Diffusion in Heterogeneous Environments

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In this proposal, using the classical diffusive Lotka-Volterra competition systems as one of the basic examples for illustrations, Professor Wei-Ming Ni intends to systematically study the interactions between diffusion and spatial variations, with particular emphasis on the effects of varying diffusion rates - including general shadow systems; that is, the limiting systems when one of the diffusion rates tends to infinity. Moreover, to handle ecologically more realistic models, directed movements as well as cross-diffusion systems, created by either spatial heterogeneity or inter-specific population pressures, will be investigated. Spatial variations reflect inhomogeneous or non-uniform distributions of resources in our environments. In this proposal, Professor Wei-Ming Ni will first explore the roles of different dispersal rates and spatial variations, as well as their interactions, in population dynamics of two or more competing species. Then, directed movements, due to spatial heterogeneity and/or inter-specific population pressures, will be studied in order to understand mathematically the basic mechanisms involved, and thereby enhance the ability of ecologists in modeling ecologically more realistic situations.