This meeting was envisaged by the organisers as the first of a continuing series of short, informal conferences within the general area of applied stochastic processes. The twenty-seven participants were drawn mostly from statistics and mathematics departments of British Universities, but also included a small number of biologists and geographers. The meeting was notable for its pleasantly informal atmosphere, and the consequent lively interchange of ideas, although for future meetings in the series a better balance between statisticians and applied scientists should, we feel, be effected.

In this paper we develop a model for describing the spread of the parasitic disease bilharziasis. The detailed model first developed (Model A) is made more tractable by replacing infecting population sizes by their expected values in the transition probabilities, giving Model B. Using this approximating model it is possible to examine the nature of the spread of the disease and the effects of control measures. The effects of a particular control measure are examined in order to compare Models A and B.

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.

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

Detection of the different scales of pattern in plant communities is an important area of plant ecological research, and various tests of pattern have been devised. The method of pattern detection which is ecologically most meaningful is that due to Greig-Smith (1952) but, until now, this has suffered from the lack of valid tests of significance for the individual scales of pattern, once the overall departure from a random distribution has been established. Various tests which partially or completely overcome this deficiency are discussed and exemplified and their small sample distributional properties examined. It is concluded that a set of tests, based on randomisation arguments, provides a fully valid method of testing simultaneously for pattern at various scales.

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.

Aggregated spatial patterns may be generated by a clustering process (see, for example, Bartlett (1964)) in which ‘parent’ events are distributed completely at random, and produce, independently, random numbers of ‘offspring’ according to some distribution Pn; the position of each offspring relative to its parent is governed, independently, by a given bivariate distribution. Parents and offspring are assumed indistinguishable. For such a process, Bartlett (1974) shows that the distribution function F of the distance, X say, from a randomly selected point to the nearest event is given by where ρ denotes the mean number of parents per unit area, A is the circle with centre the origin and radius x, and E(ds) denotes the event ‘no offspring in A from parent in ds’.

Some Wiener-Hopf type results are collected, related and given more direct proofs. Spitzer's random-walk method for the Wiener-Hopf integral equation also produces his factorisation relating functionals of maxima and minima. Transform equations are interpreted as decompositions of time-changed processes. Discrete- and continuous-time versions are related. Prabhu's factorisation for generators is equivalent to Fristedt's Lévy measure factorisation and to process decomposition.

A point process is obtained by considering a square lattice of side p½, 0 < p ≦ 1, and supposing that each lattice point contains a point of the process or does not independently, with probabilities p and 1 – p. The family of processes thus includes the deterministic lattice for p = 1, and has the Poisson as a limit as p → 0.

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

In their paper [2] Linnik and Zinger proved a theorem which has an important role in both theoretical and applied statistics. For the proof of their theorem the authors used a condition which greatly narrows down its applicability. In this paper we prove the theorem of Linnik and Zinger without this condition, and also give other extensions of it.

The practical problems involved in the maximum likelihood estimation of the parameters in a binary Markov field on a lattice are investigated. Methods using whole and conditional data are compared.

A statistical method for estimating the factorisation of the spectral density of a stationary point process is presented and asymptotic properties of the resulting estimates are derived. The estimated functions are of interest in the analysis of a self-exciting process and more generally in the problem of linear prediction, and can be viewed as an alternative second-order analysis of the process. Some examples are discussed.

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.