Demographic stochasticity

Demographic stochasticity refers to the variability in population growth rates arising from random differences among individuals in survival and reproduction within a season. This variability will occur even if all individuals have the same expected ability to survive and reproduce and if the expected rates of survival and reproduction don't change from one generation to the next. Even though it will occur in all populations, it is generally important only in populations that are already fairly small.⁵

To make this concrete, let's compare two populations, one of size 10, one of size 10,000. We'll asssume that individuals produce 2 offspring, on average, but that the actual number of offspring any one individual produces is a Poisson random variable, i.e.,

$$P(N=n) = {\lambda^n e^{-\lambda} \over n!} \qquad ,$$

where $\lambda = 2$. We'll also assume that the offpring have a 50% chance of survival. The combination of an average of two offspring per individual and 50% survival means that the population size doesn't change, on average, i.e. $\bar R = 0$.

• Population of 10,000
  • Expected number of offspring in next generation is $20,000$

  • Variance in number is $20,000$. Standard deviation is $141$.

  • Therefore, $20,000 \pm 300$ offspring produced, half of which (on average) survive.

  • Results of a little simulation.
    • Generate 1000 random numbers from a Poisson distribution with a mean of 20,000, matching the number of offspring produced in this hypothetical example.

    • For each of these 1000 numbers, randomly determine whether each individual survives to reproduction with probability $p=0.5$.

    • Here's the code in R⁶

      > n <- rpois(1000, 20000)
      > n.surv <- numeric(0)
      > for (i in 1:1000) n.surv[i] <- rbinom(1, n[i], 0.5)
      > mean(n)
      [1] 20002.28
      > var(n)
      [1] 19284.28
      > mean(n.surv)
      [1] 9999.959
      > var(n.surv)
      [1] 9370.934
      > quantile(n.surv)
            0%      25%      50%      75%     100% 
       9734.00  9931.75  9999.00 10070.00 10322.00
      

      So in this simulation the maximum decline (from 10,000 to 9734) was a little less than 3%, and the median outcome was indistinguishable from stability.

      Another way of summarizing these results is to say that $\mbox{Var}(R_t) = \frac{1}{N_t^2}\mbox{Var}({\tt n.surv})\approx 1.0\times 10^{-4}$.

• Population of 10
  • Expected number of offspring in next generation is $20$

  • Variance in number is $20$. Standard deviation is $4.5$.

  • Therefore, $20 \pm 9$ offspring produced.

  • Simulation results:

    > n <- rpois(1000, 20)
    > n.surv <- numeric(0)
    > for (i in 1:1000) n.surv[i] <- rbinom(1, n[i], 0.5)
    > mean(n)
    [1] 20.102
    > var(n)
    [1] 19.01261
    > mean(n.surv)
    [1] 10.016
    > var(n.surv)
    [1] 10.46621
    > quantile(n.surv)
      0%  25%  50%  75% 100% 
       2    8   10   12   23
    

    So in this simulation the maximum decline (from 10 to 2) was 80%, even though the median outcome was stability and the individuals in this population had exactly the same reproductive potential as those in the population of 10,000.

    Another way of summarizing these results is to say that $\mbox{Var}(R_t) \approx 1.0\times 10^{-1}$, three orders of magnitude greater than the variance when $N=10,000$.

Like genetic stochasticity, demographic stochasticity is likely to be important only in populations that are already small.⁷It may pose an additional threat to species that are already endangered, but it is unlikely to cause the endangerment of those with reasonably large populations.