Dynamics To Some Reaction-Diffusion Models From Biology

This thesis deals with dynamics of solutions to some reaction-diffusion models from biology. The topics include the existence and the stability of steady states, the persistence of solutions for two un-stirred chemostat models, and the long-time behavior of solutions for a three-species predator-prey model with cross-diffusion, as well as a Keller-Segel model for chemotaxis with prevention of overcrowding. Firstly, we consider a multiple food chain model for two resources in un-stirred chemostat. We derive the existence of the positive steady solutions and the persistence of solutions.Secondly, we concern a competition model for two resources in un-stirred chemostat to prove the conditions for the existence of positive steady solutions and the persistence for solutions. Thirdly, we study the long-time behavior of solutions for a three-species predator-prey model with cross-diffusion. We prove the existence of a global attractor to the system. Finally, we investigate a Keller-Segel model. We obtain the decay and the long-time behavior of solutions to the system.Chapter 1 is to summarize the background of the related issues and to briefly intro-duce the main results of the thesis.Chapter 2 is concerned with a multiple food chain model for two resources in un-stirred chemostat. Firstly, Combining with the fixed point theory and the eigenvalue theory, we get conditions for the existence of positive solutions, and then find the global structure of positive steady solutions under some conditions by the global bifurcation theory. Meanwhile, we study how the parameters affect the extinction or persistence of the predator and prey, and how the prey affect the predator.Chapter 3 is devoted to a competition model for two resources in un-stirred chemo-stat. On the basis of the existence of positive solutions, we study the global attracting conditions for one of the populations, and derive the extinct and persistent conditions for the two (or one) populations. Furthermore, regarding the diffusion rates as parameters, we consider the stability, persistence and asymptotic behavior for the two populations. We find that the two populations will go to extinct when both possess large diffusion rate. If just one of them spreads fast with the other one diffusing slower, then the related semitrivial steady state will be global attracting. Chapter 4 deals with large-time behavior of solutions for a three-species predator-prey model with cross-diffusion. By using the infinite-dimensional dynamical systems theory and the a priori estimate method, we prove that the system admits a global attractor if m,l≥2. This extends the known results on the global existence of solutions.Chapter 5 studies a Keller-Segel model for chemotaxis with prevention of overcrowd-ing. We prove a crucial decay result for arbitrary diffusion rateε>0, and then obtain the asymptotic behavior of solutions, where the stronger assumptionε>1/4 (or∫nρ0(x)dx≤Co(ε, n)) in the related literature is unnecessary any more.