Extinction Probability

Definition

Let $\mu =E[{\xi }_i]$ — the expected number of offspring per individual, i.e., the mean of the offspring distribution $P_j$.

• If $\mu \le 1$: ${\pi }_0=1$ (certain extinction, unless $P_1=1$)
• If $\mu >1$: ${\pi }_0$ is the smallest positive solution to:

${\pi }_0={\sum }_{j=0}^{\infty }{\pi }_0^j{P}_j$

Interpretation

• ${\pi }_0$ (left side): the extinction probability we’re solving for
• $j$: number of offspring a single individual produces
• $P_j$: probability of producing exactly $j$ offspring
• ${\pi }_0^j$ (right side): probability that all $j$ offspring lines eventually go extinct (each offspring independently has extinction probability ${\pi }_0$, and there are $j$ of them)

The right side sums over all possible offspring counts: probability of having $j$ offspring × probability all $j$ lineages die out.

This is a self-consistency equation — ${\pi }_0$ appears on both sides because extinction of the whole population requires extinction of every sub-lineage.

Related

• Branching Process