Dispersion in Spacio-Temporally Heterogeneous Environments

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Mischaikow 0107396 The goal of this project is to better understand the relationship between dispersal rates and the spatio-temporal heterogeneity of environments. In particular, the investigator would like to understand if there are any fundamental relationships upon which a framework for this theory can be developed. Therefore, to minimize extraneous effects the investigator continues to study what is perhaps the simplest continuous model that explicitly incorporates a spatial variable: a system of reaction diffusion equations with Lotka-Volterra reaction terms, where the reaction term of each species is identical, and where the birth rate is spatially and temporally heterogeneous. This model allows for a variety of perturbations through which one can study the impact of various factors in the evolution of dispersal rates. Of course, this model has several limitations. The first is the assumption that leads to the dispersal being represented as simple diffusion. The investigator wishes to understand the effects of taxis on the relationship between dispersal rates and spatial heterogeneity. On a more general level the investigator intends to replace diffusion by an integral kernel. That ecology and evolution are fundamentally influenced by the spatial characteristics of the environment is well accepted. As an example of this one may consider the paradox of diversity. Simple mathematical models that do not include any spatial component give rise to the principle of competitive exclusion: when two species compete for the same limited resource one of the species usually becomes extinct. On the other hand it is commonly observed that in a wide variety of habitats a multitude of species coexist. This can be explained, at least in part, by including spatial effects. Of course, once spatial components are introduced, dispersal rates become a central feature. Unfortunately, our understanding of cause and effect in this more general situation is poor. The reasons for this appears to be fourfold. First, the number of variables in realistic ecological and environmental models is enormous. Second, spatial heterogeneities occur at all scales of the environment. Third, obtaining precise data for these variables from field studies is extremely difficult. Finally, current mathematical techniques for handling models that incorporate both spatial and dynamical properties seem to be inadequate. Given this state of affairs, a simple model is thoroughly investigated in the hopes of elucidating the basic biological principles and identifying the fundamental mathematical issues.