The evaluation of the chance of survival of an advantageous mutant gene in an age-structured population is of interest since, as pointed out by Charlesworth (1973), the usual evolutionary arguments concerning the long-term relative effectiveness of natural selection on genes acting at different ages (Cole (1954), Hamilton (1966)) should be translated into statements about the survival probabilities of genes with age-specific effects. We shall be concerned with this quantity for the case of a mutant introduced into a large population with discrete age-classes (Leslie (1945)). Let unity be the index of the birth age-class, b that of the age of first reproduction, and d that of the age of last reproduction. Let px be the probability of survival of individuals from age x to x + 1, and let lx be the probability of surviving from conception to age x (lx = πy=1x-1px). Let mx be half the expected number of offspring at age x (demographic differences between the sexes are assumed to be absent; the environment is assumed constant). Since a new mutant gene is carried largely in heterozygotes for a long time after its first appearance, the problem reduces to that of finding the chance of survival of the line descended from a single individual of age-class 1, with the parameters of the heterozygote for the mutant gene. Let the probability generating function for the distribution of heterozygous offspring by a heterozygote aged x be hx(z). If offspring distributions at different ages are independent, the p.g.f. for the total offspring produced at ages b to x by an individual aged x (x ≧ b) is given as
Large population limit and time behaviour of a stochastic particle model describing an age-structured population
