Two Types Of Biological Models Of Equilibrium And Long-time Behavior

Chemostat model and Lotka-Volterra model are two kinds of the most significant models in mathematical biology. The two kinds of models play a very important role in microbiology and ecology, respectively. Therefore, it has aroused extensive concern to domestic and overseas experts and scholars, and many significant research results have been obtained. Base on the recent work on these two kinds of biological models, mainly applying the theory and method of reaction-diffusion equations and correspond-ing elliptic equations, we have profoundly investigated the equilibrium state and the dynamical behavior of two unstirred chemostat models with Beddington-DeAngelis functional response and two (cross-) diffusion Lotka-Volterra model with functional response. The main tools used in this thesis include super-sub solutions method, com-parison principle, monotone system of topology, Lyapunov-Schmidt procedure, per-turbation technique and persistence theory. The obtained results include coexistence, multiplicity, uniqueness, stability of positive steady-states and the longtime behavior of species.The main contents in this dissertation are as follows:Chapter 1 firstly introduce some research background and current stage of chemo-stat model and Lotka-Volterra model. Secondly, we list some primary theory such as fixed-point index theory, eigenvalue problem and bifurcation theory, etc.Chapter 2, a competition model between plasmid-bearing and plasmid-free or-ganisms in the unstirred chemostat is researched. Firstly, the sufficient condition of existence on the coexistence solutions are obtained by applying fixed-point index theory. Secondly, global structure of bifurcation solutions and local stability are inves-tigated by global bifurcation theory and eigenvalue perturbation technique. Thirdly, uniqueness, multiplicity and linear stability of coexistence solutions depending on pa-rameter ki(i=1,2) are discussed by fixed-point index theory, perturbation technique and linear stability method. Finally, the longtime behavior of the system is determined by monotone method.Chapter 3, the positive solutions of a Beddington-DeAngelis food chain unstirred chemostat model is investigated. Firstly, the existence and stability of the trivial,semi-trivial,strongly semi-trivial solutions of the model are obtained. Secondly, the index of the trivial,weakly semi-trivial,strongly semi-trivial solutions are calculated by fixed-point index theory. Finally, the sufficient condition of the existence to positive solutions are given. Chapter 4 investigate a strongly couple prey-predator model with Michelis-Menten functional response in a bounded domain under Dirichlet boundary condition. some priori estimates for steady-state solutions are obtained by maximum principle. The sufficient condition of the existence for positive solutions is given, next, the coexistence region of positive solutions is analyzed by implicit function theorem, intermediate value theorem and Dini's theorem. The relation of the coexistence regionΣand the parameters a and b is considered. Moreover, we can prove that the coexistence region∑spreads (narrows) asβ(α) increases. Finally, local bifurcation solution and global bifurcation solution are obtained by bifurcation theory.Chapter 5 deals with the positive solution and asymptotic behavior of a prey-predator model with predator saturation and competition. Firstly, global structure of bifurcation solutions and local stability are investigated by global bifurcation theory and eigenvalue perturbation technique. Secondly, the unique existence and stability of bifurcation solution which bifurcates from double multiplicity eigenvalue are proved by Lyapunov-Schmidt procedure. Thirdly, by fixed-point index theory and perturbation, we may show that if parameter d is sufficient small or m or k is sufficient large, then the model exist at least two positive solutions. Especially, if parameter m is sufficient large and a satisfies some condition, the model have exactly two positive solutions which belongs to two different types, one asymptotically stable and the other unstable. Finally, the asymptotic stability and persistence of positive solutions are obtained by monotone method and persistence theory.