Dispersion in Spacio-Temporally Heterogeneous Environments

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Mischaikow0107396 The goal of this project is to better understand therelationship between dispersal rates and the spatio-temporalheterogeneity of environments. In particular, the investigatorwould like to understand if there are any fundamentalrelationships upon which a framework for this theory can bedeveloped. Therefore, to minimize extraneous effects theinvestigator continues to study what is perhaps the simplestcontinuous model that explicitly incorporates a spatial variable:a system of reaction diffusion equations with Lotka-Volterrareaction terms, where the reaction term of each species isidentical, and where the birth rate is spatially and temporallyheterogeneous. This model allows for a variety of perturbationsthrough which one can study the impact of various factors in theevolution of dispersal rates. Of course, this model has severallimitations. The first is the assumption that leads to thedispersal being represented as simple diffusion. The investigatorwishes to understand the effects of taxis on the relationshipbetween dispersal rates and spatial heterogeneity. On a moregeneral level the investigator intends to replace diffusion by anintegral kernel. That ecology and evolution are fundamentally influenced bythe 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 spatialcomponent give rise to the principle of competitive exclusion:when two species compete for the same limited resource one of thespecies usually becomes extinct. On the other hand it is commonlyobserved that in a wide variety of habitats a multitude ofspecies coexist. This can be explained, at least in part, byincluding spatial effects. Of course, once spatial components areintroduced, dispersal rates become a central feature.Unfortunately, our understanding of cause and effect in this moregeneral situation is poor. The reasons for this appears to befourfold. First, the number of variables in realistic ecologicaland environmental models is enormous. Second, spatialheterogeneities occur at all scales of the environment. Third,obtaining precise data for these variables from field studies isextremely difficult. Finally, current mathematical techniques forhandling models that incorporate both spatial and dynamicalproperties seem to be inadequate. Given this state of affairs, asimple model is thoroughly investigated in the hopes ofelucidating the basic biological principles and identifying thefundamental mathematical issues.