On the Evolution of Dispersal in Patchy Landscapes

Abstract

To better understand the evolution of dispersal in spatially heterogeneous landscapes, we study difference equation models of populations that reproduce and disperse in a landscape consisting of k patches. The connectivity of the patches and costs of dispersal are determined by a $k\times k$ column substochastic matrix S, where $S_{ij}$ represents the fraction of dispersing individuals from patch j that end up in patch i. Given S, a dispersal strategy is a $k\times 1$ vector whose ith entry gives the probability $p_i$ that individuals disperse from patch i. If all of the $p_i$'s are the same, then the dispersal strategy is called unconditional; otherwise it is called conditional. For two competing populations of unconditional dispersers, we prove that the slower dispersing population (i.e., the population with the smaller dispersal probability) displaces the faster dispersing population. Alternatively, for populations of conditional dispersers without any dispersal costs (i.e., S is column stochastic and all patches can support a population), we prove that there is a one parameter family of strategies that resists invasion attempts by all other strategies.