Global existence, persistence, spreading speeds, and forced waves in chemotaxis models

Abstract

Chemotaxis models are widely used to describe the movements of biological species or living organisms in response to certain chemicals in their environments. This dissertation is devoted to the study of various dynamical aspects of a parabolic-elliptic chemotaxis model in shifting environments and a parabolic-parabolic chemotaxis model with logistic source on the whole space.

Concerning parabolic-elliptic chemotaxis models in shifting environments, we study persistence, spreading speeds and existence of forced waves in two different shifting environments. In particular, in the case favorable environment and unfavorable environment are separated, we prove that if the shifting speed of the environment is large, the biological species with compactly supported initial distribution will die out in the long run; if the shifting speed of the environment is not large, the species will persist and spread along the shifting habitat at a fixed asymptotic spreading speed. We also prove that there is a forced wave with speed $c$ which coincides with the shifting speed of the environment connecting two points provided that $c$ is large. In the case favorable environment is surrounded by unfavorable environment, we show that if the generalized principle eigenvalue of the linearized system at the trivial solution is positive, the species will persist surrounding the good habitat; if the generalized principle eigenvalue is negative and the degradation rate of the chemical substance is large, the species will become extinct in the habitat. We also show that there is a forced wave connecting $(0,0)$ and $(0,0)$ with the speed agreeing to the shifting speed of the environment provided that the chemotactic sensitivity is sufficiently small and the generalized principle eigenvalue is positive. Some numerical simulations are also carried out in both cases. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.

Regarding parabolic-parabolic chemotaxis models with logistic source on the whole space, we first prove the local existence and uniqueness of classical solutions for given initial functions. We then prove the global existence and boundedness of classical solutions for given initial functions under the assumption that the logistic damping is large relative to the product of the chemotactic sensitivity and the production rate of the chemical substance. Next, we study the asymptotic behavior of the global classical solutions with strictly positive initial functions. We show that under further conditions on parameters, the nonnegative constant solution is globally stable in some sense. Finally, we investigate the spreading speeds of global classical solutions with nonempty compact supported initial functions and front like initial functions. We prove that the spreading speed of such global classical solutions of the parabolic-parabolic chemotaxis model with logistic source is the same as that of Fisher-KPP equation under the same assumption of the existence of global classical solutions. Note that when there is no chemotaxis, the parabolic-parabolic chemotaxis model reduces to famous Fisher-KPP equation. Hence, under the same assumption of the existence of global classical solutions, the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation. As a by-product of spreading speeds, we show that persistence phenomena occurs, that is, any globally defined bounded classical solution with strictly positive initial function is bounded below by a positive constant independent of its initial function when time is large.