Properties Of Solutions For Predator-Prey Systems With Diffusion

Since partial differential equations (PDE) were used to describe biological regulations and phenomena, a great many scholars and specialists have been paying more attention to PDE and many new subjects have been established which have more reality backgrounds. One kind of classic models in the dynamics of biological populations is reaction diffusion equations, which is a kind of semi-linear parabolic equations, that isHere D(x, U) = diag(d1(x, U),…, dm(x, U)). The usage of reaction diffusion equations in the dynamics of biological populations mainly reflects that the ecological equations have a diffusion term. In ecological systems, biological populations spontaneously migrant or diffuse from locations of high concentration to locations of low concentration by their own diffused rates di for the sake of effects of space, foods and other competitive factors. A series of processes: species birth, growth or death; predator-prey, competition or cooperation among species, etc., can be presented in the reaction term. Therefore, studies on the coexistence, stability and persistence for predator-prey systems have very important practical significance to equilibrium of ecology, ecological environments preservation and even saving the rare and precious creature on the brink of extinction.The whole thesis is made up of three sections to investigate the properties of solutions for three kinds of predator-prey systems, respectively.One classical model of predator-prey systems in ecology is Volterra - Lotka model. As an extension, we get one kind of predator-prey systems with diffusion, and Modell is the steady-state equations of this model, i.e.Here Ω is a bounded domain in Rn(n ≥0) with sufficiently smooth boundary Ω. u,v are the densities of two predator-prey species, b(x) ≥ 0, 0 on Ω, and the functions satisfy < 0, > 0. With the usage of degree theory in [1], calculating indices of fixed points of compact maps in cones of [2-10] and combining with bifurcation theories[11-19], maximum principles[17][19], lower-upper solutions methods[20-22] and spectrum analysis of operators[23-24], we get some results of the coexistence solutions for Modell in the first section. The results can be seen extensions of [7] [23].Model2 is also concerned with steady-state equations of an extended class of predator-prey systems with diffusion, i.e.This model is less sophisticated than Model1, but we investigate it far more deeply. In the second section, by means of local bifurcation theories[17], we have proved the system generate bifurcation at the semi-trivial solutions (uo,0), (0, v0). Furthermore, using stability theory[16][17], we analyze the stability of the positive solutions near the bifurcating points and get ideal results. A fundamental question of the theory of reaction diffusion equations, stability, has been solved successfully.In addition, the fact that ecological systems may occur some fluctuations for the seasonal or environmental factors can be reflected on the equations by coefficients which effect the properties of solutions. For an instance, the various coefficients of Volterra -Lotka predator-prey systems[30] depend on both x and t modeling the fact that effects vary in both time and space; the periodicity of coefficients models seasonal fluctuations: occurring periodic solutions. In the third section, we explore a kind of two-dimension periodic predator-prey systems with diffusion, i.e. Mode/3By employing the theory of periodic parabolic operators [38], the estimates of Schauder[39], decoupling method[30] and bifurcation theory, we get the results of coexistence of time-periodic solutions. A necessary and sufficient condition for coexistence of positive periodic solutions is obtained. The result we get can be seen an extension of [30].In this paper, we partially bring new ideas in the model establishing and some methods of proof. We get Mode/1 and Mode/3 by reasonable extending the existing models. We give more general conditions for priori estimates of solutions in the fi...