Codimension Two Bifurcations And Spatiotemporal Patterns Of Predator-prey Reaction-diffusion Systems

There are various interactions among biological populations in the real world.Among them,predation is one of the most common relationships,which plays a vital role in the survival and development of biological populations.Therefore,by using linear stability analysis,center manifold theory and normal form method,this thesis investigates codimension two Turing-Hopf bifurcation,Turing-Turing bifurcation and Bogdanov-Takens bifurcation of several types of predator-prey systems,as well as spatiotemporal patterns revealed by these bifurcations,to help ones better understand and interpret spatiotemporal dynamics of biological populations.The main work of this thesis is as follows:(1)Turing-Hopf bifurcations of a predator-prey system with Crowley-Martin functional response and of a predator-prey system involving intra-specific competition among predators,are investigated respectively,to discuss spatiotemporal patterns revealed by Turing-Hopf bifurcation.By analyzing characteristic equations,conditions for the occurrences of several bifurcations are established,and the first Turing bifurcation curves are determined at the same time.Utilizing normal form method,the third-order truncated normal forms of these two systems at Turing-Hopf singularity points are attained,respectively.Based on center manifold theory and dynamics of normal forms,spatiotemporal patterns revealed by Turing-Hopf bifurcations of type Ib and type II,are investigated.And,spatiotemporal phenomenon that a stable spatially homogeneous periodic solution and a pair of stable nonconstant steady states coexist,as well as bi-stable spatiotemporal pattern that a pair of stable spatially inhomogeneous periodic solutions coexist,is found.(2)Spatial patterns revealed by Turing-Turing bifurcation are investigated.Based on normal form method developed by Faria et al,normal form of Turing-Turing bifurcation for partial functional differential equations with discrete time delay is obtained.Also,concise formulas for calculating coefficients of the third-order normal form are given,which are explicitly expressed by system parameters of the original differential equations and could also be used to calculate coefficients of the third-order normal form of TuringTuring bifurcation for partial differential equations.By choosing one-dimensional domain and Neumann boundary conditions,this normal form is further simplified under different spatial resonance conditions.To investigate spatial dynamics revealed by Turing-Turing bifurcation,according to center manifold theory and dynamics of normal forms,spatial dynamics of a diffusive predator-prey system with Crowley-Martin functional response are investigated,and the existences of bi-stable superposition patterns,tri-stable spatial patterns and quad-stable spatial patterns are proved.(3)For partial functional differential equations with discrete time delay,the thirdorder normal form of Bogdanov-Takens bifurcation and the formulas for calculating coefficients of the third-order normal form,are established by utilizing normal form method.These formulas could also be used to calculate coefficients of the third-order normal form of Bogdanov-Takens bifurcation for partial differential equations,and they are explicitly expressed by system parameters of the original differential equations.Based on center manifold theory and dynamics of normal form,spatiotemporal dynamics near BogdanovTakens singularity point of a diffusive predator-prey system with nonlocal prey competition are discussed,to investigate spatiotemporal patterns revealed by Bogdanov-Takens bifurcation.And,tri-stable spatiotemporal pattern that a stable spatially inhomogeneous periodic solution and a pair of stable nonconstant steady states coexist,is found.