Coexistence And Asymptotic Behavior Of Two Kinds Of Biological Models

Lotka-Volterra model and chemostat model are two kinds of the most significant models in Mathematical biology. Lotka-Volterra model is the nuclear contents of population dynamics. This model plays a very important role in ecology, especially in protection of plants and creatures and in the control and exploiture of environment. The chemostat is a laboratory apparatus used for the continuous culture of microorganisms. It is used as an ecological model of a simple lake, as a model of the growth of unicellular phytoplankton in lake and sea. Moreover, it has been widely applied to the commercial production of microorganisms, biological pharmacy, food manufacture and the management and prediction of the ecology system, particularly the marine ecology, and the control of the environment pollution.In the light of the recent work in these two kinds of biological models, mainly using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of reaction-diffusion equations and corresponding elliptic equations, we have systematically studied the dynamical behavior of the unstirred chemostat model with inhibitor and Lotka-Volterra model with nonmonotonic conversion rate, such as coexistence, multiplicity, stability of positive steady states and the longtime behavior of species. The tools used here include super-sub solutions method, comparison principle, monotone system theory, global bifurcation theory, fixed-point theory of topology, Lyapunov-Schmidt procedure and perturbation technique. The main contents and results in this dissertation are as follows:i) The standard unstirred chemostat model is studied. The global attractivity of the positive steady-state solutions of the original system is established by the comparison principle and super-sub solutions method. Moreover, the effects of the growth rate on the unique positive equilibrium of the single population model are studied in detail by means of super-sub solutions method, Sobolev embedding theorem and the properties of eigenvalues.ii) An unstirred chemostat model with an internal inhibitor is discussed. First, the elementary stability and asymptotic behavior of solutions of the system are determined. The existence of positive steady-state solutions is given by monotone system theory. Second, the effects of the inhibitor are considered carefully by making use of the degree theory,...