Stochastic Population Modelling
QSCI/ Fish 454
Stochastic vs. deterministic
• So far, all models we’ve explored have been
“deterministic”
– Their behavior is perfectly “determined” by the model
equations
• Alternatively, we might want to include
“stochasticity”, or some randomness to our
models
• Stochasticity might reflect:
– Environmental stochasticity
– Demographic stochasticity
Demographic stochasicity
• We often depict the number of surviving individuals
from one time point to another as the product of
Numbers at time t (N(t)) times an average survivorship
• This works well when N is very large (in the 1000’s or
more)
• For instance, if I flip a coin 1000 times, I’m pretty sure
that I’m going to get around 500 heads
(or around p * N = 0.5 * 1000)
• If N is small (say 10), I might get 3 heads, or even 0
heads
– The approximation N = p * 10 doesn’t work so well
Why consider stochasticity?
• Stochasticity generally lowers population
growth rates
• “Autocorrelated” stochasticity REALLY lowers
population growth rates
• Allows for risk assessment
– What’s the probability of extinction
– What’s the probability of reaching a minimum
threshold size
Mechanics: Adding Environmental
Stochasticity
• Recall our general form for a dynamic model
N
 f (N )
t
dN
 f (N )
dt
• So that N(t) can be derived by
– Creating a recursive equation (for difference
equations)
– Integrating (for differential equations)
Mechanics: Adding Environmental
Stochasticity
• In stochastic models, we presume that the
dynamic equation is a probability distribution,
so that :
N (t )
 f ( N (t ))  v(t )
t
• Where v(t) is some random variable with a
mean 0.
Density-Independent Model
• Deterministic Model:
N (t )
 (b  d ) N (t )
t
N (t  1)  (1  b  d ) N (t )
N (t  1)  N (t )
• We can predict population size 2 time steps into
the future:
2
N (t  2)  N (t  1)  N (t )   N (t )
• Or any ‘n’ time steps into the future:
N (t  n)   N (t )
n
Adding Stochasicity
• Presume that  varies over time according to
some distribution
N(t+1)=(t)N(t)
• Each model run
is unique
• We’re interested
in the distribution
of N(t)s
Why does stochasticity lower overall
growth rate
• Consider a population changing over 500
years: N(t+1)=(t)N(t)
– During “good” years,  = 1.16
– During “bad” years,  = 0.86
• The probability of a good or bad year is 50%
• N(t+1)=[tt-1t-2…. 2 1 o]N(0)
• The “arithmetic” mean of  (A)equals 1.01
(implying slight population growth)
Model Result
There are exactly
250 “good” and
250 “bad” years
This produces a
net reduction in
population size
from time = 0 to
t =500
The arithmetic
mean  doesn’t
tell us much
about the actual
population
trajectory!
Why does stochasticity lower overall
growth rate
•
•
•
•
•
N(t+1)=[tt-1t-2…. 2 1 o]N(0)
There are 250 good  and 250 bad 
N(500)=[1.16250 x 0.86250]N(0)
N(500)=0.9988 N(0)
Instead of the arithmetic mean, the population size at
year 500 is determined by the geometric mean:


G     (t ) 

t

1
t
• The geometric mean is ALWAYS less than the
arithmetic mean
Calculating Geometric Mean
• Remember:
ln (1 x 2 x 3 x 4)=ln(1)+ln(2)+ln(3)+ln(4)
So that geometric mean G = exp(ln(t))
It is sometimes convenient to replace ln() with r
Mean and Variance of N(t)
• If we presume that r is normally distributed
with mean r and variance s2
• Then the mean and variance of the possible
population sizes at time t equals
N (t )  N (0) exp( r t )
s N2 (t )  N (0) 2 exp( 2r t )exp(s r2t )  1
Probability Distributions of Future
Population Sizes
r ~ N(0.08,0.15)
Application:
• Grizzly bears in the greater
Yellowstone ecosystem are a
federally listed species
• There are annual counts of
females with cubs to provide
an index of population trends
1957 to present
• We presume that the
extinction risk becomes very
high when adult female
counts is less than 20
Trends in Grizzly Bear Abundance
• From the N(t),we
can calculate the
ln (N(t+1)/N(t)) to
get r(t)
• From this, we can
calculate the mean
and variance of r
• For these data,
mean r = 0.02 and
variance σ2 = 0.0123
Apply stochastic population model
• This is a result of 100
stochastic
simulations, showing
the upper and lower
5th percentiles
• This says it is unlikely
that adult female
grizzly numbers will
drop below 20
But wait!
That simulation
presumed that we knew
the mean of r perfectly
95% confidence interval for
r = -0.015 – 0.58
We need to account for uncertainty
in r as well (much harder)
Including this uncertainty leads to a
much less optimistic outlook (95%
confidence interval for 2050 includes
20)
Other issues: autocorrelated variance
• The examples so far assumed that the r(t) were
independent of each other
– That is, r(t) did not depend on r(t-1) in any way
• We can add correlation in the following way:
r (t )  r r (t  1)  r   r  v(t )
v(t ) ~ N (0, s 2 )
• r is the “autocorrelation” coefficient.
r  0 means no temporal correlation
Three time series of r
• For all, v(t) had mean 0 and variance 0.06
Density Dependence
• In a density-dependent model, we need to
account for the effect of population size on
r(t) (per-capita growth rate)
• Typically, we presume that the mean r(t)
increases as population sizes become small
– This is called “compensation” because r(t)
compensates for low population size
• This should “rescue” declining populations
Compensatory vs. depensatory
• Our general model:
N
 f ( N )  Nf ' ( N )
t
• f’(N) is the per capita growth rate
• In a compensatory model f’(N) is always a
decreasing function of N
• In a depensatory model, f’(N) may be an
increasing function of N
– Also sometimes called an “Allee effect”
Per-Capita Growth Rate, f’(N)
Compensatory vs. depensatory
N
 f ( N )  Nf ' ( N )
t
Population Size (N)
Below this point, population growth rate will be negative
Lab this week
• Create your own stochastic densityindependent population model and evaluate
extinction risk
• Evaluate the effects of autocorrelated variance
on extinction risk
• Evaluate the interactive effect of stochastic
variance and “Allee effects” on extinction risk