Study On The Dynamics Behavior Of The Reaction-diffusion Predator-prey System

The research of dynamics of reaction–difusion predator–preysystem plays an important role on the population ecology. In thisthesis, by using the theory of bifurcation and diferential equationsand via numerical simulations, we investigate dynamical behaviorsof a reaction–difusion predator–prey system, including the local andglobal asymptotic stability of the equilibrium, Turing pattern for-mation, the existence and direction of Hopf bifurcation around thepositive equilibrium. There are four aspects in this thesis.In chapter1, we give a brief survey on the developments ofreaction–difusion predator–prey systems, and introduce the back-ground of our problems and the main contents.In chapter2, we investigate the complex dynamics induced byAllee efect in a spatial predator-prey model. In the case without Alleeefect, there is nonexistence of difusion-driven instability. While inthe case with Allee efect, the positive equilibrium can be unstable un-der certain conditions. This instability is induced by Allee efect anddifusion together. Via numerical simulations, the model dynamicsexhibits both Allee efect and difusion controlled pattern formationgrowth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures,spots replication.In chapter3, with homogeneous Neumann boundary conditions,we give the dynamical analysis of a delayed reaction-difusion predator-prey system, involve the stability of the nonnegative equilibria andthe existence of Hopf bifurcation by analyzing the characteristic equa- tions. The direction of Hopf bifurcation and the stability of bifurcat-ing periodic solution are also discussed by using the normal formtheory and the center manifold reduction.In the last chapter, some conclusions and discussions are given.