Hopf Bifurcation And Steady-state Solutions Of Reaction-diffusion Predator-prey Systems

Reaction-diffusion equations can be used to describe the natural phenomena of physics, chemistry, ecology and medicine etc. and has become an essential part in the study of modern mathematics. Hopf bifurcation is an important subject in the study of long time behaviors of reaction-diffusion systems (equations). Since the long time behaviors of reaction-diffusion systems have a close relationship with the corresponding steady-state problems, studies on the non-constant steady-state solutions and it’s qualitative properties have an important significance both in theory and reality.Firstly, we study a diffusive Holling-Tanner predator-prey model subject to ho-mogeneous Neumann boundary condition. By choosing the ratio of intrinsic growth rates of predator to prey A as the bifurcation parameter, the model undergoes spa-tially homogeneous and inhomogeneous Hopf bifurcation when A crosses through a sequence of critical values, i.e., spatially homogeneous and inhomogeneous periodic solutions will bifurcate from the positive constant equilibrium. In addition, we con-sidered bifurcation of non-constant steady-state solution both for simple eigenvalues and double eigenvalues. If the former, a smooth curve of positive solution of the system bifurcates from the positive constant equilibrium when A crosses through a sequence of steady-state bifurcation values, and the curve is contained in a glob-al branch of the positive solutions. If the latter, the existence of local bifurcation of steady-state solution is proved by using space decomposition and the implicit-function theorem.Secondly, we are concerned with a diffusive three-species Lotka-Volterra food chain model with homogeneous Neumann boundary condition. Xie [128] proved that the positive constant equilibrium is globally asymptotically stable, which im-plies that the model does not exhibit Hopf bifurcation and the corresponding elliptic system can’t create non-constant steady-state solutions. We considered time delay r in the model, by taking τ as the bifurcation parameter, the model undergoes patially homogeneous and inhomogeneous Hopf bifurcation when r crosses through a sequence of critical value, which means that spatially homogeneous and inhomo-geneous periodic solutions will bifurcate from the positive constant equilibrium. In addition, sufficient conditions for global asymptotic stability of the positive constant equilibrium are obtained, and an algorithm determining the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcated periodic solutions is given.Thirdly, we studied the effects of cross-diffusion on the above diffusive food chain model. Taking parameter p which is contained in cross-diffusion terms as bi-furcation, the model undergoes only spatially inhomogeneous Hopf bifurcation when p crosses through critical value p*, i.e., spatially inhomogeneous periodic solution will bifurcate from the positive constant equilibrium. In addition, by using the Leray-Schauder degree theory, the existence of non-constant steady-state solutions are obtained.Finally, we investigated bifurcation of positive solutions for the above diffusive food chain model with homogeneous Dirichlet boundary condition. By regarding the intrinsic growth rate of prey as bifurcation parameter, we proved that con-tinuum of positive of the corresponding elliptic system bifurcated from both the semi-trivial solution (ulri,O,u3r3) and (0,u2,u3) and the continuum tends to∞in ExX. Moreover, we constructed a asymptotic expression of positive solution by using the implicit function theorem.