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 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

9. 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)

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

11. 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)=[1.16 250 x 0.86 250 ]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 ( 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

13. 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

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 σ 2 = 0.0123

17. Apply stochastic population model This is a result of 100 stochastic simulations, showing the upper and lower 5 th 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:  is the “autocorrelation” coefficient.  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