Spatial Structure, Environmental Heterogeneity, and Population Dynamics: Analysis of the Coupled Logistic Map

Abstract

Spatial extent can have two important consequences for population dynamics: It can generatespatial structure, in which individuals interact more intensely with neighbors than with more distant conspecifics, and it allows forenvironmental heterogeneity, in which habitat quality varies spatially. Studies of these features are difficult to interpret because the models are complex and sometimes idiosyncratic. Here we analyze one of the simplest possible spatial population models, to understand the mathematical basis for the observed patterns: two patches coupled by dispersal, with dynamics in each patch governed by the logistic map. With suitable choices of parameters, this model can represent spatial structure, environmental heterogeneity, or both in combination. We synthesize previous work and new analyses on this model, with two goals: to provide a comprehensive baseline to aid our understanding of more complex spatial models, and to generate predictions about the effects of spatial structure and environmental heterogeneity on population dynamics. Spatial structure alone can generate positive, negative, or zero spatial correlations between patches when dispersal rates are high, medium, or low relative to the complexity of the local dynamics. It can also lead to quasiperiodicity and hyperchaos, which are not present in the nonspatial model. With density-independent dispersal, spatial structure cannot destabilize equilibria or periodic orbits that would be stable in the absence of space. When densities in the two patches are uncorrelated, the probability that the population in a patch reaches extreme low densities is reduced relative to the same patch in isolation; this “rescue effect” would reduce the probability of metapopulation extinction beyond the simple effect of spreading of risk. Pure environmental heterogeneity always produces positive spatial correlations. The dynamics of the entire population is approximated by a nonspatial model with mean patch characteristics. This approximation worsens as the difference between the patches increases and the dispersal rate decreases: Under extreme conditions, destabilization of equilibria and periodic orbits occurs at mean parameter values lower than those predicted by the mean parameters. Apparent within-patch dynamics are distorted: The local population appears to have the wrong growth parameter and a constant number of immigrants (or emigrants) per generation. Adding environmental heterogeneity to spatial structure increases the occurrence of spatially correlated population dynamics, but the resulting temporal dynamics are more complex than would be predicted by the mean parameter values. The three classes of spatial pattern (positive, negative, and zero correlation), while still mathematically distinct, become increasingly similar phenomenologically.