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

• So that N(t) can be derived by– Creating a recursive equation (for difference

equations)– Integrating (for differential equations)

)(

)(

Nfdt

dN

Nft

N

Mechanics: Adding Environmental Stochasticity

• In stochastic models, we presume that the dynamic equation is a probability distribution, so that :

• Where v(t) is some random variable with a mean 0.

)())(()(

tvtNft

tN

Density-Independent Model

• Deterministic Model:

• We can predict population size 2 time steps into the future:

• Or any ‘n’ time steps into the future:

)()1(

)()1()1(

)()()(

tNtN

tNdbtN

tNdbt

tN

)()()1()2( 2 tNtNtNtN

)()( tNntN n

Adding Stochasicity

• Presume that varies over time according to some distributionN(t+1)=(t)N(t)

• Each model run is unique

• We’re interested in the distributionof 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)=[tt-1t-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)=[tt-1t-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:

• The geometric mean is ALWAYS less than the arithmetic mean

t

tG t

1

)(

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 2

• Then the mean and variance of the possible population sizes at time t equals

1)exp()2exp()0(

)exp()0()(222

)(

ttrN

trNtN

rtN

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:

• is the “autocorrelation” coefficient. 0 means no temporal correlation

),0(~)(

)()1()(2

Ntv

tvrrtrtr

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:

• 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”

)(')( NNfNft

N

Compensatory vs. depensatory

)(')( NNfNft

N

Per-

Capi

ta G

row

th R

ate,

f’(N

)

Population Size (N)

Below this point, population growth rate will be negative

Lab this week

• Create your own stochastic density-independent 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