Pair Approximations for Lattice-based Ecological Models

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Abstract
Introduction In recent years, the effects of spatial configuration on population and evolutionary processes have been the subject of intensive research efforts in ecology and evolutionary biology. One reason for this surge of interest is the observation of intricate, natural spatio-temporal structures. Such structures are exemplified, for instance, by wave regeneration of fir forests, in which trees show a large-scale, wavelike pattern of regeneration with many stripes of dieback zones moving slowly downwind at a constant rate (Iwasa et al. 1991; Satō and Iwasa 1993; Jeltsch and Wissel 1994). Another example of spatio-temporal structure studied in forest ecology is the formation and closure of canopy gaps (Kubo et al. 1996; see Chapter 13). It is now acknowledged that considering spatio-temporal structures spontaneously formed by demographic processes and ecological interactions is sometimes essential for understanding population and evolutionary dynamics, and that traditional modeling in theoretical ecology assuming complete spatial mixing often fails to capture these dynamics. A simple and useful method for modeling population and evolutionary dynamics in a spatially explicit way is to use a lattice, or cellular automaton, model. Typically, one considers a large regular lattice, which may be linear, square, hexagonal, or triangular, in which an individual occupies one vertex, or lattice site. We assume that each individual interacts directly and strongly only with its nearest neighbors. Most ecological interactions between individuals – for example, competition for resources, disease transmission, cooperative interaction such as attracting common pollinators, and reproduction – occur at spatial scales much smaller than that of the whole population.

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