# The purpose of this next problem is to determine the effects that different forms of mortality…

The purpose of this next problem is to determine the effects that different forms of mortality have on the stability of a population. We define stability as the probability of indefinite survivorship = 1 — probability of eventual extinction. In the absence of the additional mortality we'll consider momentarily, the offspring X of a single individual has the probability distribution

Suppose that the mean of the distribution is m and that all offspring in the population are independent and identically distributed. We consider 3 types of mortality. In each case, the probability of an individual surviving is p), but the form the survivorship takes differs among the cases. Assume

(a) Mortality on Individuals: Independent of what happens to others, each individual survives with probability p. That is, given an actual litter size or number of offspring of X, the effective litter size has a binomial distribution with parameters (X, p). This type of mortality might reflect predation on adults,

(b) Mortality on Litters: Independent of what happens to other litters, each litter survives with probability p and is wiped out with probability g = 1 —p. That is, given an actual litter size of X, the effective litter size is X with probability p and 0 with probability q. This type of mortality might reflect predation on juveniles, or on nests and eggs in the case of birds.

(c) Mortality on Generations: An entire generation survives with probability p and is wiped out with probability q. This type of mortality might represent environmental catastrophies such as forest fire, flood, etc. Give the equations for determining 1 — Stability = Pr {Eventual Extinction} in each of these cases. Which population is the most stable? Which is least stable? Can you prove this?