Types Of Ecological And Chemical Model Of Qualitative Analysis And Numerical Simulation

In recent decades, reaction diffusion equations are widely used in biological and chemical research. By establishing mathematical models, mathematicians use the rich mathematical theories and methods to study the problems of ecology and chemistry. Lotka-Volterra model and Belousov-Zhabotinskii reaction model are very important mathematical models in the field of ecology and chemistry. So they have been investigated extensively by many scholars and many significant research results have been gained.Based on the current researches of the above ecological and chemical models, mainly using the theories and methods of the reaction diffusion equations and the corresponding elliptic equations, we have systematically investigated two kinds of Lotka-Volterra models with Beddington-DeAngelis functional response and a kind of Oregonator model in the Belousov-Zhabotinskii reaction, such as the existence, multiplicity, uniqueness and stability of positive steady-states. The tools used here include comparison principle, upper and lower solution methods, fixed points in-dex theory, regularity theorem, bifurcation theory, stability theory and numerical simulation.The main content and structure of this paper are as follows:Chapter1introduces some research background and current situation of Lotka-Volterra model and Belousov-Zhabotinskii reaction model. Some basic theories that will be useful in the forthcoming chapter are given.In chapter2, a kind of Lotka-Volterra competition model with Beddington-DeAngelis functional response is studied. Firstly, by calculating the fixed point index, some sufficient conditions of existence of positive steady-state are obtained. Moreover, combined with the regularity theory and perturbation theory, the multi-plicity, uniqueness and stability of positive solution are discussed. The results show that when the intraspecific interference parameter a of the species υ is large enough, the model either has at least two positive solutions or has a unique positive solution which is asymptotically stable. When the interspecific interference parameter β of two species is sufficiently small, the model has a unique positive solution which is asymptotically stable. Furthermore, using bifurcation theory and stability theory of linear operator, the global trend and the local stability of the steady-state bifurca-tion solution are considered. We present some numerical simulations to verify and complement our theoretical analysis with the help of Matlab software.In chapter3, we consider a predator-prey model with Beddington-DeAngelis functional response subject to the homogeneous Neumann boundary condition. The local stability of the positive constant solution is discussed by the stability theory of linear operator. The global stability of the positive constant solution is proved by the iterative method. Using the energy integral method, the nonexistence of nonconstant positive steady-state is proved. It is shown that prey and predator cannot coexist when the diffusion coefficient d1or d2is large enough. Furthermore, by regarding d1as a bifurcation parameter and applying to local and global bifurcation theorem, the sufficient conditions of nonconstant positive steady-state are given. Finally, we make use of some numerical simulations to verify the theory results that have been obtained. Based on the numerical simulations, we put forward some conjecture on the property of nonconstant positive steady-state to the model,which guide the direction for the further study in the future.In chapter4, we still discuss the predator-prey model of the chapter3. The dif-ference is that we study on the equilibrium system under the homogeneous Dirichlet boundary conditions. Treating the intrinsic growth rate δ of the predator as bi-furcation parameter, by local bifurcation theory, the existence of local bifurcation solution to the model is gained. We prove that the local bifurcation solution can extend to a global one by means of the global bifurcation theorem and establish a sufficient condition of the existence of positive solution. The stability of local bi-furcation solution is proved by the stability theory and verified in combination with examples of numerical simulations.In chapter5, the Oregonator model is investigated in the homogeneous Neu-mann boundary condition case. By the stability theory of linear operator, the sta- bility of positive constant solution is discussed. By using bifurcation theory and regarding diffusion coefficient d1as bifurcation parameter, the local and global bi-furcation from the positive constant steady-state are investigated. The sufficient conditions of the existence of nonconstant positive steady-state solution are gained. It is shown that the model has nonconstant positive solution when diffusion coeffi-cient d1belongs to the open interval from0to bifurcation point d1j and is not equal arbitrary bifurcation point. By using numerical simulations, we analyze and verify the theoretical results of this chapter. Moreover, we also find some conclusions that have yet to be proved.