Global Bifurcation And Stability Of Steady-state Solutions Of The Three-species Prey-predator Model

In this master's dissertation, we mainly consider the co-existence states population ecology. Population ecology is an important branch of ecology. Mathematics is wildly applied and developed systematically in this branch of ecology. This subject involved in the study of population dynamical property and structure, and also involved in the study of the interaction of given populationand related population ecology. The mathematical model and methods based on this subject is not only improving the development of ecology, but also make effect on other field in mathematical biology. In recent years, many scholars have devoted themselves to the research of co-existence states, It's result is a guide of real life. Therefore, co-existence states of ecological models have attracted considerable attention. A number of studies have been made on the existence of steady-state solutions of the system of ecology. In fact, the problem on the existence of steady-state solutions of the system of ecology is the problem on the existence of positive solutions of elliptical system.This paper deals with the research a bout one type of population dynamic system-having functional reaction on a predator and two preys by means of locally and global bifurcation theory, we have proved the existence of non-trivial positive solutions and also given the local stability results for the coexistence states. This presentation is divided in four chapter:In Chapter 1, the background and history of current research situation about the related work and major work of this presentation are introduced. In Chapter 2, we introduce the background of mathematical ecology mode and some preliminaries. In Section 3.1 of Chapter 3 we regard c and a as a bifurcation parameter, respectively, and discuss the bifurcation solutions of (A₂) which are relative to the unique semi-trivial solution of the form (u_a, v_b, 0). We prove that, if the condition is satisfied, then two prey, a predator can co-exist, when c is bifurcation parameter, then the birth-rate of predator has the limitations. In Section 3.2, we assume conditions under which the predator - prey subsystem (u≡0) has a unique positive solution (0, (?),(?)), and using a as a bifurcation parameter, we obtain a continuum of positive solution of (A₂) emanating from (a^*,0,(?),(?)) tending to 00. In Section 3.3, we regard a and b as a bifurcation parameter, respectively, and discuss the bifurcation solutions of (A₂) which are relative to the unique semi-trivial solution of the form (0,0, w_c).Finally, in Chapter 4, we discuss the local stability of the bifurcation solutionsin the neighborhood of bifurcation points.