Dynamic Analysis Of Three Types Of Ecological Model With Diffusion

Ecological dynamics research has been a hot topic. The rich research methods and achievements not only bring the innovation of science and technology on the technical level, but also greatly improve the living environment and quality of life of human beings. With the development of the nonlinear analysis and theory of nonlinear partial differential equations (the theory of reaction diffusion equations, especially), and the intervention of the computer simulation technology, quantita-tive/qualitative research work on the ecological model has stepped into a new stage, more profounder and practical application value research has been made.By using the reaction diffusion equation theory and combining with numeri-cal simulation technology, this article will study the ecological dynamics of three types of ecological models with homogeneous Dirichlet boundary conditions. The main research contents include a priori estimates, existence, uniqueness, stability of positive equilibrium solution, and the long time behavior of time-dependent t posi-tive solutions. The methods involve the comparison principle, upper lower solution method, bifurcation theory, fixed point method, stability theory, theory of mono-tone dynamical systems and numerical simulation technology based on platform of MATLAB(?).Chapter1presents firstly the research background and present situation of the Variable-Territory prey predator model, Plant-pollinator population dynamics model as well as Spider-insect predation model. Next, the main works of this paper are introduced, including the research plan. Finally, some preliminaries are given, which will be useful to study in this paper.In chapter2, the properties of steady-state positive solutions of Variable-Territory predator-prey model are discussed in the one-dimensional case. Firstly, v/u is ex-tended continuously on the boundary to deal with the singularity of the system on the boundary. Meanwhile, a priori estimates of positive solutions and the necessary condition for the existence of positive solutions are given. Then, by using bifurca- tion theory, the existence of positive solutions is considered. Moreover, under the continuous extension conditions, the global bifurcation curves are analyzed and a sufficient condition of the existence of positive solution is given. By applying to the stability theory, the stability of positive solutions is discussed. Furthermore, we es-tablish in detail the properties of positive steady-state solutions as m is large enough. Making use of the method of upper and lower solutions and regular perturbation theory, a sufficient condition of the existence of positive steady-state solutions is gained. Finally, the positive solutions of the Variable-Territory prey predator model are simulated. The effect of main parameters on the positive solutions is summa-rized. Combining with the theoretical research, some conclusions with biological background are provided.In chapter3, we study the Variable-Territory prey predator model in the mul-tidimensional case. We handle the singularity of the system on the boundary by adding a small constant. First, a priori estimates of positive steady-state solu-tions and the necessary conditions for the existence of positive solutions are given. Similar to the chapter2, using the bifurcation theory and the stability theory, the existence and stability of positive solutions, as m is large enough the existence of the positive steady-state solution, are investigated. Then dynamic analysis for Variable-Territory predator-prey model is given by the numerical simulation. We evaluate the two methods by comparing with the conclusions of the second chapter.The fourth chapter studies the Plant-pollinator population dynamics. Firstly, a priori estimates of positive solutions, the necessary condition of the existence of positive solutions are investigated. Secondly, using the theory of homotopy opera-tor fixed point index, a sufficient condition for the existence of positive solution is given. Especially, when a is sufficiently small, the existence, uniqueness and global attraction of the positive steady-state solution are established by the classical theo-ry of the monotone dynamical system and perturbation theory. Finally, the effects of γ,δ, α on the system are analyzed by using numerical simulation. Furthermore some unstable positive solutions of system are found in numerical simulation. The analysis shows that the mode has unique and stable ecological coexistence results when the growth rate γ of plant, the death rate δ of pollinator, and benefit coefficient a of plants satisfy certain conditions. The fifth chapter studies the spider insect predation system. The necessary conditions for the existence of and a prior estimate of positive steady-state solutions are given by using principal eigenvalue property. Then the sufficient conditions for the existence of positive steady-states solutions are built by the fixed point index theory of the homotopy operators. Moreover, we use spatial decomposition technique combined with the implicit function theorem to research two parameter bifurcation solutions bifurcated from the (γ*,δ*,θβ,0,0), and the sufficient condition of the existence and uniqueness of positive solutions are given, in which we also detailedly analyze the rationality of the conditions and give the simulation parameters rules. We also study the long time behavior of positive solutions of system by using the method of upper and lower solutions and the comparison principle of Parabolic Equations. Finally, we verify the theoretical results by numerical simulation.