The paper explores three different aspects of spatial modelling for systems in statistical equilibrium; that is, systems which are both mean and covariance stationary. Attention is primarily directed at models relating to observations for a single time period.

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.

A simple direct proof is given of Minkowski's result that the mean length of the orthogonal projection of a convex set in E3 onto an isotropic random line is (2π)–1 times the integral of mean curvature over its surface. This proof is generalised to a correspondingly direct derivation of an analogous formula for the mean projection of a convex set in En onto an isotropic random s-dimensional subspace in En. (The standard derivation of this, and a companion formula, to be found in Bonnesen and Fenchel's classic book on convex sets, is most indirect.) Finally, an alternative short inductive derivation (due to Matheron) of both formulae, by way of Steiner's formula, is presented.

A sequence of random variables η0, η1, …, ηn, … is defined by the recurrence formula ηn = max (ηn–1 + ξn, 0) where η0 is a discrete random variable taking on non-negative integers only and ξ1, ξ2, … ξn, … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.

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.

The merits of a conditional probability (or Markovian) approach to the analysis of local spatial interaction lie in its intuitive appeal and its ability to cope with non-Gaussian situations. Whilst relationships between plausible spatial-temporal models and Markov fields may sometimes be obtained, this does not remove the need for a purely spatial approach to be available. In addition to promoting the above point of view, some Monte Carlo simulation results, due to Mr. D. H. Green, were presented. Analytical progress on coding efficiency was reported.

We establish the estimation equations corresponding to a system in which the state is a process with jumps and the observation is a point process. We then solve two problems: the estimation of the state of a queue when its output is observed, and the estimation of the time of disorder of a machine when the ‘complaints process’ is observed.

Intra-specific plant competition has been studied (Ford (1975)) by planting individuals in regular lattice arrangements and then observing during growth both the frequency distributions of plant size and the occurrence of different sized plants across the lattice. Lattice distributions were studied through comparison with direct simulation of the random case and by the method of Krishna-Iyer (1950). A time-lapse film has been produced with one exposure every 10 minutes over a period of 6 weeks, to illustrate plant competition in lattice arrangements.

A type of service system of particular interest in teletraffic theory has the property that calls (customers) originating from different sources have access to different subsets of a single group of devices (servers). One may or may not allow the formation of queues. Thierer [11], [12] has studied such a system with queues assuming it to be reversible. We investigate the conditions under which this assumption is exact.

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.

This paper is concerned with the identification and estimation of dynamic space-time models for the evolution of a single random variable Y across space and through time. Suppose Y takes on values yit for each geographic zone i = 1, …, n in successive time periods t = 1, …, T.

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).