growth and decline

8/12/2019 Growth and Decline

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Exponential growth

pop. size

at time

t+t

=pop. size

at time t+

growth

increment

N(t+ t) = N(t) + N

Hypothesis: N = r N t

r - rate constant of growth


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Differential equation for exponential growth

rNdt

dN

)exp()( 0 rtNtN


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Exponential growth in discrete time

Nt+1= Nt + r Nt

Nt+1= (1+r) Nt

Nt= (1+r)t N0


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Exponential decline

rNdt

dN

)exp()( 0 rtNtN

r - mortality rate


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0 2 4 6 8 10 12 14 16 18 20

0

20

40

60

80

100

120

Timet

N(t)

Exponential decline r=0.1


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Limited growth

Factors that affect population dynamics

reproduction (growth rate)

mortality

environmental capacity


9/22

Monomolecular model for limited

growthFirst order chemical reaction: A P

Areactant, Pproduct, R(t)reactant concentration

kreaction rate

kRdt

dR - Exponential decay

C(t)product concentration

)( CAkdt

dC A = R(0)


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0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

Conc

entrations

time

Product C(t)=A(1-exp(-kt))

Reactant R(t)=A exp(-kt)

Monomolecular growth


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Logistic growth model

Relies on the hypothesis that population

growth is limited by environmental capacity

K

NrN

dt

dN1

Kenvironmental capacity


12/22

)exp(11

)(

0

rtNK

KtN

0 2 4 6 8 10 12 14 16 18 200

50

100

150

time

N(t)


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Logistic growth with time delay

Factor that limits growth acts after some time TD

No analytical solution

K

TtNtrN

dt

tdND)(

1)()(


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600

)(1)(5.0

)(DTtN

tNdt

tdN

0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

1800


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Discrete logistic model

K

iNiRNiN )(1)()1(


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Growth of individual organisms


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Von Bertalanffys model

Postulates:

Gain in weight is proportional to the surface

area of the organism

Loss in weight is proportional to the weight

of the organism

Organism maintain the same shape while

growing


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Von Bertalanffys model

CWHSdt

dW

Ssurface area

Wweight

L - length

H,C - parameters3

32

2 , LaWLaS

)( max LLk

dt

dL

)exp(1)(

max

0max kt

L

LLtL

(monomolecular growth)


19/22

Richards family of models

1)1(

1

max

m

W

W

m

kW

dt

dW

mktAWtW 11

exp1)( max

Has all of previous models as special cases


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Allometric growth

Allometrystudy of relative sizes of different

parts of organisms X, Y

Hypothesis:dt

dX

Xb

dt

dY

Y

11

bAXY


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Computations

Matlab script files and functions

Simulink block diagrams


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Computations

Matlab functions:

exp(x) - exponential

plot(x,y) - plot

ode45compute solution to ODE

X=A\B - least squares (help slash)

fmins - minimize function over arguments

growth and decline

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