Dispersion in Spacio-Temporally Heterogeneous Environments

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.