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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Two species compete for territory along their mutual boundary. The species are fairly matched and the result of conflict is the invasion by one of the species of territory held by the other. A simple stochastic model for this process is described and rules are given for the calculation, as a function of time, of the probabilities that individual territories and groups of territories are held by a single species. Asymptotic results are given for the pattern of territories held by a single species, and here a remarkable distinction is seen between the cases of one-dimensional and two-dimensional competition and the case of three-dimensional competition. The process of invasion is contrasted with two relatively benign processes, that in which an exchange of territories is arranged and that in which a spatial alternation of species may be beneficial to the community. The implications of the long-term behaviour of such processes are discussed. It is suggested further that, under certain ideal conditions, it may be possible to estimate the duration of a spatial struggle for life by observing at some time the configuration of territories held by each species.

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Modified tests of association based on the correlation coefficient or the covariance between two spatially autocorrelated processes are presented. These tests can be used both for lattice and nonlattice data. They are based on the evaluation of an effective sample size that takes into account the spatial structure. For positively autocorrelated processes, the effective sample size is reduced. A method for evaluating this reduction via an approximation of the variance of the correlation coefficient is developed. The performance of the tests is assessed by Monte Carlo simulations. The method is illustrated by examples from geographical epidemiology.