SPATIAL POINT PROCESSES, WITH APPLICATIONS TO ECOLOGY

The term 'point processes ', referring to stochastic processes in which events occur at more or less irregular intervals and which are represented by points on the time-axis, is of comparatively recent origin, although the existence of such processes has in fact been well known for a long time. They have been discussed fairly extensively in such diverse applications as the counting of radioactive impulses, telephone calls and cases of contagious diseases. Wold (1949) developed a statistical theory for treating processes of this type, and also mentioned briefly how the events could take place in a two-dimensional or higher field. Such a generalization, from events with no time extension to those with no 'space' extension (i.e. specifically of a point character), has a suitable field of application in the ecological study of the distributional pattern of plants. If we can assume to a first approximation that the plants have the dimensions of a point, then we shall see that it is possible to discuss precisely probability relationships between the numbers of plants in different areas of the region under investigation. The main aims of quantitative ecology are the precise description of a community of plants with interpretations in terms of the biology of the species, and the correlation of vegetational and environmental data, and ecologists have used several methods in an attempt to achieve these aims. In most of the initial work on field sampling for ecological data, the procedure was to take 'quadrats' (sample areas small in relation to the total area of the region) scattered at random over the area, and study statistics derived from the frequency distribution of the numbers of plants per quadrat. While this approach is useful to some extent, in that any given type of distribution function may be fitted to the data, it does not necessarily furnish the kind of information required by an ecologist. It will not give any evidence of trends, or indicate the pattern of the distribution over the area or the way in which this pattern may have arisen, all factors of prime importance in the study of the structure of a plant community. We only have to cite the negative binomial distribution, which is known to arise in at least four different ways, all based on widely differing assumptions, to illustrate this point. In recent years ecologists have become aware of the need for a more satisfactory approach to the problem, and Greig-Smith (1952) provided a potentially great advance on the statistical side when he recommended the use of a grid of contiguous quadrats over some portion or portions of the region. The advantage, of course, in arranging the quadrats in a grid is that the analysis of variance technique may be employed, either for the detection of trends, or, more importantly, for the detection of a mosaic variation in density (due to ecological causes connected with the spread of the plants) by a 'nested sampling' type of analysis of variance, associating the quadrats into successively larger blocks and comparing the component block variances. The details and applications of this method are described at length by Greig-Smith, together with the results from sampling experiments on artificial