Dynamics of Chemotaxis Models in Heterogeneous Environments

Abstract

Chemotaxis describes the oriented movements of biological cells or organisms in response to chemical gradients in their environments and is crucial for many aspects of behaviour such as the location of food sources, avoidance of predators and attracting mates, slime mold aggregation, tumor angiogenesis, and primitive streak formation. Chemotaxis is also crucial in macroscopic process such as population dynamics and gravitational collapse. In 1970, Keller and Segel introduced a celebrated mathematical model to describe chemotaxis. Since then a tremendous effort has been dedicated to understand the classical chemotaxis model and its various variants. But there are still a lot of open interesting problems in the understanding of chemotaxis models. In particular, to the best of our knowledge, there has been no study of chemotaxis models in heterogeneous environments. This dissertation aims to study the dynamics of chemotaxis models of both one and two species in bounded heterogeneous environments.

Regarding chemotaxis models of one species in heterogeneous environments, we first investigate and prove the local existence and uniqueness of classical solutions. Next under some natural conditions on the parameters, we prove the boundedness of classical solutions and the existence of positive entire solutions. Finally, under some further conditions on the parameters, we establish the uniqueness and stability of positive entire solutions. Our results on the existence, uniqueness and stability of positive entire solutions are new and original. Important new techniques have been established to prove those results.

Concerning chemotaxis models of two species in heterogeneous environments, we first find various conditions on the parameters which guarantee the global existence and boundedness of classical solutions. Next, we find further conditions on the parameters which establish the persistence of the two species. Furthermore, under the same set of conditions for the persistence of the two species, we prove the existence of coexistence states. We then prove the extinction phenomena in the sense that one of the species dies out asymptotically and the other reaches its carrying capacity as time goes to infinity. Finally, we study the asymptotic dynamics of two species competition systems with/without chemotaxis in heterogeneous media and find conditions on the parameters for the uniqueness and stability of positive coexistence states of such systems. The persistence in general two species chemotaxis systems is studied for the first time. Several important techniques are developed to study the persistence and coexistence of the two species chemotaxis systems. Many existing results on the persistence, coexistence, and extinction on two species competition systems without chemotaxis are recovered. The established results on the asymptotic dynamics of two species competition systems are new even for the two species competition systems without chemotaxis but with space dependent coefficients.