Demography of chance extinction

When we're managing a threatened species, our primary focus is identifying strategies to prevent it from going extinct. The primary conceptual tools for organizing our thinking are related to extinction, specifically the probability that a population goes extinct within a specified time period (typically 100 years) or the expected time to extinction. We'll talk more about this when we discuss population viability analysis next week.

Probabilists have studied birth-and-death processes for many years. Individuals die with a given probability between seasons. Surviving individuals give birth to a random number of individuals. For a population of size $N$, the expected time to extinction is

$$T_{(N)} = \sum_{x=1}^N\sum_{y=x}^{N_m} {2 \over {y[yV_{(y)}... ...-1} {{V_{(z)}z + r_{(z)}} \over {V_{(z)}z - r_{(z)}}} \quad ,$$

where $r_{(n)}$ is the mean per capita growth rate and $V_{(n)}$ is its variance when the population size is $n$ [2]. $N_m$ is maximum possible population size - the population ceiling.

If all variance in population growth rates were due to events that affect individuals independently and all individuals have the same probability distribution governing death rates and fecundity rates then

$$V_{(n)} = {V_1 \over N} \quad ,$$

where $N$ is the population size and $V_1$ is the variance of the per capita offspring production rate. This approach incorporates only demographic stochasticity.

Alternatively, all variance in population growth rates may be due to population-wide variation in the vital rates, with all individuals behaving identically. Then

$$V_{(n)} = V_e \quad ,$$

where $V_e$ is a measure of environmental stochasticity.⁷ This approach incorporates only environmental stochasticity.

To incorporate both

$$V_{(n)} = V_e + {V_1 \over N} \quad .$$ (2)

Note: This ignores any covariance between $V_1$ and environment, which is likely to inflate the variance. From equation (2) we would expect demographic stochasticity to have a large influence only when population sizes are very small. This entire approach ignores age/stage structure within the population, but analysis of these types of models still provide some indications of the qualitative features of the extinction process.

1. Persistence time increases greatly as population ceiling is increased: Management implication, small reserve areas have small population ceilings (at least for large animals), therefore extincition is more likely in small reserves than large ones.

2. Demographic stochasticity is unimportant in populations with more than about 50 reproductive individuals:⁸ Management implication, direct manipulation of reproduction is unlikely to be needed except in very small populations

   Figure 1: Effect of demographic stochasticity on persistence times. Solid line has $r=0.2$. Dotted line has $r=0.05$. Individual variance in reproductive success is equal to 1.

   

3. Persistence time is drastically shortened by environmental stochasticity:⁹ Management implication, populations must be very large to buffer environmental stochasticity without direct intervention

   Figure 2: Effect of environmental stochasticity on persistence times. $r=0.05$ for all lines. $V_e = 1/30$ for the solid line, $1/20$ for the dotted line, and $1/15$ for the dashed line.

   

4. Distribution of persistence times is roughly geometric: If mean persistence time is $\bar t$ probability of extinction after $n$ generations is
   
   $$\left({1 \over\bar t}\right)\left(1 - {1 \over\bar t}\right)^{n-1}$$
   
   
   Management implication, there is a greater than 50% of extinction before $\bar t$.
   1. 63% chance of extinction by $\bar t$

   2. 50% chance of extinction by $0.7\bar t$

5. Persistence time increases exponentially with carrying capacity if $\bar R > \frac{s^2_R}{2}$, but only logarithmically with carrying capacity if $\bar R < \frac{s^2_R}{2}$: Management implication, increasing the size of a managed population has less impact on its long-term persistence than reducing the variability in growth rate. Only when variability in population growth rate is small are isolated populations likely to persist without frequent management intervention.