Spatial structure and transient periodicity in biological dynamics.

Abstract

Structure, in its many forms, is a central theme in theoretical population ecology. At a mathematical level, it arises as nonuniformities in the topology of nonlinear dynamical systems. I investigate a mechanism wherein a chaotic time series can have episodes of nearly periodic dynamics interspersed with more 'typical' irregular dynamics. This phenomenon frequently appears in biological models, and may explain patterns of alternating biennial and irregular dynamics in measles epidemics. I investigate the interaction between spatial structure and density-dependent population regulation with a simple model of two logistic maps coupled by diffusive migration. I examine two different consequences of spatial structure: scale-dependent interactions ("nonlocal interactions") and spatial variation in resource quality ("environmental heterogeneity"). Nonlocal interactions allow three general dynamical regimes: in-phase, out-of-phase, and uncorrelated. With environmental heterogeneity, the dynamics of the total population size can be approximated by a logistic map with the mean growth parameter of the two patches; the dynamics within a single patch are often less regular. Adding environmental heterogeneity to non-local interactions has little qualitative effect on the dynamics when the differences between patches are small; when the differences are large, uncorrelated dynamics are most likely to be seen, and there are interesting consequences for the stability of source-sink systems. A third type of structure arises when individuals differ from one another. Accurate prediction of extinction risk in small populations requires that a distinction be made between demographic stochasticity (variation among individuals) and environmental stochasticity (variation among years or sites). I describe and evaluate two tests to determine whether all the variation in population survivorship can be explained by demographic stochasticity alone. Both tests have appropriate probabilities of type I error, unless the survival probability is very low or very high. Small amounts of environmental stochasticity are often not detected by the tests, but the hypothesis of demographic stochasticity alone is consistently rejected when environmental stochasticity is large. I also show how to factor out deterministic sources of variability, such as density-dependence. I illustrate these tests with data on a population of Acorn Woodpeckers.