The sex of a zygote is often decided by an XX/XY sex-determination mechanism. One important property of this system is the approximate numerical equality of the two sexes (at least at fertilization) due to mendelian segregation. Does the obvious evolutionary stability of the XX/XY sex-determination system pose a difficult problem to population genetics? We study the fate of new mutations occurring in populations with an XX/XY system, where the mutations are such that they change the sex-determination system. If such mutations occur frequently and the conditions for their initial increase are weak, then the evolutionary stability of the XX/XY system is hard to understand.

The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is A more general form (Bishop and Cannings (1978)) has and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT, then there exist constants 0 < a, C1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C1n).

From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn(c) <∞, where Nn(c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

The paper develops a formula for inbreeding probabilities in the multi-locus case, this formula being analogous to that given by Malecot for single-locus chromosomes. A detailed example provides a comparison between the present approach and that by Malecot and it is found that the two methods produce slightly different results. This slight difference is then shown to be due to certain paths in pedigrees which seem to have been overlooked by Malecot. An indication of calculations necessary for higher-locus cases concludes the paper.

Let μ (F) be the time constant of first-passage percolation on the square lattice with underlying distribution function F. Two theorems are presented which show, under some restrictions, that μ varies continuously in F with respect to weak convergence. These results are improvements of existing continuity theorems.

A class of stochastic processes useful in the investigation of the deterioration and replacement of equipment, as well as in the study of group arrival/batch service queues, storage systems, epidemics and computer software, is introduced. These so-called nested renewal processes consist of ordinary and cumulative renewal processes sequentially nested within a hierarchy. The main properties of these processes are discussed with emphasis on the asymptotic results. An example of their application to tyre wear is considered.

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I1, I2, ···, In where Ij is the strategy ‘play value mj′ and m1 < m2 < ··· < mn. The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[Ii, Ij] is the expected payoff for playing Ii against Ij. We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

The model of a population which grows logistically, except for random disasters that reduce the population by a given amount at the instants of a Poisson process, which was examined by Hanson and Tuckwell (1978), is generalised in two ways:

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.