For many years animal behaviourists, sociologists and the like have been studying how animals and humans form groups, and how the groups change in time. A brief review was given of some of the models devised for such group changing situations, and some new models were described.

We examine birth and death processes where the birth/death rates depend on the particular configuration of the population. The easiest case is when the configurations form the subsets of some finite lattice (thus the state space is finite). In this case we look at the relationships between time-reversibility, nearest neighbour interactions, and the equilibrium state being a Markov random field. A more interesting case is when the entities which can be born or die can do so at any point in some bounded region of space (or the plane). This gives us a pure jump process, whose general properties are well-known, (see for example, Feller (1966) Vol. II Chapter X.3). We examine some particular examples and compute equilibrium distributions.

This paper is a survey of the solutions of various first-passage time problems in the theory of queues.

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.

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.

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

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.

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 relationship between the Wiener-Hopf factorisation of matrices and the solution of systems of certain operator equations is discussed in an algebraic setting. It is shown that the study of the waiting time process of the nth arriving group of the general single server bulk queue leads to equations of that type. This system of equations may be considered as an extension of Lindley's waiting-time equation.

This paper studies a controlled queueing system in which the decisionmaker may change servers according to rules which depend only on the queue length. It is proved that for a given control policy a properly normalised sequence of these controlled queue length processes converges weakly to a controlled diffusion process as the queueing systems approach a state of heavy traffic.

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.