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The goal of this investigation was to study the effects of spatially structured habitat heterogeneities on locally dispersing single‐species populations. In this investigation, the environmental heterogeneities were not randomly distributed, but rather were clustered by specifying probabilities of small local configurations of the landscape, as in local structure or pair approximations. This allows the study of landscapes with the same amount of habitat loss but different levels of fragmentation or clustering. I describe a simple algorithm for generating such structured landscapes. Spatially explicit simulations of population models on these landscapes were performed using stochastic cellular automata and compared to predictions from mean‐field and pair approximations, for which detailed derivations are presented. For populations with local dispersal, I show that the spatial correlations of habitat types completely determine equilibrium population density on suitable sites and that the amount of suitable habitat has no effect, precisely the opposite of what the mean‐field approximation predicts. When habitat types are randomly distributed on the landscape, the two approximations do almost equally well, and thus the additional complexity of the pair approximation is not justified. However, when habitat types are not randomly distributed, the mean‐field approximation gives qualitatively incorrect predictions for population response to varying habitat heterogeneity. Thus, pair approximations combine some of the best features of spatially explicit and implicit models and serve as a useful supplement to those methods for understanding spatially structured ecological systems, especially where environmental heterogeneities are spatially correlated.

A discrete stochastic spatial model for a single species is examined. First, detailed spatial simulations are performed using stochastic cellular automata. Then, several analytic approximations are made. First, two versions of mean field theory are presented: the infinite-dispersal mean field approximation, which is a metapopulation-like model, and the localdispersal mean field approximation, which incorporates the locality of the cellular automaton model but assumes that no spatial correlations develop in the lattice. Next, the local-dispersal mean field theory is generalized into several varieties of local structure theory, in which one assumes that groups of nearby sites in the lattice are correlated, and tracks such correlations under the action of the cellular automaton rule. Assuming such local correlations allows one to predict patch occupancy as well as the degree of clustering in the cellular automaton model much more accurately than mean field theory, especially in parameter regimes where mean field theory does poorly. Simulation and mean field theory are seen to be two opposite extremes of an entire spectrum of methods that may be used to investigate discrete spatial models.

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