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.

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.

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’.

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 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.

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.

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.

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.

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.

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

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?

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.

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.

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.

The model is just the natural extension to n dimensions of the classical model for an epidemic with removals: however, the deterministic version does show a richer variety of threshold behaviour. In the stochastic model, some results will be discussed for the embedded Markov chain, and also a possible control policy involving the removal of entire communities.

The General Stochastic Epidemic model is extended to allow for outside infection. The likelihood is derived for small households, and the difficulties for larger populations are discussed.

Variability in space of plankton can arise from stochastic effects on growth and predation, and should be smoothed by horizontal diffusion of the water. The interactions of these factors within a non-linear system should permit small-scale perturbations to appear as larger-scale patches. A simple simulation illustrates this possibility.

Over the last four years the Forestry Commission has conducted sample surveys to estimate the spread of Dutch Elm disease in England and Wales. The data were described and some of the problems of making inferences from them were indicated. These include the definition of an individual elm tree, the two main modes of infection, the complexity of the spatial distribution of elms and the existence of at least two strains of the fungus C. ulmi with different pathogenicity.