Catastrophes

In thinking about the influence of catastrophes on population persistence, it is useful to think of them as events that happen very rarely, but eliminate a fixed proportion of the population whenever they occur.

Definitions [1]:

$$\begin{eqnarray*} \alpha &=& \parbox[t]{4.25in}{\rm probability of a birth in th... ...e} \\ n &=& \parbox[t]{4.25in}{\rm initial population size} \\ \end{eqnarray*}$$

$$P(\hbox{population extinction} \le t) \approx \exp\left[-\exp\left(-{{t-a_n} \over b_n}\right)\right] \qquad ,$$

where $b_n = 0.7797\sigma_n$, $a_n = \mu_n - 0.5772b_n$, and $\mu_n$ and $\sigma^2_n$ are the mean and variance of the extinction time.¹⁰

$$\mu_n = -(\ln n)/(\alpha - \beta + \gamma\ln p) \quad ,$$ (3)

$$\sigma^2_i = -(\ln n)[\gamma(\ln p)^2]/(\alpha - \beta + \gamma\ln p)^3$$ (4)

This formula can only be used if $\alpha < \beta - \gamma\ln p$.

Mean lifetime for an individual	$\beta^{-1}$
Mean number of offspring for an individual	$\alpha\beta^{-1}$
Mean time between catastrophes	$\gamma^{-1}$