A Prey - Predator-prey System With Stage Model Analysis

Recently, mathematical biology becomes a popular and widespread subject. A lot of mathematical models are established in various life phenomena, which is studied by using modern mathematical tools and produced many valuable results. As a result, these not only promote a development of a branch of mathematics disciplines, but also these mathematical results can be directly fed back to the phenomenon itself, contributing to the social production and pest prevention and control.This thesis is concerned with a predator-prey reaction-diffusion model with stage structure for the predator and its corresponding steady-state problemThe structure of this thesis is as follows.In chapter one, we introduce the background of predator-prey models and stage structure model and some works in the related field. We also state some basic concept and some classical results, such as eigenvalue problems, global bifurcation theorem and some definition of dynamical systems and persistence. These are the basic parts that will be very useful in the forthcoming chapters.In chapter two, the long time behavior of time-dependent solutions are consid-ered. Firstly, using the comparison principle on a single equation and quasi-increase equations, some estimates and existence of global solution are obtained, and semi-trivial steady-state is globally asymptotically stable under some certain condition; by analysing the eigenvalue value of linearization system, local asymptotic stability at the three constant steady-state is given. Secondly, the reaction-diffusion system has a global attractor by applying geometry theory of semi-linear parabolic equation and several important inequalities. Thirdly, by the theory of persistence of dynamical sys-tem, the robust persistence of the reaction-diffusion system is proved, and the positive constant steady-state is a global attractor under some conditions by the methods of Liapunov function.In chapter three, we deal with the existence of non-constant steady-states. Firstly, with the help of a maximum principle, the positive solutions has a prior upper bound estimates and then has a positive lower bound by applying the standard elliptic the-ory, and there is no non-constant positive solution for large diffusion by using the energy integral methods. Secondly, we give some spectrum theories of a class of ellip-tic operators, such as the range, kernel and the algebraic multiplicity of the operator, and then obtain a global bifurcation result for a more general model. Thirdly, using simple eigenvalue bifurcation theory, we obtain that the local bifurcation at the three constants steady-states the structure of the bifurcation solutions and the existence of the non-constant positive solutions. Fourthly, using global bifurcation theorem, we show that the global bifurcation can be occurred at the positive constant solution. In one dimension of the space various, we get a good description of the global bifurca-tion curve, and obtain the existence of non-constant positive solutions if the diffusion coefficient is less than a given constant.In chapter four, we do some numerical simulations for reaction-diffusion model and prove the global asymptotical stability of the corresponding ODE models under a special condition depending only on the parameters.