steady states can be determined by the fol-

lowing theorem.

Theorem 5.2.7. Let f ∈ C1(R+ × R+) and satisfy the assumptions given in (5.40).

Let g be a strictly decreasing function in the C1(R+) , where the constant K∞ is the

unique solution for g = 0. With Hc = N∗f(N∗, K∞), the number of steady states in the

model (5.39) can be determined by the harvesting value as follows:

(i) If H = Hc, then there exists a unique steady state in the model.

(ii) If H > Hc, then there are no steady states in the model.

(iii) If H < Hc, then there are exactly two steady states in the model.

Proof. Case (i). When H = Hc, we have L(N∗;H) = 0 in (5.43). Hence N∗ is also

the steady state N∗ = N∞ with corresponding K∞. As N∗ is unique, N∞ is also

unique.

Case (ii). First we can write L(x;H) ≤ L(N∗;H) for 0 < x < K∞ since

L(N∗;H) is the global maximum. Then, from (5.43), we have L(N∗;H) < 0 since

H > Hc. Combining these two statements implies L(x;H) < 0 for 0 < x < K∞.

Furthermore, for x > K∞, we use assumption (i) to see that xf(x,K∞) < 0

since x is positive, that leads to L(x;H) = xf(x,K∞)−H < 0. From the results of

these two regions of x, we can infer that L(x;H) < 0 for all x > 0. This means L

never reaches zero, which also means no steady states appear in the model.

139

Case (iii). For small x, the sign L in (5.41) is seen to be negative since as-

sumption (iii) says limx→0+ xf(x,K∞) = 0. Likewise, for large x > K∞, we use

assumption (i) to acquire xf(x,K∞) < 0, which implies L(x;H) < 0. Meanwhile,

in the interval 0 < x < K∞, we can pick x = N∗ such that from (5.43) we obtain

L(N∗;H) > 0 since H < Hc

Next, we separate the domain x into two regions: the left region 0 < x <

N∗ and the right region x > N∗. Starting from the left region, we already have

L(0+;H) < 0 < L(N∗;H). Thus Bolzano’s Intermediate Value Theorem can be

applied here, which results that there exists N∞,1 in the interval (0, N∗) such that

L(N∞,1;H) = 0. Also for the right region, we recall that L(N∗;H) > 0 > L(x;H)

for large x > K∞, that implies there exists N∞,2 > N∗ such that L(N∞,2;H) = 0.

Since L is concave,N∞,1 on the left region is unique, as well as the uniqueN∞,2 on

the right region. Hence, we can conclude that there are exactly two steady states

for this case.

The stability of the steady states that appear in model (5.1) can be examined

by the following theorem.

Theorem 5.2.8. Let f be a functions in the class C1(R+ × R+) with some assumptions

given in (5.40) and g be a strictly decreasing function in the class C1(R+). With Hs =

−N2∞D1f(N∞, K∞), where (N∞, K∞) denotes the steady states in model (5.39), and

Hc defined in (5.42), the stable condition for the steady states can be determined by the

harvesting value as follows:

(i) The unique steady state obtained when H = Hc is unstable.

(ii) One of the two steady states which has greater value is stable when H <

min(Hs, Hc), while the other steady state is unstable.

Proof. We start with the trace and determinant of the Jacobian matrix of (5.39) at

the steady state, J∞, and express them, respectively as

tr (J∞) =H

N∞+N∞D1f(N∞, K∞) +K∞g

′(K∞)

140

and

det(J∞) =K∞

N∞Hg′(K∞) +N∞K∞D1f(N∞, K∞)g′(K∞).

Since we have uncertain signs for the trace and determinant even using the as-

sumptions, the conditions that must be imposed to have a stable steady state are

that the tr J∞ < 0 and det J∞ > 0 . Thus we can write those as

H < −N2∞D1f(N∞, K∞)−N∞K∞g

′(K∞).

from the trace, and

H < −N2∞D1f(N∞, K∞)

from the determinant. With these results, we can infer that the stable condition

for the steady state can be obtained by having the intersection between the trace

and determinant conditions, that is

H < −N2∞D1f(N∞, K∞). (5.44)

Case (i). Recall the proof of Case (i) in Theorem 5.2.8 that stated when H = Hc,

then N∗ = N∞. Thus, by referring to (5.42) we can write

H = Hc = N∞f(N∞, K∞). (5.45)

For this case, we can also notice that L′(N∞, H) = L′(N∗, H) = 0 since N∗ is the

maximiser of L. Hence we can find the first derivative of L in (5.41) at N∞, then

equating it to zero to obtain

f(N∞, K∞) = −N∞D1f(N∞, K∞)

Multiplying N∞ to both sides of the latter equation, then subsituting into (5.45)

yields

H = −N2∞D1f(N∞, K∞)

141

which does not satisfy the stability condition stated in (5.44). In other words, the

steady state is unstable.

Case (ii). Recall that we have two steady states in this case, these are N∞,1

and N∞,2 such that N∞,1 < N∞,2. As explained in the proof in Case (iii) Theo-

rem 5.2.7, that L is concave and L(0+;H) < 0 < L(N∗;H), with N∞,1 is in the

interval (0, N∗). With this reason, L in increases at N∞,1, so that we can write

L′(N∞,1;H) = f(N∞,1, K∞) + N∞,1D1f(N∞,1, K∞) > 0, or we can also reexpress

this as

−N2∞,1D1f(N∞,1, K∞) < N∞,1f(N∞,1, K∞)

Subtracting both sides to H we now have

−N2∞,1D1f(N∞,1, K∞)−H < N∞,1f(N∞,1, K∞)−H = 0,

which results in

H > −N2∞,1D1f(N∞,1, K∞),

which does not satisfy (5.44). So (N∞,1, K∞) is unstable.

By doing the same way to N∞,2, and we also already inspect in the proof of

Case (iii) in Theorem 5.2.7 that L′(N∞,2;H) < 0, we eventually find the condition

H < −N2∞,2D1f(N∞,2, K∞)

which fulfils (5.44). Therefore the population steady state N∞,2 which is greater

than N∞,1 is stable. Since H < Hc is the requirement for the steady state to exist,

we now have

H < min(−N2

∞,2D1f(N∞,2, K∞), Hc

)as the condtion for the stablity steady state (N∞,2, K∞).

Example 5.2.3. We use the Gompertz model for the population per capita growth

142

rate, f(N,K) = r log(K/N), and the model from Meyer & Ausubel for the carry-

ing capacity growth rate, g(K) = b − cK. Thus we can expressed the model as

dN

dt= rN log

(KN

)−H,

dK

dt= K(b− cK).

(5.46)

Note that f verifies the assumption in (5.40) and g can be easily observed that to

be a strictly decreasing function. In this example we perform some simulations

and we choose the parameter values r = 0.2, a = 0.1, b = 0.3, c = 0.05 and H =

0.1. From these values we obtain the critical harvesting rate as Hc = 0.44 and

two steady states: (0.13, 6) and (5.48, 6). Since H = 0.1 > min(0.0005, 0.44) for

0 10020 40 60 8010 30 50 70 90

4

6

3

5

3.5

4.5

5.5

6.5

Figure 5.9: The steady state exists and stable in the model (5.46) with the chosenH < min(Hs, Hc)

N∞ = 0.13, and H = 0.1 < min(0.99, 0.44) for N∞ = 5.48, then the chosen H = 0.1

fulfils the stable condition of the steady state (5.48, 6) as stated in Theorem 5.2.8.

Therefore the population and carrying capacity grow toward the limiting value

(5.48, 6) as shown in Figure 5.9. Meanwhile whenH is chosen atH = 0.5 such that

H > min(0.1, 0.44), then the population decreases, reaching zero and negative

values, and no steady state is found. This is shown by Figure 5.10.

143

0 20 40 60 8010 30 50 70

0

10

−2

2

4

6

8

Figure 5.10: No steady state exists in the model (5.46) with the chosen H >min(Hs, Hc)

In general, it can be informed that the number of steady states that appear

in model (5.46) is driven by the varying value H as presented in Figure 5.11. The

graphs show that two steady states are obtained only when H is chosen at value

less than Hc=0.44. Meanwhile, one steady state will appear when H = 0.44 and

no steady state will be yielded when H > 0.44 is taken.

0 0.2 0.4 0.60.1 0.3 0.5

0

2

4

6

8

1

3

5

7

Figure 5.11: Bifurcation diagram N∞,1 and N∞,2 versus H for model (5.46).

144

5.3 Conclusion

In this chapter we analysed the steady states for the general harvesting popula-

tion model with variable carrying capacity given in (5.7). Some assumptions were

given for f(N,K) such that they fit the character used for most population mod-

els, such as logistic and Gompertz models, as well as the models introduced by

Smith and Glipin & Ayala. Likewise, g(N,K) was assumed to have some proper-

ties such that it covers the properties of the population models, given by Safuan

et al. and Thornley & France. With these assumptions, we could determine how

many steady states appear in the model as well as their behaviour influenced by

the value H (below, exceeds or at critical value of H). From here we found that

the harvesting model experiences a saddle-node bifurcation with respect to the

parameter H .

The carrying capacity per capita growth rate model was then broken down

into two special types. First, when it depends on population only and second,

when it depends on carrying capacity only. With these modified models, the as-

sumptions also changed. For the first type, we obtained a unique steady state

when H is less than Hc, but it disappears when H was made to increase beyond

Hc. For the second type, like in the main carrying capacity model, the saddle-

node bifurcation also appears. With these two types of carrying capacity models,

steady states are always obtained with the condition H < Hc and another condi-

tion needs to be applied to obtain the stable behaviour.

An example was given for each model. For the main model we introduced

our own form that satisfies the assumptions, while the other two special models

used the carrying capacity per capita model from Safuan et al. and Thornley &

France. Unlike the second type, the first type has different dynamics compared to

the main model for either the population or carrying capacity, where the solutions

oscillate before reaching the limiting value.

145

Chapter 6

Optimisation of A Harvesting

Model

In this chapter we present the optimal harvesting problem in fish populations

with a controllable effort, where the fish carrying capacity is defined as a vari-

able. The growth rate of the fish population is represented by a general function

so that we can use any type of growth rate model other than the logistic func-

tion. Likewise, the carrying capacity growth rate is also expressed as a general

function, in which the functions given in the previous chapter will be used here.

Furthermore, the model is then modified by expressing the carrying capacity in

terms of the food availability for fish which is also harvested. In other words,

there is one additional effort as a control variable that is used for this optimisa-

tion problem.

6.1 Model of one harvesting effort

In this section we use the same objective function for the optimal harvesting prob-

lem given in (2.67), i.e. maximising the discounted profit by controlling the effort

146

value E(t), governed by

J =

∫ T

0

e−δt(pqN − c1 −

c22E)E dt. (6.1)

As defined earlier in Section 2.6, p denotes the price for the harvest unit, q is the

catchability and ρ is the discount rate. Meanwhile c1 and c2 represent the cost of

harvesting. But since the carrying capacity is set as a variable, the constraint given

in this problem is not only the dynamics of the fish population biomass N(t)

but also its carrying capacity K(t). We also construct the models of population

and carrying capacity growth rate by two general functions, denoted by f(N,K)

and g(N,K), respectively. Therefore, the constraints for obtaining the optimised

discounted profit can be specified as

dN

dt= f(N,K)− qEN, N(0) = N0,

dK

dt= g(N,K), K(0) = K0.

(6.2)

The most common population growth rate used in a fisheries management is the

Schaefer model [107], where f(N,K) is governed by logistic growth. It should

be noted that the effort value is set so that it does not exceed the upper limit, say

Emax, to avoid the negative equilibrium population size due to overharvesting.

We can write this condition as

0 < E < Emax <∞,

which can be investigated in the first equation in (6.2).

There are several methods to solve this optimisation problem, two of which

are the calculus of variations and the Hamiltonian method. But since an overde-

termined system of ODEs occurs when calculus of variations is applied, we will

instead apply the Hamiltonian method to solve the problem. Nevertheless, here

we do not compare these two methods to solve the optimisation problem, but we

147

focus on constructing some population models.

In this problem two state variables are used, and hence the current-value

Hamiltonian function defined in (2.61) is applied here and written as

H(N,K,E) =(pqN − c1 −

c22E)E + λ1

[f(N,K)− qEN

]+ λ2g(N,K), (6.3)

where λ1 and λ2 define the lagrange multipiers that depend on time t. As stated

in (2.63), the following conditions need to be fulfilled to solve the problem;

(i) ∂H/∂E = 0.

We differentiate H with respect to E in (6.3) then equate to zero to get

pqN − c1 − c2E − λ1qN = 0.

Hence

E(t) =

[p− λ1(t)

]qN(t)− c1

c2. (6.4)

(ii) dλ1/dt = δλ1(t)− ∂H/∂N .

This equation can be expanded to express the first Lagrange multiplier growth

rate asdλ1dt

=

(–∂f

∂N+ qE + δ

)λ1 −

∂g

∂Nλ2 − pqE.

(iii) dλ2/dt = δλ2(t)− ∂H/∂K.

Likewise, for the second lagrange multiplier we obtain

dλ2dt

=

(δ − ∂g

∂K− 1

)λ2 −

∂f

∂Kλ1.

Eventually, we can re-establish the constraint, provided in (6.2), by adding two

more ODEs for the rates of lagrange multipliers, specified in (ii) and (iii). Thus,

148

the constraints now become

dN

dt=f(N,K)− qEN,

dK

dt=g(N,K),

dλ1dt

=

(–∂f

∂N+ qE + δ

)λ1 −

∂g

∂Nλ2 − pqE,

dλ2dt

=

(δ − ∂g

∂K− 1

)λ2 −

∂f

∂Kλ1,

(6.5)

with the boundary conditions given by

N(0) = N0, K(0) = K0, λ1(T ) = 0, λ2(T ) = 0. (6.6)

Observe that the constraints in (6.5) contain the effort E(t), at the same time E(t)

has been assigned in (6.4). Therefore, the boundary value problem in (6.5) can be

solved numerically by first substituting E(t) given in (6.4) into (6.5). Once this is

solved, we go back to (6.4) and use N(t) and λ1(t) to calculate the optimal effort.

Simulations and Discussion

Since the terminal points λ1(T ) and λ2(T ) are given instead of the initial condi-

tions, the system in (6.5) is classified as a multipoint boundary value problem.

Several numerical methods can be applied to solve this type of problem such as

Invariant Embedding [1, 12] or the method proposed in [2]. Since the model ex-

pression is complicated, and we failed to try to use several boundary value prob-

lem methods, we then decided to use an alternative way, that is converting the

boundary value problem into an initial value problem by choosing the initial val-

ues λ1(0) and λ2(0) to obtain the terminal values λ1(T ) = 0 and λ2(T ) = 0.

In this simulation, the logistic growth function is applied to f(N,K). With

this function, we can specify Emax = r/q as a common definition used by [27,

48]. This can be inspected from the first ODE with regards to the equilibrium of

the population size. However, Emax can also be specified differently as has been

considered in [25, 89], where Emax depends on a capital investment as well as the

149

maximum capacity of the fishing fleet.

Meanwhile, we choose the function g(N,K) which was given in the previous

chapter. However, since the two functions of g(N,K) used in Example 5.2.3 and

Example 5.2.1 have similar results, especially when the population size is large, in

these simulations we only consider the model in (6.5), where g(N,K) is governed

by the functions given in Example 5.2.2 and Example 5.2.1. For comparison, the

numerical result for the model that Suri has worked on in [111] with a constant

carrying capacity K is also shown here. Thus, the three harvesting models with

logistic growth can be compiled as:

(a) The model in (6.5), with g(N,K) = K(b− cN) introduced in [105] and given

in Example 5.2.2.

(b) The model in (6.5), where g(N,K) = K[b− c(K − e−N)

]proposed in Exam-

ple 5.2.1.

(c) The model in (2.84), with constant K.

Next we use the same parameter values taken from Table 2.1, then we assign the

two additional parameter values for b and c. Hence, all parameter values we

use in these simulations are r = 0.71, δ = 0.12, p = 0.5, q = 0.0001, T = 1, c1 =

c2 = 0.01, K = 106, b = 10, c = 10−6 with the initial values N0 = 0.5 × 106 and

K0 = 106. Meanwhile, the initial conditions for the Lagrange multipliers λ1(0)

and λ2(0) are chosen such that their terminal values pointing close zero (up to six

decimal places). Here we use the pairs (λ1(0), λ2(0)) for the models (a), (b) and

(c) as (0.1935, 0.0145), (0.0475, 0.2913) and (0.1935, 0.33), respectively.

Applying these values we obtain the population dynamics as can be seen in

Figure 6.1. We can notice from the graphs that the population sizes for models

(a) and (b) grow faster and higher than model (c). This behaviour occurs due to

the differences in the type of carrying capacity model used. Since models (a) and

(b) utilise a carrying capacity that changes over time and increases as shown in

150

Figure 6.2, the population size increases rapidly following the increasing carrying

capacity. On the other hand, the population size for model (c) shows a slow

0 0.2 0.4 0.6 0.8 1

5e05

5.5e05

6e05

6.5e05

7e05

Model a

Model b

Model c

Figure 6.1: The numerical solutions for population size implementing theHamiltonian method

growth since the carrying capacity is of a constant-type. Hence the population

can only grow between the values 0.5× 106 to 106. Now, comparing the carrying

capacity dynamics between models (a) and (b), illustrated in Figure 6.2, model

(a) does not seem to grow to a bounded value, unlike model (b) that shows the

”S-shape” growth to a limiting value of approximately 107.

Due to the abundant fish population, the effort required to carry out fish

harvesting is even higher. This correlation between the population and effort can

be observed from Figure 6.1 and Figure 6.3. From the two figures, we can notice

that the efforts for models (a) and (b) tend to increase in line with the increasing

population. Meanwhile, the effort for model (c) does not experience a significant

escalation as the population increases slowly.

Now inspecting the correlation between the effort and the present value of

net profit specified in (6.1), Table 6.1 shows the optimal effort range, as well as

151

0 0.2 0.4 0.6 0.8 1

1e06

2e09

4e09

6e09

8e09

1e10

1.2e10

(a)

0 0.2 0.4 0.6 0.8 1

1e06

3e06

5e06

7e06

9e06

(b)

Figure 6.2: The carrying capacity solutions for models (a) and (b).

0 0.2 0.4 0.6 0.8 1

2000

2500

3000

3500

Model a

Model b

Model c

Figure 6.3: The numerical solutions for population size implementing theHamiltonian method.

the average effort value and the profit earned throughout the interval [0, 1]. As

can be seen in the table, the more average effort is made to harvest the fish pop-

ulation, the more profit is gained, where model (a) generates the highest average

effort compare to the other two models, hence earning the highest profit. Con-

152

Model Interval E(t) Average E(t) Max ProfitModel(a) [2350, 3672] 2868.86 44085Model(b) [1785, 3633] 2596.65 43208Model(c) [2011, 2754] 2364.49 32869

Table 6.1: The results of the optimum effort and maximum profit for three dif-ferent models

versely, when model (c) is used, the smallest profit will be earned in proportion to

the lowest harvestng effort made. With these simulations, we do not determine

which model is the best one, but it depends on the fishery industries to make a

decision. We can only suggest using model (a) if the industry wants to get the

best profit. But if the industry wants to get a good profit and at the same time

wants to avoid overpopulation, then model (b) is the best choice. If neither of the

two is the purpose of what the industry wants, but prefer to gain a fairly good

profit with little effort, then the best choice is model (c). We point out that this

model uses a constant carrying capacity, which may not be realistic.

6.2 Model with two harvesting efforts

Now, suppose that fish and some other animal ocean life occupy a marine ecosys-

tem, where seaweed assemblages are the most essential nursery places to live in

as they play a vital role in capturing carbon dioxide and releasing high amounts

of oxygen as well as producing food and energy for the surrounding living things.

They also absorb the excessive nutrients that enter the ocean, some of which are

harmful to the marine ecosystems. Kelp forests which can be found in the shal-

low waters of the Pacific Ocean are one of the brown algae-types seaweed assem-

blages that form a habitat and shelter, as well as a food source for various kinds

of marine life including crab, opaleye fish and halfmoon fish. In other words,

the function of seaweed assemblages is so important that it provides a habitat for

marine biota and contributes to maintaining the ecosystem balance.

Seaweed can be used in pharmaceutical and cosmetic industries, and is ex-

153

tensively consumed as food in coastal cuisines around the world. But nowadays

they are also used as an ingredient for making pastas, breads and beef patties [75,

97, 109]. The presence of seaweed in everyday human life has improved the econ-

omy of a country and the world. From 2007 to 2008, Malaysia had produced

more than 118,298 tons of seaweed which gave the increasing market value at

97.3% [67]. Meanwhile, the total annual global seaweed harvesting production

in 2014 had reached 28.5 million tonnes [38]. Indeed, seaweed has experienced

popularity and escalated the global demand for industrial and economic reasons.

However, over-harvesting to satisfy commercial demand may or even has led to

deterioration of seaweed beds in some regions [19]. Therefore, harvesting strate-

gies need attention and management such that disruption to marine ecology can

be prevented.

For this reason, we introduce a model of optimal harvesting problem of ma-

rine life populations between seaweed, i.e. kelp forest, and fish species that con-

sume kelp (i.e opaleye fish, halfmoon fish, etc) such that this seaweed assem-

blages are mainly considered as the habitats for these fish species for feeding,

growing, breeding and as a shelter for their living. Then, both fish and seaweed

are harvested for economic purposes to gain maximum profit by controlling har-

vesting efforts. In other words, the objective of the harvesting problem is to de-

termine the optimal harvesting effort of fish and seaweed populations such that

the desired maximum net profit is earned.

First we need to specify each net revenue for fish and seaweed. For fish, we

use the same concept of gaining net profit given in (2.66), only the notations are

changed for p1, q1 and E1 instead of p, q, and E. For net profit of seaweed, we

define it similarly and specified by

Π(I(t), E2(t)

)= p2q2E2(t)I(t)− d1E2(t)−

d22E2(t)

2. (6.7)

The constant p2 denotes the seaweed unit price and q2 is the catchability, while

154

E2(t) and I(t) represent the harvesting effort and seaweed biomass, respectively.

Here, we still use word ”catch” for seaweed, although it certainly has a different

perception with the fish catch. The way ”catch” of seaweed can be implemented

is using a range of techniques and cuttings. The seaweed harvesting can be made

using boats, rakes or by diving, as well as by hand with scissors or knife at shore

during low tide [98]. For the cost of harvesting, it comprises of two constants d1

and d2 calculated by d1E2(t) + d2/2E2(t)2.

Thus, incorporating the net revenue for the fish harvest given in (2.66) and

for the seaweed given in (6.7), the objective function that maximises the present

value of the discounted profit can be expressed as

Maximise J =

∫ T

0

e−ρt

[(p1q1N−c1−

c22E1

)E1+

(p2q2I−d1−

d22E2

)E2

]dt. (6.8)

Since we assume the fish are strongly dependent on seaweed to grow and re-

produce, we assume that this seaweed represents the carrying capacity of the

fish. This way of thinking is the same as the concept given in Chapter 3, where

the carrying capacity of fish K(t) is a function of the food availability I(t), i.e.

the seaweed, or written as K(t) = f(I(t)). Once the carrying capacity and its

growth rate are specified, then the seaweed growth rate can be established and

then followed by harvesting. Thus, the population dynamics comprising fish and

seaweed regulated by general functions can be written as

dN

dt= f(N, I)− q1E1N,

dI

dt= g(N, I)− q2E2I.

(6.9)

where

N(0) = N0 and I(0) = I0.

The function f(N, I) represents the logistic growth, Smith function or any rea-

sonable population growth function. Meanwhile for g(N, I), we do not specify it

155

directly, but this will be explained further before the simulation section.

Here we do not implement the Hamiltonian method like in the previous sec-

tion since there will result six differential equations before we obtain maximum

profit: two populations, two efforts and two Lagrange multipliers. Thus we ap-

ply the calculus of variations instead, outined Section 2.4.

Now let us consider the variation of two state variables and two efforts

as

N = N∗ + x, I = I∗ + y, E1 = E∗1 + s1, E2 = E∗

2 + s2,

where x, y, s1 and s2 are small. Substituting these variations to (6.8), then taking

the difference of J we obtain

∆J =

∫ T

0

e−ρt[p1q1(N

∗ + x)− c1 −c22(E∗

1 + s1)](E∗

1 + s1)

+[p2q2(I

∗ + y)− d1 −d22(E∗

2 + s2)](E∗

2 + s2)dt

−∫ T

0

e−ρt

[(p1q1N

∗ − c1 −c22E∗

1

)E∗

1 +(p2q2I

∗ − d1 −d22E∗

2

)E∗

2

]dt,

or can be written approximately as

∆J =

∫ T

0

e−ρt[p1q1(E

∗1x+N

∗s1)−c1s1−c2E∗s1+p2q2(I∗s2+E

∗2y)−d1s2−d2E∗

2s2

]dt,

since x, y, s1 and s2 are small. After that, we can separate this difference equation

into four integral terms as

∆J =

∫ T

0

e−ρt(p1q1E∗1)x dt+

∫ T

0

e−ρt(p2q2E∗2)y dt

+

∫ T

0

e−ρt(p1q1N∗ − c1 − c2E

∗1)s1 dt+

∫ T

0

e−ρt(p2q2I∗ − d1 − d2E

∗2)s2 dt.

(6.10)

Next, we need to reduce and eventually remove all small variations by first sub-

stituting s1 and s2 in terms of x or y, which can be obtained from the model (6.9).

156

Starting from the first ODE, we may write

dx

dt+

dN∗

dt= f(N∗, I∗) +

∂f

∂N(N∗, I∗)x+

∂f

∂I(N∗, I∗)y − q1E

∗1N

∗ − q1(E∗1x+ s1N

∗),

then it is reduced to

dx

dt=

∂f

∂N(N∗, I∗)x+

∂f

∂I(N∗, I∗)k − q1(E

∗1x+ s1N

∗).

For notational convenience, we write

∂f

∂N∗ =∂f

∂N(N∗, I∗) and

∂f

∂I∗=∂f

∂I(N∗, I∗).

This also applies to function g(N∗, I∗).

Next, from the latter equation we obtain

s1 =1

q1N∗

(∂f

∂N∗x+∂f

∂I∗y − q1E

∗1x−

dx

dt

).

The same calculation applied to the second ODE in (6.9) yields

s2 =1

q2I∗

(∂g

∂N∗x+∂g

∂I∗y − q2E

∗2y −

dy

dt

).

Next, we substitute the equations for s1 and s2 to (6.10) to rewrite

∆J =

∫ T

0

e−ρt

[p1q1E

∗1 +

(p1 −

c1q1N∗ − c2E

∗1

q1N∗

)( ∂f

∂N∗ − q1E∗1

)+(

p2 −d1q2I∗

− c2E∗2

q2I∗

) ∂g

∂N∗

]x dt+

∫ T

0

e−ρt

[p2q2E

∗2 +

(p1 −

c1q1N∗ − c2E

∗1

q1N∗

) ∂f∂I∗

+(p2 −

d1q2I∗

− d2E∗2

q2I∗

)( ∂g∂I∗

− q2E∗2

)]y dt

−∫ T

0

e−ρt(p1 −

c1q1N∗ − c2E

∗1

q1N∗

) dx

dtdt−

∫ T

0

e−ρt(p2 −

d1q2I∗

− d2E∗2

q2I∗

) dy

dtdt.

157

Integrating the last two integrals by parts, the equation becomes

∆J =

∫ T

0

e−ρt

[p1q1E

∗1 +

(p1 −

c1q1N∗ − c2E

∗1

q1N∗

)( ∂f

∂N∗ − q1E∗1 − ρ

)+(p2 −

d1q2I∗

− d2E∗2

q2I∗

) ∂g

∂N∗ − c1q1

(1

N∗

)′

− c2q1

(E∗

1

N∗

)′]x dt

+

∫ T

0

e−ρt

[p2q2E

∗2 +

(p1 −

c1q1N∗ − c2E

∗1

q1N∗

) ∂f∂I∗

+(p2 −

d1q2I∗

− d2E∗2

q2I∗

)( ∂g∂I∗

− q2E∗2 − ρ

)− d1q2

(1

I∗

)′

− d2q

(E∗

2

I∗

)′]y dt

− e−ρt(p1 −

c1q1N∗ − c2E

∗1

q1N∗

)x∣∣∣T0− e−ρt

(p2 −

d1q2I∗

− d2E∗2

q2I∗

)y∣∣∣T0,

(6.11)

where (·)’ represents the first derivative with respect to t.

To have ∆J equals to zero each integral should also be zero since x and y

are arbitrary. Now for the first and second integral, we apply the Fundamen-

tal Lemma of Variational Calculus to acquire the growth rate E1 and E2 as fol-

lows.

dE∗1

dt=p1q

21N

∗E∗1

c2+(p1q1N∗

c2− c1c2

− E∗1

)( ∂f

∂N∗ − q1E∗1 − ρ

)+(p2q1N∗

c2− d1q1N

∗

c2q2I∗− d2q1E

∗2N

∗

c2q2I∗

) ∂g

∂N∗ +(c1c2

+ E∗1

)(f − q1E1N∗)

N∗ ,

dE∗2

dt=p2q

22I

∗E∗1

d2+(p1q2I∗

d2− c1q2I

∗

d2q1N∗ − c2q2E∗1I

∗

d2q1N∗

) ∂f∂I∗

+(p2q2I∗

d2− d1d2

− E∗2

)( ∂g∂I∗

− q2E∗2 − ρ

)+(d1d2

+ E∗2

)(g − q2E2I∗)

I∗.

(6.12)

For the third of four terms of (6.11), we have N∗(0) = N(0) = N0 since N(0) is

fixed. This implies that x(0) = 0. Then equating this term to zero we get

[p1 −

c1q1N∗(T )

− c2E∗1(T )

q1N∗(T )

]x(T ) = 0.

As N∗(T ) is free, hence x(T ) = 0 which implies the boundary condition for the

158

fish harvesting effort is expressed as

E∗1(T ) =

p1q1N∗(T )− c1c2

. (6.13)

This also occurs for the last term of (6.11), so that we obtain the boundary condi-

tion for seaweed as

E∗2(T ) =

p2q2I∗(T )− d1d2

. (6.14)

In conclusion, we now have the growth rate model with four ODEs. Two ODEs

are the efforts given in (6.15) and the other two ODEs are fish and seaweed pop-

ulations that have been specified earlier in (6.9). Thus we can rewrite the growth

rate model as

dE∗1

dt=p1q

21N

∗E∗1

c2+(p1q1N∗

c2− c1c2

− E∗1

)( ∂f

∂N∗ − q1E∗1 − ρ

)+(p2q1N∗

c2− d1q1N

∗

c2q2I∗− d2q1E

∗2N

∗

c2q2I∗

) ∂g

∂N∗ +(c1c2

+ E∗1

)(f − q1E1N∗)

N∗ ,

dE∗2

dt=p2q

22I

∗E∗1

d2+(p1q2I∗

d2− c1q2I

∗

d2q1N∗ − c2q2E∗1I

∗

d2q1N∗

) ∂f∂I∗

+(p2q2I∗

d2− d1d2

− E∗2

)( ∂g∂I∗

− q2E∗2 − ρ

)+(d1d2

+ E∗2

)(g − q2E2I∗)

I∗,

dN∗

dt=f(N∗, I∗)− q1E

∗1N

∗,

dI∗

dt=g(N∗, I∗)− q2E

∗2I

∗,

(6.15)

with N∗(0) = N0 and I∗(0) = I0, while E∗1(T ) and E∗

2(T ) are defined as in (6.13)

and (6.14), respectively.

Before we show the results of the simulations, we assume the fish carrying

capacity has a linear proportional relation with the seaweed. Hence, we may

write K(t) = αI(t). Then we choose the fish population growth rate f(N∗, I∗) as

159

the logistic function written as

f(N∗, I∗) = rN∗(1− N∗

αI∗

).

Meanwhile the seaweed growth rate g(N, I) is specified based on the two carry-

ing capacity-population related functions given previously in Chapter 5. There-

fore, the seaweed dynamics can be governed by the following model.

(a) The model which is derived from [105] and used in Example 5.2.2, that is

dK

dt= K(b− cN).

Substituting K(t) = αI(t) to this equation and then incorporating the har-

vesting term we obtain

dI

dt= I(b− cN)− q2E2I. (6.16)

Comparing this equation to (6.9), we have

g(N, I) = I(b− cN).

(b) The model which was proposed in Example 5.2.1, expressed as

dK

dt= K

[b− c

(K − e−N

)].

Implementing the linear proportionality between K(t) and I(t), and then

inserting the harvesting term, the latter equation becomes

dI

dt= I

[b− c

(αI − e−N

)]− q2E2I. (6.17)

160

As a result we have

g(N, I) = I[b− c

(αI − e−N

)].

The parameter b in (6.16) and (6.17) denotes the development rate of seaweed

assemblages, while c represents the seaweed depletion rate due to the fish con-

sumption.

Simulations and Discussion

Next, some simulations are demonstrated to illustrate the model dynamics in (6.15).

Here we use the same parameter values given earlier in Table 2.1. But since two

harvesting efforts are made, we have two parameters for the catchability, two

harvest prices per unit and two constant harvesting costs for the two popula-

tions which we assign the same values. For more details, all of those parame-

ter values are given in Table 6.2. Meanwhile the initial fish population is given

at N(0) = 5 × 105 tonnes (which is half smaller than the seaweed, which is at

I(0) = 106 tonnes) and the terminal time is set at T = 1.

Parameter Description Value Unit

ρ Discount rate 0.1 year−1

r Intrinsic fish growth rate 0.71 year−1

p1 and p2 Unit harvest price 0.5 $ tonnes−1

q1 and q2 Catchability coefficient 0.0001 SFU−1 year−1

c1 and d1 Unit effort cost coefficient 1 0.01 $ SFU−1 year−1

c2 and d2 Unit effort cost coefficient 2 0.01 $ SFU−2 year−1

α fish carrying capacity and seaweed ratio 0.2 –

b Seaweed development rate 0.15 year−1

c Seaweed depletion rate 10−6 tonnes−1 year−1

Table 6.2: Parameter values used to obtain the numerical solution of the modelin (6.15).

Implementing these values, Figure 6.4 depicts the population dynamics for

161

models (a) and (b). Both population graphs for model (a) show us the declining

trend, which has the same pattern to that of model (b). Likewise, the optimal

0.0 0.2 0.4 0.6 0.8 1.0

2e05

4e05

6e05

8e05

1e06

(a)

0 0.2 0.4 0.6 0.8 1

2e05

4e05

6e05

8e05

1e06

(b)

Figure 6.4: Comparing the population dynamics for models (a) and (b).

efforts exhibit the downward trend as pictured in Figure 6.5. Comparing the

graphs in Figure 6.4 and Figure 6.5, we can infer that the less effort is claimed to

be the optimal effort to harvest the small population.

0.0 0.2 0.4 0.6 0.8 1.0

1000

2000

3000

4000

(a)

0 0.2 0.4 0.6 0.8 1

1000

2000

3000

4000

(b)

Figure 6.5: Comparing the optimal efforts for models (a) and (b)

Now when comparing the solutions for models (a) and (b), they seem to

162

have very similar results if we look at them graphically. We therefore look further

through data shown in Table 6.3.

Model Average E∗1(t) Average E∗

2(t) Max J

Model (a) 1522 SFU 3530 SFU $21,683

Model (b) 1542 SFU 3809 SFU $23,551

Table 6.3: The results of the optimum effort and maximum profit for two differ-ent models

The table provides the average data of optimal harvesting efforts with the

corresponding maximum profits. From this data, it can be concluded that model (b)

suggests using the average harvesting effort for fish at 1542 SFU and 3809 SFU for

seaweed such that the industries earns $23,551 of profit. Meanwhile model (a)

suggests making less effort than model (b) at the respective average efforts 1522

SFU and 3530 SFU, but the maximum profit gainings is only $21,683.

6.3 Conclusion

In this chapter we solved the optimisation problem by implementing the carry-

ing capacity with one and two effort control variables. Two different methods

were applied to find the solution, namely the Hamiltonian method for solving

the problem with one effort, and calculus of variations for the problem with two

efforts. Here, we do not make a comparison between these two methods, but in-

stead we investigate the profit earnings with the corresponding effort where the

carrying capacity constraint takes the general form. Then, the carrying capacity

models given in Chapter 5 are used to replace this general function so that the

simulations can be performed. With these simulations we produced results re-

garding maximum profit with the corresponding effort for different types of car-

rying capacity models such that a fisheries industry can make a decision about

which carrying capacity model is right to use.

163

The first optimisation problem in Section 6.1 investigated the maximum profit

controlled by a single effort. Two carrying capacity models are applied and one

model used constant carrying capacity. These three models agree that profit is

proportional to the effort dynamics, the more efforts are deployed the more prof-

its earned. However, the models with variable carrying capacity suggest to de-

ploy more efforts than the model with constant carrying capacity to obtain higher

profits.

This problem was then modified and developed in Section 6.2 by adding

one more effort control variable, so that these efforts are now not only had a

part that affects the fish population dynamics but also had a part that affects car-

rying capacity. In this case, the carrying capacity of fish is considered as food

availability (seaweed assemblages) with a linear proportional correlation, where

a harvesting was also carried out. Two carrying capacity models were used to

obtain maximum profit with the corresponding optimal efforts. Based on the

simulation results, these models also agree that the profit is proportional to the

effort. Furthermore, we can compare the fish population dynamics between the

single harvesting effort problem in Figure 6.1 and the two harvesting efforts prob-

lem in Figure 6.4. For the single effort problem, the population with variable

carrying capacity tends to grow quickly, especially for model (a) that is feared

to face overpopulation. Therefore harvesting against the fish carrying capacity,

which is seaweed, also needs to be made such that an unexpected growth can be

avoided.

164

Chapter 7

Summary and Future Directions

7.1 Summary

In Chapter 1, we presented some models that are commonly used to describe the

population dynamics, where they can be formed as a general population model

that contains a function of population growth rate. Kolmogorov models were also

provided here as a basis for finding analytical solutions, as well as for investigat-

ing the qualitative behaviour of the population models with variable carrying

capacity.

Chapter 2 provided basic theories that support our work in solving problems

related to population models with variable carrying capacity such as integration-

based parameter estimation technique, calculus of variations and the Hamilto-

nian method. Models related to harvesting, human carrying capacity and opti-

misation problems were also described in this chapter.

In Chapter 3, we started with a model related to human carrying capacity, in

which population growth is governed by the logistic function. We modified the

model by introducing three types of carrying capacity functions represented by

the food availability I as

(a) f ′(I) > 0 for I > 0 and f(∞) = ∞.

165

(b) f ′(I) > 0 for I > 0 and 0 < f(∞) <∞.

(c) There exists I∗ > 0 such that f ′(I) > 0 for 0 < I < I∗ and f ′(I) < 0 for I >

I∗, i.e. f has a unique global maximum at I∗. Furthermore, 0 < f(∞) <∞.

For each carrying capacity model, parameter estimation was carried out to de-

termine population dynamics then compared with actual data collected from the

Food and Agricultural Organisation. Models (b) and (c) are considered to be the

feasible models since not only are their approximations close to the actual data,

but also the population reaches a limiting value when forecasting was imple-

mented.

The population models were then structured to form the Kolmogorov model,

and is presented in Chapter 4. This model is governed by a coupled system of

two ODEs which specifically comprises two general functions of population and

carrying capacity per capita growth rate. In this form, a procedure to obtain an-

alytical solutions was proposed. For the case when the per capita growth rates

are proportional, in addition to obtaining analytical solutions, the inflexion point

was also found in terms of a criterion of a fraction of the asymptotic carrying

capacity.

In Chapter 5, the general population model with variable carrying capacity

was enhanced by adding a harvesting term to the population ODE. Some as-

sumptions were assigned, either for the population or the carrying capacity per

capita growth rates, such as first derivative conditions and concavity. Here, we

focused on examining the qualitative behaviour of the models rather than ana-

lytical solutions, in particular, steady state analysis. The model was then broken

down into two special cases based on the carrying capacity per capita growth

rate: depending on the population only or the carrying capacity only. For each

special type as well as the original model, some assumptions for the general func-

tions were given differently, especially the function for the population per capita

growth rate, but they still meet the characteristics of the common population

166

models, for instance, logistic, Gompertz, Smith, and Gilpin & Ayala. Then the

number of steady states and the stability were determined with regards to the

harvesting value. For the three models of carrying capacity, they agreed that the

harvesting rate needs to be smaller than its critical value in order to have a stable

steady state with a certain condition. All these results for steady state analysis

were stated as theorems, in which we presented some graphical illustrations to

aid in understanding the proof of these theorems.

The harvesting population model with a constant harvesting rate was then

converted into a variable harvesting rate. So now the harvesting rate depends

on catchability and the effort variable. However, this model still consists of two

ODEs, but uses two general functions for population and carrying capacity grow-

th rate instead of its per capita. This model was described in Chapter 6. The pur-

pose of this chapter is to obtain the maximum profit in the fisheries industry by

controlling the harvesting effort. There were two models given in this chapter:

models with one and two controllable efforts variables. In the first model, we

used the Hamiltonian method to gain a solution of the optimised effort variable.

With this technique, two additional ODEs were obtained, namely the Lagrangian

multipliers. By solving these four ODEs numerically, we calculated the optimum

effort that maximises the profit. After that, we used a logistic function and three

different functions for the carrying capacity growth rate to perform the simula-

tions, those are

(a) The model with g(N,K) = K(b− cN) introduced in [105].

(b) The model with g(N,K) = K[b− c(K − e−N)

]as our own model

(c) The model with constant carrying capacity.

Implementing these carrying capacity models, we found that model (a) gives the

highest value of the maximum profit gain, whereas model (c) is the lowest one.

However, this value is proportional to the harvesting effort; the higher the profit

is gained, the higher the effort is made. The second model was proposed as a

167

modification of the first model so that one more controllable effort is added to the

carrying capacity ODE. Firstly, the carrying capacity was converted to a function

of the fish food source (i.e. seaweed). Then the seaweed assemblages were also

harvested in the same way (mathematically the same form model) as the fish pop-

ulation. With this model the variational calculus method was applied such that

we could establish a system of four ODEs with two of which are the ODEs for the

two optimised efforts. This optimisation problem can be solved by solving these

system of ODEs numerically. Since the carrying capacity represented by seaweed

assemblages cannot be constant (seaweed harvesting is performed), we only used

two carrying capacity models, namely models (a) and (b), while omitting model

(c). Unlike the first optimisation model, here model (b) was considered the best

in gaining the maximum profit.

7.2 Future directions

This thesis presented several population dynamical problems, as well as their

solutions where the carrying capacity is assigned as a variable instead of a con-

stant. The research described in the main chapters (Chapter 3, 4, 5 and 6) appears

to have promising avenues for undertaking other useful work in the future. The

following are some directions for researchers who are interested in the field of

population dynamcs with variable carrying capacity to work more on some prob-

lems which might be more complicated than that given in the thesis.

Periodic solutions

It rarely happens that a population experiences smooth growth, then reaches its

carrying capacity and stays on the limiting value. Instead, due to some external

factors such as the availability of food or the presence of predators, the popula-

tion may exhibit a periodic or regular cycle of increasing and decreasing growth

over time. In a model that is formed by differential equations, inspecting for

periodic solutions is very important, although finding out whether a differen-

168

tial equation has such solutions or not is a difficult question. In planar systems,

the Bendixson-Dulac criterion can be implemented in order to obtain a sufficient

condition such that the system has no periodic orbits. In this criterion, an aux-

iliary function (called the Dulac function) with specific properties is required.

Recall the Bendixson-Dulac criterion for planar system given in the following

theorem.

Theorem 7.2.1 (Bendixson-Dulac criterion). Let f(x, y), g(x, y) and h(x, y) be func-

tions C1 in a simply connected domain Ω ⊂ R2 such that ∂(fh)/∂x + ∂(gh)/∂y does

not change sign in Ω and vanishes at most on a set of measure zero. Then the system

dx

dt= f(x, y),

dy

dt= g(x, y), (x, y) ∈ Ω

does not have periodic orbits in Ω

The function h in this theorem denotes the Dulac function. However, spec-

ifying this function is not an easy task. With the general harvesting population

model given by (5.7) in Chapter 5, one can modify the assumptions of the func-

tions f(N,K) and g(N,K), also specifying the Dulac function which can be a

challenging task such that any given condition for a constant harvesting rate H

never causes periodic cycles for the population size.

Meanwhile, Hopf bifurcation theory can also be applied to investigate pe-

riodic solutions of a system. Hopf bifurcation is a phenomenon in a nonlinear

system in which the phase trajectory converges to a node under certain parame-

ter conditions, and then the trajectory is switched to a periodic motion for a small

changes in the parameter. In other words, no new steady states arise, but periodic

solutions emerge as the parameter passes the bifurcation value. Thus the assump-

tions for the general functions f(N,K) and g(N,K) in the harvesting population

model can also be altered and the bifurcation value of constant harvesting rate H

can be determined such that appearance and dissapearance of periodic solutions

occur.

169

Constant effort of general harvesting model

Another problem to be investigated is regulating the harvesting rate in the gen-

eral harvesting model by effort. Therefore the general harvesting model with

variable carrying capacity can be constructed as

dN

dt= Nf(N,K)− EN,

dK

dt= Kg(N,K),

(7.1)

As usual, the maximum harvesting yield Y (E) = EN is inspected here, but the

result will refer to either the function f(N,K) or g(N,K).

We begin with the steady state solution (N∞, K∞), which is obtained by solv-

ing two equations from (7.1), those are

f(N∞, K∞) = E and g(N∞, K∞) = 0. (7.2)

First, we may assume the existance of a unique solution K∞ = q(N∞) from

g(N∞, K∞) = 0 that implies f(N∞, q(N∞)) = E. Then we may use the Inverse

Function Theorem to find the inverse function of f in a neighbourhood of the

population steady state N∞. Thus we can express the population size N that

includes N∞ as an inverse function that depends on the effort E, denoted by

N∞(E) = f−1(E). This function influences the carrying capacity steady state K∞

which now depends on effort, written as K∞ = q(N∞(E)).

Next, the steady state behaviour needs to be inspected to gain a stable be-

haviour which is important in analysing the maximum yield, specified as

Y (E) = EN∞(E).

The dynamics of the yield Y is examined in such a way that a concave curve is

expected to occur. As Theorem A.0.4 in Appendix A says the concavity can be

examined from the second derivative, we can express the second derivative of

170

the yield as

Y ′′(E) = 2N ′∞(E) + EN ′′

∞(E). (7.3)

This implies that N ′∞(E) and N ′′

∞(E) need to be calculated and inspected prior

to obtaining the concave curve of Y . Performing the first derivative of the first

equation in (7.2), we can write

D1f(N∞, K∞)N ′∞(E) +D2f(N∞, K∞)q′(N∞)N ′

∞(E) = 1,

to obtain

N ′∞(E) =

1

D1f(N∞, K∞)N ′(E) +D2f(N∞, K∞)q′(N)N ′(E).

Then from here we can calculate the second derivative as

N ′′∞(E) =−

[N ′

∞(E)]3[

D21f(N∞, q(N∞)) + 2D12f(N∞, q(N∞))q′(N∞)

+D22f(N∞, q(N∞))

[q′(N∞)

]2+D2f(N∞, q(N∞))q′′(N∞)

].

Substituting N ′∞(E) and N ′′

∞(E) into (7.3), we obtain a condition which produces

a concave curve of Y as

(∂2f

∂N2+ 2

∂2f

∂N∂K

dg

dN+∂2f

∂K2

( dg

dN

)2

+∂f

∂K

d2g

dN2

)< 2

(∂f

∂N+∂f

∂K

dg

dN

).

Thus, we need some assumptions for the functions f and g which are not only

the first derivative but also the second one in order to satisfy this concavity re-

quirement. For instance, the asssumptions for function f could be set as

D1f(N,K) < 0, D1f(N,K) > 0,

D11f(N,K) ≥ 0, D12f(N,K) ≥ 0, D22f(N,K) < 0,

which meets the logistic function characteristic, but we may say that these as-

171

sumptions are too strong. Therefore the proper assumptions for f as well as g

need to be defined efficiently to determine the optimal effort that maximises the

yield.

Stochastic models

In some situations, a population may face an undetermined condition. This means

a stochastic differential equation can be utilised here. We may refer to the stochas-

tic population model proposed by Kloeden & Platen [69] and Mendez et al. [80],

given by

dN(t) = rN(t)

[1− N(t)

K

]+ rσN(t)2dW (t).

SinceK in this model is a constant, we can change it to variable, and use the three

types of carrying capacity models as proposed in Chapter 3. Then we can start

analysing and comparing their qualitative behaviours.

However, a stochastic environment of population with variable carrying ca-

pacity is in fact not new study. One such model has been established by Anderson

et al. [5], with the proposed model

dN(t)

dt= rN(t)

[1− N(t)

K(t)

],

dK(t) = −γ[K(t)−K1] dt+ σ dW (t),

where W denotes a Wiener process. As we can notice, the rate of change of car-

rying capacity is independent of N(t). Thus by adding its dependence also on

population size, we can construct an alternative model to see the population and

carrying capacity dynamics in an non-deterministic environment. For instance,

the function proposed in Chapter 5, specifically in Example 5.2.1, can be used to

govern the carrying capacity rate of change.

172

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183

Appendix A

Convex Theory

Some definition, theorem and other theories with regards to convexity used in

the thesis is based on the book written by Boyd & Vandenberghe [15] and Hiriart-

Urrut & Lemarechal [51].

Definition A.0.1 (Convex set). A set Ω is convex if the line segment between any

two points in Ω lies in Ω, i.e., if for any x, y ∈ Ω and any θ with 0 ≤ θ ≤ 1, we have

θx+ (1− θ)y ∈ Ω.

Definition A.0.2 (Convex function). A function F : Rn → R is convex if its do-

main Ω is a convex set and if for all x, y ∈ Ω , and θ with 0 ≤ θ ≤ 1, we have

F (θx+ (1− θ)y) ≤ θF (x) + (1− θ)F (y). (A.1)

A function F is said to be strictly convex when (A.1) holds the strict inequality

for x = y and 0 < θ < 1. A function F is said to be concave if −F is convex, and

it is strictly concave when −F is strictly convex. Thus the definition of concavity

can be given as follows.

Definition A.0.3 (Strictly concave). A function F is strictly concave if its domain

184

Ω is a convex set and if for all x, y ∈ Ω and x = y, and θ with 0 < θ < 1, we have

F (θx+ (1− θ)y) > θF (x) + (1− θ)F (y). (A.2)

Notice that when x = y, we have F (θx + (1 − θ)y) = θF (x) + (1 − θ)F (y).

This means a function which is strictly concave (convex) is also concave (con-

vex).

Now suppose F (x) = xf(x) is strictly concave then L(x,H) = F (x)−H with

constant H is also strictly concave. See below as the proof.

L(θx+ (1− θ)y

)− θL(x)− (1− θ)L(y)

= F(θx+ (1− θ)y

)−H − θ(F (x)−H)− (1− θ)(F (y)−H) [L is expanded]

= F(θx+ (1− θ)y

)− θF (x)− (1− θ)F (y) [H is canceled]

> 0. [using concave definition for F (x)]

Thus

θL(x) + (1− θ)L(y) < L(θx+ (1− θ)y

)which we conclude that L is also strictly concave.

The following is the theorem derived from [51] which relates to the concave func-

tion.

Theorem A.0.4. Let F be twice differentiable on an open convex set Ω ⊂ ℜ2. Then

(i). F is concave on Ω if and only if F ′′(x) is negative semi-definite for all x0 ∈ Ω.

(ii). If F ′′(x) is negative definite for all x ∈ Ω, then F is strictly concave on Ω.

185

Appendix B

Programming Code: Carrying

Capacity as Food Availability

B.1 One parameter estimation

clc();clear;

//World Population 1962-2014. We use the data from xls file.

SheetPop = readxls(’D:\RESEARCH PHD\Population.xls’);

DataPop = SheetPop(1);

cnty_id = 262; //data from particular country, 262 for world.

cnty_name = DataPop(cnty_id,1);

denum = 10ˆ9; //population in billion.

Ndat = DataPop(cnty_id,7:59)/denum;

N_in = Ndat(1);

//Food Production Index 1962-2014 from xls file.

SheetFood = readxls(’D:\RESEARCH PHD\Food Index.xls’)

DataFood = SheetFood(1);

Idat = DataFood(cnty_id,7:59);

in_year = 1962;

I_in = Idat(1);

m = size(Ndat,2);

//n first data points for parameter estimate where n < m

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n = 30;

t = (0:1:n-1);

//Model fitting on n data points for food production index I(t)

ln_I = log(Idat(1:n));

c2 = (n*sum(t.*ln_I)-sum(t)*sum(ln_I))/(n*sum(t.ˆ2)-sum(t)ˆ2)

c1 = exp((sum(t.ˆ2)*sum(ln_I)-sum(t.*ln_I)*sum(t))/(n*sum(t.ˆ2)-...

sum(t)ˆ2))

I_LS = c1*exp(c2*t);

N(1:n) = Ndat(1:n);

I(1:n) = Idat(1:n);

//choose which carrying capacity model to estimate

function g = funcI(a,I)

if a == 1 then //model(a)

g = I;

end

if a == 2 then //model(b)

g = I./(1+I);

end

if a == 3 then //model(c)

g = I.*(1+I)./(1+I.ˆ2);

end

endfunction

//weight function (here we use exponential form)

function f = W(b,t)

f = exp(-b*t);

endfunction

// derivative of weight function

function df = dW(b,t)

df = -b*exp(-b*t)

endfunction

s0 = [0; 0.01; 0.1] //three values of s0 are chosen

num_s = size(s0,1)

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r = 0.03; //r is known;

printf(’\nCarrying capacity models with K(I)=a*g(I)\n’)

printf("1. Model (a): g(I)=I\n");

printf("2. Model (b): g(I)=I/(1+I)\n");

printf("3. Model (c): g(I)=I(1+I)/(1+Iˆ2)\n");

opt=input("choose the function g(t) : ")

printf("Number data used for approximation = %d\n",n)

printf(’Estimating alpha only with fixed r = %.2f \n\n’, r)

//parameter estimate of r using integration-based method

for j = 1:num_s

a_p = intsplin(t,W(s0(j),t).*N);

b_p = -intsplin(t,W(s0(j),t).*N.ˆ2./funcI(opt,I));

c_p = W(s0(j),t($))*N($)-W(s0(j),t(1))*N(1)-...

intsplin(t,N.*dW(s0(j),t));

alp = r*b_p/(c_p-r*a_p);

printf(’s0=%.2f,\t alpha = %f,\t’,s0(j),alp);

dt = 0.01;

tend = m;

num_each = 1/dt;

numpoint = (tend-1)*num_each+1;

t2 = (0:1:tend-1);

tt = linspace(t2(1),t2($),numpoint);

It = c1*exp(c2*tt); //food index of m data points

Nt = zeros(1,numpoint);

Nt(1) = N(1);

for i = 1:numpoint-1 //population data estimation

Kt = alp*funcI(opt,It(i));

Nt(i+1) = Nt(i)+r*Nt(i)*(1-Nt(i)/Kt)*dt;

end

//Root Mean Square Error

RMSE = sum((Nt(n*num_each+1:num_each:(m-1)*num_each+1)-...

Ndat(n+1:$)).ˆ2)/(m-n);

printf("RMS Error = %f\n",RMSE)

end

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B.2 Two parameter estimation

clc(); clear;

//World population 1962-2014. We use the data from xls file.

SheetPop = readxls(’D:\RESEARCH PHD\Population.xls’);

DataPop = SheetPop(1);

cnty_id = 262; //data from particular country, 262 for world.

cnty_name = DataPop(cnty_id,1);

denum = 10ˆ9;

Ndat = DataPop(cnty_id,7:59)/denum;

N_in = Ndat(1);

//Food production index 1962-2014 from xls file.

SheetFood = readxls(’D:\RESEARCH PHD\Food Index.xls’);

DataFood = SheetFood(1);

Idat=DataFood(cnty_id,7:59);

in_year = 1962;

I_in = Idat(1);

m = size(Ndat,2);

//n first data points for parameter estimate where n < m.

n = 30;

t = (0:1:n-1);

//Model fitting on n data points for food production index I(t).

ln_I = log(Idat(1:n));

c2 = (n*sum(t.*ln_I)-sum(t)*sum(ln_I))/(n*sum(t.ˆ2)-sum(t)ˆ2)

c1 = exp((sum(t.ˆ2)*sum(ln_I)-sum(t.*ln_I)*sum(t))/(n*sum(t.ˆ2)-...

sum(t)ˆ2))

I_LS = c1*exp(c2*t);

N(1:n) = Ndat(1:n);

I(1:n) = Idat(1:n);

//choose which carrying capacity model to estimate.

function g = funcI(a,I)

if a == 1 then //model (a).

p = 4;

g = I.ˆ(1/p);

end

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if a == 2 then //model (b).

g = I./(1+I);

end

if a == 3 then //model (c).

g = I.*(1+I)./(1+I.ˆ2);

end

endfunction

//weight function (here we use exponential form).

function f = W(b,t)

f = exp(-b*t);

endfunction

// derivative of weight function.

function df = dW(b,t)

df = -b*exp(-b*t)

endfunction

s = [0 0.01; 0 0.1; 0.05 0.1];//three pairs are chosen.

num_s = size(s,1)

printf(’\nCarrying capacity models with K(I)=a*g(I)\n’)

printf(’1. Model (a): g(I)=Iˆ(1/4)\n’);

printf(’2. Model (b): g(I)=I/(1+I)\n’);

printf(’3. Model (c): g(I)=I(1+I)/(1+Iˆ2)\n’);

opt = input("choose the function g(I): ")

printf("Number data used for approximation = %d\n",n)

printf(’Estimating r and alpha\n\n’)

//estimating r and alpha using integration-based method

for j = 1:num_s

for i = 1:2

a_p(i) = intsplin(t,W(s(j,i),t).*N);

b_p(i) = -intsplin(t,W(s(j,i),t).*N.ˆ2./funcI(opt,I));

c_p(i) = W(s(j,i),t($))*N($)-W(s(j,i),t(1))*N(1)-...

intsplin(t,N.*dW(s(j,i),t));

end

printf(’Beta(%.2f,%.2f)\n’,s(j,1),s(j,2));

A = [a_p(1) b_p(1);a_p(2) b_p(2)];

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B = [c_p(1);c_p(2)];

par_sol = inv(A)*B;

r = par_sol(1);

alp = r/par_sol(2);

printf("r = %f, \t alpha = %f,\t",r,alp);

dt = 0.1;

tend = m;

num_each = 1/dt;

numpoint = (tend-1)*num_each+1;

t2 = (0:1:tend-1);

tt = linspace(t2(1),t2($),numpoint);

It = c1*exp(c2*tt); //food index of m data points

Nt = zeros(1,numpoint);

Nt(1) = N(1);

//population data estimation

for i = 1:numpoint-1

Kt = alp*funcI(opt,It(i));

Nt(i+1) = Nt(i)+r*Nt(i)*(1-Nt(i)/Kt)*dt;

end

//Root Mean Square Error

RMSE = sum((Nt(n*num_each+1:num_each:(m-1)*num_each+1)-...

Ndat(n+1:$)).ˆ2)/(m-n);

printf("RMSE = %f\n\n",RMSE)

end

191