Steady State And Bifurcation Of Some Predator-prey Models With Reaction And Diffusion

Reaction-diffusion equations are widely used to investigate many kinds of problems in physics, chemistry, and biology. The contents and methods of investigations are diverse, and the results have important theoretical and practical significance. The steady state solution is one of the basis investigations. Taking into account the close relationship between the long time behavior of the solutions to reaction-diffusion equations and its steady state solution, the research of the steady state solution to reaction-diffusion equations is very necessary. The thesis focuses on stationary pattern and Hopf bifurcation of some predator-prey models with reaction-diffusion and actual backgrounds. Biologically speaking, Hopf bifurcation can reflect the periodic variation of populations.Firstly, we propose a general Schnakenberg model and consider the impact of selfdiffusion coeffcients on steady state solutions. By the maximum principle of elliptic equations, we obtain a priori estimates(i.e., positive upper and lower bounds) for the positive steady state solutions of the model. By energy estimates and implicit function theorem, we conclude the non-existence of non-constant positive steady state solutions.Existence of non-constant positive steady state solutions is deduced by degree method.The conclusions of the model indicate that self diffusions can create stationary pattern.Secondly, we deal with a three species predator-prey model with cross diffusion. We focus on the existence of non-constant positive steady state solutions of the model. By virtue of the maximum principle and Harnack inequality, we get a priori estimates for the positive steady state solution. Using degree theory and bifurcation theory, we prove the existence and the bifurcation phenomenon of non-constant positive steady state solutions dependent of cross diffusion coeffcient. The conclusion indicates that only cross diffusion can create stationary pattern for the model.Finally, a diffusive predator-prey model with Holling II prey harvesting is considered. For the problem with the homogeneous Neumman boundary condition, we deduce the stability of constant equilibria, especially the global asymptotical stability of a positive constant equilibrium by iterative technique. We also give the long behavior of the solutions to the model. Choosing the parameters concerning with prey harvesting as Hopf bifurcation parameter, we obtain the existence of Hopf bifurcation. Using the center manifold theorem and normal form theory, we compute the expressions of the determination of direction and stability. Numerical simulations are carried out to illustrate our results. The existence of non-constant positive steady state solution is also discussed. The conclusions show that self diffusion can create stationary pattern, and the introduction of prey harvesting can create the periodic variation of populations. For the problem subject to the homogeneous Dirichlet boundary condition, we study the existence of coexistence steady state solutions by the upper and lower solutions method. And we obtain the existence, asymptoticity, stability and bifurcation of coexistence steady state solutions using degree theory. For the delayed predator-prey model with reaction and diffusion, using the upper and lower solutions method, we get the global asymptotical stability of positive constant equilibrium. By analyzing the corresponding characteristic equations respectively, we give the suffcient conditions of the emergence of Hopf bifurcation of functional differential equations and reaction-diffusion equations, and carry out the analysis of Hopf bifurcation.