Reaction-diffusion systems involving cross-diffusions.

摘要

This thesis is devoted to study Reaction-Diffusion systems with cross-diffusions arising from mathematical biology and mathematical ecology. In particular, we study models of two types, Keller-Segel and Lotka-Volterra, which are used to model Chemotaxis and Species competition respectively. Questions of our interests include, but are not limited to, whether or not the systems allow global-in-time solutions, what are the mechanisms of cross-diffusions on the formations nonconstant solutions, and under what scenarios do we expect that the nonconstant solutions have striking structures, like spikes, layers, etc. We could then use these structures to interpret interesting biological and/or ecological phenomena, such as, cell aggregations, species segregation, etc.;Through semigroup theories, we establish the existence of global solutions to a modified Keller-Segel system, provided that the chemotactic coefficient is not too large. Furthermore, we construct a solution to the corresponding stationary system that has a boundary spike for small chemical diffusion rate. In particular, this boundary spike is supported on a platform and it approaches the most curved part of the boundary if the chemical diffusion rate shrinks to zero.;For the Lotka-Volterra competition system with cross-diffusions, we establish nonconstant positive solutions through bifurcation theories by taking one of the cross-diffusion rates as the bifurcation parameter. The advantage of bifurcation analysis is that we can find the exact range of the parameter that tends to create nontrivial solutions. It turns out that if the cross-diffusion is sufficiently large this system can be approximated by two shadow systems. We also obtain nonconstant positive solutions for the first shadow system and then proceed to show that it generates solutions with boundary layers in a 1D domain. Furthermore, we give formal descriptions of solutions to the second system that allow transition layers in 1D if one of the diffusion rates goes to zero.