Reaction-diffusion-advection models for the effects and evolution of dispersal

Chris Cosner

doi:10.3934/dcds.2014.34.1701

Abstract

This review describes reaction-advection-diffusion models for the ecological effects and evolution of dispersal, and mathematical methods for analyzing those models. The topics covered include models for a single species, models for ecological interactions between species, and models for the evolution of dispersal strategies. The models are all set on bounded domains. The mathematical methods include spectral theory, specifically the theory of principal eigenvalues for elliptic operators, maximum principles and comparison theorems, bifurcation theory, and persistence theory.

Highlights

Dispersal is an important aspect of the life histories of many, perhaps most, organisms
The ecological effects of dispersal create selective pressures that influence the evolution of dispersal strategies
There are many approaches that have been used to model the process of dispersal and its ecological effects and evolution

Summary

Introduction

Dispersal is an important aspect of the life histories of many, perhaps most, organisms. Reaction-diffusion, population dynamics, dispersal, ecology, evolution, bifurcation, persistence theory. Reaction-advection-diffusion models can explain how the dispersal behavior of individuals influences the ecological interactions of populations, and in turn how those ecological interactions impose forces of evolutionary selection on the behavior of individuals. This review will discuss the derivation, analysis, and interpretation of such models It will be focused on the dynamics of populations inhabiting finite regions. The topics treated will include derivations of models from assumptions about individual movement behavior, mathematical background on methods for analyzing the models, and results about various specific ecological and evolutionary questions. The material is organized into sections on single species models, models addressing ecological issues for interacting species, and models addressing issues related to the evolution of dispersal. The last section gives a brief description of some alternatives to reaction-advection-diffusion models, namely discrete diffusion models and intergro-differential models

Single species models

Interacting species–ecological issues

Interacting species–evolutionary issues