Following the success of the first meeting in Sheffield in April 1973, a second symposium was held in the School of Biological Sciences, University of Sussex, on 26th and 27th March 1974. (See Adv. Appl. Prob., 6, 1–20, for abstracts of Sheffield meeting.) The objective was, as before, to provide an opportunity for the presentation of papers on mathematical aspects of genetics.

Balanced, autosomal chromosome mutations do not change the genotype of animals that are hetero- or homozygotes for the mutation. However, in the gametes produced by a heterozygote animal, there normally exists a fraction with unbalanced chromosome complements. Such gametes are functionally effective in mammals and produce embryos with unbalanced genotypes, which leads to zygotic loss. The fertility of chromosomally heterozygote animals is thus lower than the fertility of homozygote animals. This fertility decrease of heterozygotes has been shown in a number of studies on translocations in mice, cattle and humans. The problem is, how such mutations can ever increase in a population until all animals are homozygotes for the chromosome change.

The classical models for genetic drift of Wright (1931) and Moran (1958) have been generalized by Karlin and McGregor (1965), and Chia and Watterson (1969), this last treatment including the other three. The basic technique for the generalized models was to consider the change of certain expectations of the underlying random variables, the generalization being in terms of the conditional branching process.

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

The three-locus symmetric viability model has been analysed. The symmetric equilibria, namely those with equality of complementary chromosomes and hence with all allele frequencies equal to ½, have been found and their local stability studied. It was shown that there are fifteen symmetric equilibria, of which six may be stable for a given set of selection and recombination parameters.

Maynard Smith and Price (1973) have shown that the methods and terminology of Game Theory can be used to study the evolution of various strategies in animal species. They write aij as the mean ‘value’ to the user of strategy i when his opponent uses strategy j; these values aij are related to the fitness of the user of i, in the sense that, if aij is large, it is advantageous to use strategy i when one's opponent uses strategy j, and this advantage will be reflected in an increase in one's reproductive potential. For example, these may be a contest for territory, or dominance rights.

In populations of species such as cattle and sheep which are being selected for commercially important quantitative traits like milk yield and growth rate, the breeder cannot avoid overlapping generations since the female reproductive rate is so low. There is a classical theory for predicting the rate of response to selection in continuing programmes in which the selection differentials and age structure of the population do not change (Rendel and Robertson, (1950)). In a new programme, however, these rates of response are approached only asymptotically and may not be a useful approximation for several years. Recurrence relations have been developed by several authors to enable predictions of short term responses, but these can be simplified with Leslie's (1945) matrix formulation. Some basic principles will be given here; for details see Hill (1974).

In the location of genes affecting quantitative characters in Drosophila, Thoday has made extensive use of a technique of examining the products of crossovers between tester chromosomes and those from selected lines. The paper considers the validity of inferences which can be drawn from such experiments. The multiplicity of possible models makes this difficult and some efforts have been made, simulating results from given known situations. It seems probable that the shape of the final conclusions would be in terms of the power of the method to discover groups of genes, covering a certain length of chromosome and with a given sum total of effects.

In the quantitative inheritance of finger-ridge counts, it is conventional to take the largest of the ulnar and radial ridgecounts as a measure of the pattern size. In the case of a loop, there is but one measurement, a radial count on an ulnar loop, and an ulnar count on a radial loop. In a whorl there are (usually) two counts, and it is in this instance when the convention is used. In the case of an arch, both ulnar and radial counts are zero.

Various attempts have been made to develop a theory of the genetical effects of assortative mating, using various models. The earliest and most famous is that of Fisher (1918). However, this has run into a difficulty in that his paper is far from easy to understand even despite the efforts of Kempthorne (1957) and Moran and Smith (1966) to simplify his presentation. A key paragraph is the following (which we give in abbreviated form).

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

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.

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.

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.

We obtain results connecting the distribution of the random variables Y and W in the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1, EYβ and EWβ converge or diverge together and regular variation of the tail of one of Y, W with non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

The techniques used by Doeblin and Chung to obtain ordinary limit laws (central limit laws, weak and strong laws of large numbers, and laws of the iterated logarithm) for Markov chains, are extended to obtain analogous functional limit laws for stochastic processes which have embedded processes satisfying these laws. More generally, it is shown how functional limit laws of a stochastic process are related to those of a process embedded in it. The results herein unify and extend many existing limit laws for Markov, semi-Markov, queueing, regenerative, semi-stationary, and subordinated processes.

Multidimensional sampling of real data, e.g., in space and time, often requires observations at non-uniformly spaced intervals. A discrete transform of a multidimensional stationary stochastic process transforms a multivariate problem into an asymptotically univariate one if the spacing is uniform in at least one dimension. For both uniform and non-uniform sampling and a model of ‘signal’ imbedded in a ‘noise’ process, asymptotic normality and independence justifies statistical testing in each cell of the transformed domain of the hypothesis ‘noise alone’ versus the alternate ‘signal plus noise’.