Stochastic Population Modelling

1. Stochastic Population Modelling QSCI/ Fish 454

2. 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

3. 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

4. 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

5. 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)

6. 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.

7. Density-Independent Model • Deterministic Model: • We can predict population size 2 time steps into the future: • Or any ‘n’ time steps into the future:

8. Adding Stochasicity • Presume that l varies over time according to some distribution N(t+1)=l(t)N(t) • Each model run is unique • We’re interested in the distributionof N(t)s

9. Why does stochasticity lower overall growth rate • Consider a population changing over 500 years: N(t+1)=l(t)N(t) • During “good” years, l = 1.16 • During “bad” years, l = 0.86 • The probability of a good or bad year is 50% • N(t+1)=[ltlt-1lt-2…. l2l1lo]N(0) • The “arithmetic” mean of l (lA)equals 1.01 (implying slight population growth)

10. 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 l doesn’t tell us much about the actual population trajectory!

11. Why does stochasticity lower overall growth rate • N(t+1)=[ltlt-1lt-2…. l2l1lo]N(0) • There are 250 good l and 250 bad l • 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

12. Calculating Geometric Mean • Remember: ln (l1 x l2 x l3 x l4)=ln(l1)+ln(l2)+ln(l3)+ln(l4) So that geometric mean lG = exp(ln(lt)) It is sometimes convenient to replace ln(l) with r

13. 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

14. Probability Distributions of Future Population Sizes r ~ N(0.08,0.15)

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

16. 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

17. 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

18. 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)

19. 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 is the “autocorrelation” coefficient. r = 0 means no temporal correlation

20. Three time series of r • For all, v(t) had mean 0 and variance 0.06

21. 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

22. 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”

23. Compensatory vs. depensatory Per-Capita Growth Rate, f’(N) Population Size (N) Below this point, population growth rate will be negative

24. 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