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458
Lumped population dynamics
models
Fish 458; Lecture 2
Revision: Nomenclature
458

Which are the state variables, forcing
functions and parameters in the
following model:
Nt 1   Nt  R  Ct
N t population size at the start of year t,
 Ct catch during year t,

growth rate, and
 R annual recruitment

The Simplest Model-I
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Assumptions of the exponential model:
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No emigration and immigration.
The birth and death rates are independent of each
other, time, age and space.
The environment is deterministic.
dN
 (b  d ) N  N (t )  N 0 e rt
dt



N 0 is the initial population size, and
r is the “intrinsic” rate of growth(=b-d).
Population size can be in any units (numbers,
biomass, species, females).
The Simplest Model - II
458
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Discrete version:
N t 1  (1  r ) N t  N 0 (1  r )t
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.
Fit of the exponential model to
the bowhead abundance data
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9000
Population Size
8000
7000
6000
5000
4000
3000
2000
1000
0
1975
1980
1985
1990
Year
1995
2000
Extrapolating the exponential
model
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35000
Population Size
30000
25000
20000
15000
10000
5000
0
1940
1960
1980
2000
Year
2020
2040
2060
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Extending the exponential model
(Extinction risk estimation)
Allow for inter-annual variability in
growth rate:
N t 1  N t  ( r   t ) N t ;

 t ~ N (0; 2 )
This formulation can form the basis for
estimating estimation risk:
Prob( Nt   | t  tmax )  pcrit
(  - quasi-extinction level, tmax time
period, pcrit critical probability)
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Calculating Extinction Risk for the
Exponential Model
The Monte Carlo simulation:

1.
2.
3.
4.
5.

Set N0, r and 
Generate the normal random variates
Project the model from time 0 to time tmax and
find the lowest population size over this period
Repeat steps 2 and 3 many (1000s) times.
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).
Numerical Hint
(Generating a N(x,y2) random variate)
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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.:
X1  NormInv(Rand(),0,1)

Then compute:
R  x  y.X1
The Logistic Model-I
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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
dN
 rN (1  N / K )
dt
OR
N t 1  N t  rNt (1  N t / K )
where K is the carrying capacity.
The differential equation can be
integrated to give:
K
N (t ) 
K  N 0  rt
1
e
N0
Logistic vs exponential model
(Bowhead whales)
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Population size
15000
Which model fits the
census data better?
Which is more
Realistic??
10000
5000
0
1965
1975
1985
1995
Year
2005
2015
2025
The Logistic Model-III
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1600
No=500
1400
No=1000
No=1500
Population Size
1200
1000
800
600
r=0.1; K=1000
400
200
0
0
10
20
Year
30
40
Assumptions and caveats
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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.
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Referred to as the “Schaefer model” in fisheries.
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The Discrete Logistic Model
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Population Size
1200
1000
800
600
400
r=0.1 0.1
r=0.5 0.5
200
r=1.5 1.5
r=2 2.1
0
0
5
10
15
Year
20
25
458
Some common extensions to the
Logistic Model
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Time-lags (e.g. the lag between birth and
maturity is x):
Nt 1  Nt  rNt  x (1  Nt  x / K )
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Stochastic dynamics:
Nt 1  Nt  (r  t ) Nt {1  Nt /( K  t )}   t
Harvesting:
Nt 1  Nt  rNt (1  Nt / K )  Ct
where Ct is the catch during year t.
Surplus Production
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The logistic model is an example of a “surplus
production model”, i.e.:
Nt 1  Nt  g ( Nt )  Ct
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A variety of surplus production functions
exist:
g ( Nt )  rNt (1  nNt / nK ) the Fox model
g ( Nt )  rp Nt (1  ( Nt / K ) p ) the Pella-Tomlinson model
Exercise: show that Fox model is the limit p->0.
Variants of the Pella-Tomlinson model
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Surplus production
40
p=0
35
p=1
30
p=2.39
p=5.49
25
20
15
10
5
0
0
200
400
600
Population Size
800
1000
Some Harvesting Theory
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Consider a population in dynamic
equilibrium:
Nt  Nt 1  Ct  g ( Nt )

To find the Maximum Sustainable Yield:
dC dg ( N )

0
dN
dN

For the Schaefer / logistic model:
dC
rK
 r  2rN / K  N MSY  K / 2  MSY 
dN
4
Additional Harvesting Theory
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Find
N MSY
for the Pella-Tomlinson model
40
Surplus production
458
p=0
35
p=1
30
p=2.39
p=5.49
25
20
15
10
5
0
0
200
400
600
Population Size
800
1000
Readings – Lecture 2
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Burgman: Chapters 2 and 3.
Haddon: Chapter 2