Bifurcation Analysis In Some Reaction-diffusion Predator-prey Systems

With the development of biological mathematics theory,reaction-diffusion system,as a kind of mathematical model describing the relationship between populations,has attracted extensive attention of scientific researchers.Studying the interaction between populations can effectively predict the num-ber of populations,so as to maintain the stability of the ecosystem.This dis-sertation mainly discusses the dynamics of several kinds of reaction-diffusion predator-prey systems.The main research contents are as follows:Firstly,the research status of several kinds of reaction-diffusion predator-prey systems is introduced,and the purpose and main contents of this paper are introduced.Secondly,a class of delayed diffusive predator-prey system with food limited and nonlinear harvesting effect under Neumann boundary conditions is studied.The global existence,nonnegativity and boundedness of the so-lution are proved by using the basic theory of parabolic equation.The local stability of nonnegative constant steady-state solutions and the Turing instabil-ity induced by diffusion for without delay are discussed.The global stability of positive constant steady-state solutions of delayed system is studied by using the upper and lower solution method.Under certain conditions,the occurrence of delay driven Hopf bifurcation is considered.Based on the center mani-fold reduction and normal form theory,the discriminant formulas determining the direction and stability of bifurcation periodic solutions are derived.The corresponding numerical simulations are given to verify the rationality of the theoretical part.Again,the Gause predator-prey system with nonlinear cross-diffus-ion and general harvesting effect under Neumann boundary conditions is s-tudied.The stability of nonnegative constant steady-state solutions of non-diffusive system and the existence of Hopf bifurcation are discussed.The stability of nonnegative constant steady-state solutions of diffusive system is studied,and the conditions for steady-state bifurcation and Hopf bifurcation are given.Through the Lyapunov-Schmidt reduction method,the existence and multiplicity of the nonconstant positive steady-state solution of the system bifurcated by the constant steady-state solution are obtained.Combined with the S~1–equivariant theory,the stability and direction of the bifurcation periodic solution are derived.Then,a three species food chain self-diffusive system with strong Allee effect under Neumann boundary conditions is studied.We discuss the dynam-ics of non-diffusive system,including the well-posedness of solutions,the local stability of nonnegative constant equilibrium and the existence of Hopf bifur-cation.We study the well-posedness of the solution of the diffusive system and the stability of the positive homogeneous steady-state solution,and discuss the existence of Turing instability and codimension-2 Turing–Hopf bifurcation under certain conditions.In addition,by choosing appropriate bifurcation pa-rameter and based on the weakly nonlinear analysis,we deduce the amplitude equation of unstable modes near the Turing–Hopf bifurcation point,and an-alyze the stability of various modes.The validity of the theoretical results is verified by numerical simulation,and different spatiotemporal patterns are ob-served near the Turing–Hopf bifurcation point,including spatial nonhomoge-neous steady-state solution,spatial homogeneous periodic solution and spatial nonhomogeneous periodic solution.Besides,the Gause predator-prey system with chemotaxis and discrete delay under Neumann boundary conditions are studied.The well-posedness of the solution of the system is proved by using the regularity theory of parabolic equation.The stability of the nonnegative constant steady-state solution and the existence of bifurcation of the system without delay are discussed,including Hopf bifurcation,Turing bifurcation and Turing–Hopf bifurcation induced by chemotaxis.For delayed system,the occurrence of Hopf bifurcation and Turing bifurcation driven by delay and chemotaxis are studied.Taking chemotaxis and delay as bifurcation parameter-s,the existence of codimension-2 Turing–Hopf bifurcation is explored.Based on the center manifold reduction method,the normal form of the Turing–Hopf bifurcation of system near the bifurcation point is calculated,which is helpful to classify the dynamics of system near the Turing–Hopf bifurcation point.We give a specific example and carry out numerical simulation to show that there are abundant dynamical phenomena.Finally,we look forward to the future research work.28 figures,1 table,138 references...