Qualitative Analysis Of Solutions Of Predator-Prey Models With Two Different Functional Responses And Diffusion

In past decades, the researches of prey-predator models in the ecologyhave got some good results, and many new results of prey-predator system with re-sponse function are obtained. At present, which people utilize the reaction-diffusionequation theory to study the mathematical model in the ecology has become a quitepopular topic. The typical mathematical model in two-dimensional prey-predatorsystem is the Volterra-Lotka model, as follows:whereΩ(?)Rⁿ(n≥0) is a bounded domain with smooth boundary (?)Ω;ΔisLaplace operator; u and v, respectively, denote populaton densities of prey andpredator species, inhabit the domainΩ; a_i, b_i, c_i, D_i(i=1,2) are positive constants.the form of boundary parameter B is(?)u/(?)v is the unit outnormal direction derivatives, b₀(x)≥0(x∈(?)Ω). More atten-tion on above system is whether two species coexists or one species is persistentexistence but another species is extinct. Analyzing from mathematics angle, i.e.,solutions (u, v) are constantly positive, either u→0 or v→0, as t→+∞. Clearly,coexistent problems and the existence of positive solutions of equilibrium state sys-tem have close correlation, The relation between gradual behavior of solutions andthe properties of the steady-state solution such as stability is close. Therefore, theimportance of researchs of steady-states system become more prominent, such asexistence, stabilty of solutions.In this article, on the base of above general Volterra-Lotka model, prey-predatormodels with Holling Typeâ…¢reaction-diffusion function or Beddington-DeAngelis reaction-diffusion function, we will respectvely discuss following three specific prey-predator models:whereΩ(?)Rⁿ(n≥0) is a bounded domain with smooth boundary (?)Ω; parametersin above models are positive,αu²v/(β²+u²) is Holling Typeâ…¢reaction-diffusion function.uv/(α+bu+cv) is Beddington-DeAngelis reaction-diffusion function.The whole thesis is made up of three sections to invetigate differently properitiesof solutions for the three prey-predator model.In the first part, we discuss the properties of solutions of (0.1) system with dif-ferent diffusion coefficient and the second boundary condition. The stability of non-negative constant solutions is given, some prior-estimate of the positive steaystateare proved by using the maximum principle and lower-upper solutions, the non-existence of non-constant positive solution, the global exitence of non-constant pos-itive solution and bifurcation solution of non-constant positive steady-states areobtained.In the second part, the property of steady-states solutions of (0.2) system isdiscussed. By methods of calculating the index of fixed points of compact maps incones, in combinations with homotopy arguement, the existence and non-existenceof coexistence states are discussed. Moreover, the local existence of bifurcationsolutions, uniqueness and stability are obtained. In the third part, the properties of the steady-states solutions of prey-predatorwith Beddington-DeAngelis response and Drichilet boundary conditions are dis-cussed. Some prior-estimate of solutions are given by comparison principle, Applyingthe local bifurcation theory and the global bifurcation theory, sufficient conditionsfor the existence of local bifurcating solutions of the steay-states of (0.3) and itsstability are obtained, the existence of global bifurcating solution and its jump arealso proved.