Example

We can use these results to find the time by which the population has a 50% chance of going extinct. Suppose that in each catastrophe that occurs 95% of the population is eliminated, and that we're dealing with a species that has 2 offspring over its lifetime, lives about 20 years, and to which catastrophes occur about every 50 years. Suppose that we start with a population of 100 individuals.

Mean lifetime for an individual	20 years
Mean number of offspring for an individual	2
Mean time between catastrophes	50 years
Proportion of population surviving	0.05
Current population size	100
$\beta = 1/20$	0.05
$\alpha = \beta(2) = (1/20)(2)$	0.1
$\gamma = 1/50$	0.02
Mean time to extinction (from 3)	464
Variance of extinction time (from 4)	850,000
$b_n$	718
$a_n$	50
$t$ to 50% probability of extinction	313
Notice that $(0.7)464 \approx 325$, which is pretty close to 313.

Properties:

• Effect of catastrophes depends dramatically on their frequency (Figure 3).

  Figure 3: Effect of catastrophes depends dramatically on their frequency. ($\alpha = 0.24$, $\beta = 0.2$, $p = 0.5$, $n = 1000$)

  

• Effect of catastrophes depends dramatically on proportion of population surviving after catastrophe (Figure 4).

  Figure 4: Effect of catastrophes depends dramatically on proportion of population surviving after catastrophe. ($\alpha = 0.24$, $\beta = 0.2$, $\gamma = 0.1$, $n = 1000$)

  

• Resistance to catastrophes increases only very slowly with population size (Figure 5)

  Figure 5: Resistance to catastrophes increases only very slowly with population size ($\alpha = 0.24$, $\beta = 0.2$, $\gamma = 0.1$, $p = 0.5$).

  

These results are for density-independent population growth. It is also possible to study similar models of density-dependent population growth. Instead of expressing results in terms of persistence given the current population size, they are expressed in terms of persistence given a particular carrying capacity. The results are much more complicated, but the general pattern holds [4]:

• Persistence time is proportional to the logarithm of the current population size (in the case of density-independent population growth) or to a power $<1$ of carrying capacity (in the case of density-dependent population growth) if $r = \alpha - \beta < -\gamma\ln p$.

• Persistence time is proportional to a power $> 1$ of carrying capacity if $r = \alpha - \beta > -\gamma\ln p$.¹¹