458 Lumped population dynamics
models
Fish 458; Lecture 2
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Revision: Nomenclature Which are the state variables, forcing
functions and parameters in the following model:
population size at the start of year t,
catch during year t, growth rate, and annual recruitment
1t t tN N R C tN
tC
R
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The Simplest Model-I Assumptions of the exponential model:
No emigration and immigration. The birth and death rates are independent of
each other, time, age and space. The environment is deterministic.
is the initial population size, and is the “intrinsic” rate of growth(=b-d). Population size can be in any units (numbers,
biomass, species, females).
0( ) ( ) rtdNb d N N t N e
dt
0N
r
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The Simplest Model - II Discrete version:
The exponential model predicts that the population will eventually be infinite (for r>0) or zero (for r<0).
Use of the exponential model is unrealistic for long-term predictions but may be appropriate for populations at low population size.
The census data for many species can be adequately represented by the exponential model.
1 0(1 ) (1 )tt tN r N N r
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Fit of the exponential model to the bowhead
abundance data
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1975 1980 1985 1990 1995 2000
Year
Po
pu
lati
on
Siz
e
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Extrapolating the exponential model
0
5000
10000
15000
20000
25000
30000
35000
1940 1960 1980 2000 2020 2040 2060
Year
Po
pu
lati
on
Siz
e
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Extending the exponential model
(Extinction risk estimation)
Allow for inter-annual variability in growth rate:
This formulation can form the basis for estimating estimation risk:
( - quasi-extinction level, time period, critical probability)
21 ( ) ; ~ (0; )t t t t tN N r N N
maxProb( | )t critN t t p maxt
critp
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Calculating Extinction Risk for the Exponential Model
The Monte Carlo simulation:1. Set N0, r and 2. Generate the normal random variates3. Project the model from time 0 to time tmax
and find the lowest population size over this period
4. Repeat steps 2 and 3 many (1000s) times.5. Count the fraction of simulations in which
the value computed at step 3 is less than . This approach can be extended in all
sorts of ways (e.g. temporally correlated variates).
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Numerical Hint(Generating a N(x,y2) random
variate)
Use the NormInv function in EXCEL combined with a number drawn from the uniform distribution on [0, 1] to generate a random number from N(0,12), i.e.:
Then compute:1.R x y X
1 NormInv(Rand(),0,1)X
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The Logistic Model-I No population can realistically grow
without bound (food / space limitation, predation, competition).
We therefore introduce the notation of a “carrying capacity” to which a population will gravitate in the absence of harvesting.
This is modeled by multiplying the intrinsic rate of growth by the difference between the current population size and the “carrying capacity”.
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The Logistic Model - II
where K is the carrying capacity.
The differential equation can be integrated to give:
1(1 / ) OR (1 / )t t t t
dNrN N K N N rN N K
dt
0
0
( )1 rt
KN t
K Ne
N
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Logistic vs exponential model
(Bowhead whales)
0
5000
10000
15000
1965 1975 1985 1995 2005 2015 2025
Year
Po
pu
lati
on
siz
e
Which model fits the
census data better?
Which is moreRealistic??
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The Logistic Model-III
0
200
400
600
800
1000
1200
1400
1600
0 10 20 30 40Year
Po
pu
lati
on
Siz
e
No=500
No=1000
No=1500
r=0.1; K=1000
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Assumptions and caveats Stable age / size structure Ignores spatial, ecosystem considerations /
environmental variability Has one more parameter than the exponential
model. The discrete time version of the model can
exhibit oscillatory behavior. The response of the population is instantaneous.
Referred to as the “Schaefer model” in fisheries.
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The Discrete Logistic Model
0
200
400
600
800
1000
1200
0 5 10 15 20 25
Year
Po
pu
lati
on
Siz
e
r=0.1 0.1
r=0.5 0.5
r=1.5 1.5
r=2 2.1
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Some common extensions to the Logistic Model
Time-lags (e.g. the lag between birth and maturity is x):
Stochastic dynamics:
Harvesting:
where is the catch during year t.
1 (1 / )t t t x t xN N rN N K
1 ( ) {1 /( )}t t t t t t tN N r N N K
1 (1 / )t t t t tN N rN N K C
tC
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Surplus Production The logistic model is an example of a
“surplus production model”, i.e.:
A variety of surplus production functions exist: the Fox model the Pella-Tomlinson modelExercise: show that Fox model is the limit p-
>0.
1 ( )t t t tN N g N C
( ) (1 n / n )t t tg N rN N K
( ) (1 ( / ) )prt t tpg N N N K
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Variants of the Pella-Tomlinson model
0
5
10
15
20
25
30
35
40
0 200 400 600 800 1000
Population Size
Sur
plus
pro
duct
ion
p=0
p=1
p=2.39
p=5.49
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Some Harvesting Theory Consider a population in dynamic
equilibrium:
To find the Maximum Sustainable Yield:
For the Schaefer / logistic model:
1 ( )t t t tN N C g N
( )0
dC dg N
dN dN
2 / / 24MSY
dC r Kr rN K N K MSY
dN
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Additional Harvesting Theory
Find for the Pella-Tomlinson model
MSYN
0
5
10
15
20
25
30
35
40
0 200 400 600 800 1000
Population Size
Sur
plus
pro
duct
ion
p=0
p=1
p=2.39
p=5.49
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Readings – Lecture 2 Burgman: Chapters 2 and 3. Haddon: Chapter 2