Coexistence Analysis Of Two Biological Models With Diffusion

This paper discusses two types of biological dynamics models:a predator-prey model and a cooperative model. Coexistence in population ecology is an important branch of mathematical biology research, is also the most widely ecological research topic. By using the local bifurcation theory, global bifurcation theory, the compari-son principle, maximum principle and the method of upper and lower solutions, the existence of positive solutions of the elliptic equations corresponding to the parabolic equations are studied.In the first chapter we discusse the predator-prey model under the Dirichlet boundary conditions. Since the effects of the oncolytic virus on tumor cells can suppress or reduce the role of tumor cells, the oncolytic virus and tumor cells are regarded as the predator and prey respectively. Using the upper and lower solutions method, a priori estimate to the positive solution of this model is obtained. With the help of the local bifurcation theory and regarding the diffusion coefficient as the bifurcation parameter, the equilibrium solution branches emanating from the semi-trivial solutions are studied. By using the global bifurcation theory, the local bifurcation branch to the global is prolonged.The second chapter studies the cooperative model under the Dirichlet boundary conditions, where the parameters are the monotone and nondecreasing functions and are the periodic functions with time. First, using the upper and lower solutions method and the comparison principle, a priori estimate and the existence of the positive solutions are discussed. Next, the conditions of uniqueness of the positive solutions and the noexistence of the positive solutions are studied.