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

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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

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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