Metapopulation Dynamics of Multiple Species: The Geometry of Competition in a Fragmented Habitat

Populations of species living within a spatially fragmented habitat may be affected by interspecific competition on a regional time scale. Metapopulation theory is a potentially powerful tool for understanding the dynamics of such competing species, but metapopulation models with competition are often too complex to be useful as guides for empirical investigation. In this paper I employ two techniques from dynamical systems theory (perturbation expansion and the examination of invariant manifolds) in a qualitative and quantitative analysis of two-species competition within a metapopulation, with the goal of using geometric ideas to provide insight into how competition affects species living in a fragmented habitat. Four-state metapopulation competition models (where habitat patches may be occupied by either of two competing species, both, or neither) can be analyzed as perturbations of simpler three-state models (those without doubly occupied patches as one of the state variables). Symmetric competition for patches is always unstable in three-state Levins-type metapopulations, but generally stable in analogous mainland–island systems. The attracting manifold of a three-state system has the form of an invariant curve joining the two species' Levins equilibria. Under asymmetric competition for patches, coexistence is possible only if the inferior competitor possesses a higher equilibrium proportion of occupied patches in the absence of competition. Asymmetric competition changes the nature of the attracting manifold, with the limit case a curve that is locally tangent to a line parallel to the axis for the proportion of occupied patches of the inferior competitor. The interior equilibrium of a four-state model can be calculated as a perturbation expansion in a parameter α, the fraction of doubly occupied patches that contribute to recolonization. The perturbation method in conjunction with center manifold analysis proves that coexistence of like species within a metapopulation is possible if and only if there is recolonization from doubly occupied patches. Rescue effects may make competitive coexistence impossible via the creation of a saddle-point equilibrium having one-dimensional stable and unstable manifolds. Application of the theory to published data from several species of Daphnia in Fennoscandia shows good quantitative agreement with observations, and makes several new qualitative predictions. The results are discussed in connection with ecological dynamics of similar forms, including those in the process of speciation.