Bifurcation Problems And Turing Instability In A Class Of Predator-Prey Systems

It is well known that predator-prey systems are one of the important tools describing the real world and studying these systems has important to understand the real world. Based on this fact, in this thesis we mainly consider bifurcation problems and Turing instability of a class of predator-prey systems.Firstly, in the absence of diffusion effects, the stability and Hopf bifurcation of periodic solutions of the predator-prey system are discussed. The sufficient conditions ensuring that the positive equilibrium is asymptotically stable and the conditions guaranteeing that the system can bifurcate periodic solutions from the positive equilibrium are established. In addition, an explicit algorithm determining the direction of Hopf bifurcation, the stability of bifurcated periodic solutions is given. Meanwhile, some numerical simulations are also included.Secondly, in the presence of diffusion effects, the bifurcations and Turing instability of the corresponding diffusion system are studied. The stability of positive equilibrium is studied and the conditions under which the system undergoes a Hopf bifurcation of periodic solutions are obtained. Furthermore, by using the normal form theory and the center manifold reduction for partial differential equations, an explicit algorithm determining the direction of Hopf bifurcation of periodic solutions, the stability and period of bifurcated periodic solutions is given and the sufficient conditions ensuring that the bifurcated periodic solutions are orbitally asymptotically stable and unstable on the center manifold are also obtained. Especially, we noticed that the diffusion had a notable impact on the stability of the predator-prey system. By analyzing the bifurcation phenomena of the diffusion system and comparing with the corresponding system without diffusion, we find that under Neumann boundary condition the corresponding diffusion system can appear more bifurcation phenomena and obtain some more new conditions and conclusions regarding bifurcation phenomena and Turing instability. Meanwhile, to verify the theoretical conclusions obtained in this part, some numerical simulations are also included.