Bifurcation Analysis Of Two Reaction Diffusion Predator-prey Systems

Population dynamics is an important branch of biomathematics,it is significant to the the-ory of ecology,especially it has widely used in the field of species protection.One of the fa-mous typical interactions among population is predator-prey relationship.This relationship has high research value and application value for the development of biological resources,the management of renewable resources,the protection of ecological environment and the eco-nomic benefits obtained in fishery,forestry and wild resources.So all of these make us to im-prove and prefect the dynamic behavior of predator and prey on the basis of traditional preda-tor-prey model.In this paper,two kinds of reaction-diffusion predator-prey systems are considered,and some bifurcation analysis are done mathematically.The main work is as follows:1.For a predator-prey model with prey-dependent functional response,by linearizing the system,the characteristic equation and the characteristic roots of the system are obtained.Then we get the conclusions about Turing instability and Hopf bifurcation of the coexisting equilib-riums.Combined with the above work,the growth rate and diffusion coefficient of predator are selected as bifurcation parameters to study Turing-Hopf bifurcation.In addition,we obtain a normal form for the Turing-Hopf bifurcation.Our results demonstrate that the predator-prey model can exhibit complex spatiotemporal dynamics,including spatially homogeneous period-ic solutions,spatially inhomogeneous periodic solutions,and spatially inhomogeneous steady-state solutions.Finally,we use numerical simulations to illustrate the theoretical results.2.For a predator-prey model with predator-dependent functional response,we first discuss whether the system exists positive equilibriums or not in order to satisfy corresponding biolog-ical significance,and then obtain sufficient conditions for the stability of the positive equilibri-ums in the local region.Second,we study the Turing instability and Hopf bifurcation,and the corresponding conclusions are proved.Then,we choose ?,d1 as bifurcation parameters to discuss the existence of Turing-Hopf bifurcation.In addition,we use the methods of center manifold and normal form to reduce dimensionality and simplification,so that the normal form of Turing-Hopf bifurcation is calculated.Finally,numerical simulations are used to illustrate the theoretical results.