Beyond Diffusion: Conditional Dispersal in Ecological Models

Abstract

Reaction-diffusion models have been widely used to describe the dynamics of dispersing populations. However, many organisms disperse in ways that depend on environmental conditions or the densities of other populations. Those can include advection along environmental gradients and nonlinear diffusion, among other possibilities. In this paper I will give a survey of some models involving conditional dispersal and discuss its effects and evolution. The presence of conditional dispersal can strongly influence the equilibria of population models, for example by causing the population to concentrate at local maxima of resource density. The analysis of the evolutionary aspects of dispersal is typically based on a study of models for two competing populations that are ecologically identical except for their dispersal strategies. The models consist of Lotka-Volterra competition systems with some spatially varying coefficients and with diffusion, nonlinear diffusion, and/or advection terms that reflect the dispersal strategies of the competing populations. The evolutionary stability of dispersal strategies can be determined by analyzing the stability of single-species equilibria in such models. In the case of simple diffusion in spatially varying environments it has been known for some time that the slower diffuser will exclude the faster diffuser, but conditional dispersal can change that. In some cases a population whose dispersal strategy involves advection along environmental gradients has the advantage or can coexist with a population that simply diffuses. As is often the case in reaction-diffusion theory, many of the results depend on the analysis of eigenvalue problems for linearized models.

Mathematics Subject Classification (2010): Primary 92D; Secondary 35K