The migration and death model (Bartlett (1949)) is useful in describing animal movement among distinct areas.

Effects of sampling procedures on the probability distributions of pedigrees are illustrated. Elimination of these effects by suitable conditioning is demonstrated.

A basic class of spatial stochastic processes is that in which the phenomenon spreads between static individuals with probability of ‘infection’ falling off with distance according to a contact distribution (Mollison (1972)). The following questions then arise:

(1) How does the velocity of spread depend on the contact distribution? In particular, for which contact distributions is it finite?

(2) Does the phenomenon spread at a steady velocity, or does it spread in jumps?

In the two locus theory that has been developed, it has been assumed that amount of recombination between the two loci is the same in both sexes (Bodmer and Felsenstein, (1967); Karlin and Feldman, (1970)). Some results of assuming that the recombination value in males is rm and in females rf are reported here.

The relationships between individuals may be specified by the genes which they have in common, where two genes are considered to be the same only if they are identical by descent from some common ancestor. Relationships between two individuals have been extensively studied by, amongst many others, Cotterman (1940), Malécot (1948) and Li and Sacks (1954). Less progress has been made with the more general problem of relationships between an arbitrary number of individuals. Elandt-Johnson (1971) has considered the special case of joint genotype distributions in a sibship, and Hilden (1970) has constructed an algebraic method of combining information on several individuals to give the conditional distribution of a single unborn relative, but neither of these approaches provides a general solution to the problem.

Considerable interest has recently been shown in the spatial spread of infection between static individuals. A different approach yielding similar types of processes is to allow the individuals themselves to migrate from site to site. This movement may be physical migration, mutation from one type of organism to another, or it may even represent an approximation to the initial development of a spatial epidemic. We shall consider the population to be distributed amongst either a finite number of sites (model A), or an infinite number situated at the nodes of a lattice (model B). In this latter situation individuals are only allowed to migrate between sites which are nearest neighbours, and the transition rates are assumed to be spatially homogeneous. Individuals at each site undertake a simple birth-death-migration-immigration process.

The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.

This second meeting in the series was organised principally by Denis Mollison and Eric Renshaw with the help of their respective departments and a grant from the Royal Society. The fifty or so participants were drawn from various backgrounds, both theoretical and applied, and a diverse programme of twelve papers was presented, embracing both the development of mathematical theory and the collection and analysis of field data. Many of the non-speakers contributed to the lively discussion which prevailed. Short abstracts of the papers are appended, and the authors would be happy to answer requests for further information.

Cell kinetics deals with experiments and theories designed to determine the proportion of cells in a tissue which are engaged in replication, and the times they take to pass through the various stages of their reproductive cycle.

A finite number of colonies, each subject to a simple birth-death and immigration process is studied under the condition of migration between the colonies.

Kolmogorov's backward equations for the process are solved for some special cases, and a sequence of functions uniformly converging to the p.g.f. of the process is given for the general case. Further, a set of algebraic equations for the extinction probabilities are studied for the process without immigration, and a necessary and sufficient condition that the extinction probability be one is obtained.

For the n-dimensional birth, death and migration process with constant rates, Aksland (1975) found a necessary and sufficient condition that the extinction-probability be one. This condition is translated into the form of direct relations between the parameters of the model.

An approximation technique is described for studying the deterministic spread of an epidemic which yields the correct critical velocity, unlike the diffusion approximation. Travelling waves and the spread from an isolated infective are discussed for the simple epidemic, and the work on travelling waves is extended to the general closed epidemic.