Codimension-Two Bifurcation Analysis For Predator-Prey Models With Time Delay And Diffusion

The study of predator-prey relationship among species is helpful to predict and estimate the population number of predators and prey accurately,which is of great significance to the prevention and control of harmful populations and the protection of endangered animals.In this thesis,we mainly study the dynamics of several predator-prey models with time delay,including the local stability of constant steady states,the existence and properties of codimension-two bifurcation,among which codimension-two bifurcation include double Hopf bifurcation and Turing Hopf bifurcation,etc.The main research work is as follows:First of all,for a class of time-delayed diffusive predator-prey systems with cannibalism,the characteristic equation of the linearized system has two pairs of purely imaginary eigenvalues at a critical point,which leads to double Hopf bifurcation.An approach of center manifold reduction is applied to derive the normal form for such nonresonant double Hopf bifurcations.We find that the system exhibits very complicated dynamics,including coexistence of periodic and quasi-periodic solutions.Numerically,we show that double Hopf bifurcation is induced if the strength coefficient of the predator cannibalism term belongs to an appropriate interval.Besides,by considering the nonlocal competition effect,we improve the diffusive predator-prey model with predator cannibalism.We then give the critical conditions leading to the existence of the double Hopf bifurcation,in which the time delay and the diffusion coefficient were selected as bifurcation parameters,the center manifold reduction and normal form method are developed.Near the double Hopf bifurcation point,the stable spatially homogeneous and inhomogeneous periodic solutions coexist.The theoretical results are well demonstrated by numerical simulation.Secondly,we investigate a diffusive predator-prey model by incorporating the fear effect into prey population,since the fear of predators can visibly reduce the reproduction of prey.By introducing the mature delay as bifurcation parameter,we find this makes the predator-prey system more complicated and usually induces Hopf and double Hopf bifurcations.The formulas determining the properties of Hopf and double Hopf bifurcations by computing the normal form on the center manifold are given.Moreover,we show the existence of quasi-periodic orbits on three-torus near a double Hopf bifurcation point,leading to a strange attractor by further varying the parameter.The emergence of quasi-periodic and chaotic phenomenon may indicate that there exists complex dynamical behavior of biological system itself.We also find the existence of Bautin bifurcation numerically,then illustrate the coexistence of stable constant stationary solution and periodic solution near this Bautin bifurcation point.Finally,for a class of Holling-Tanner models with time delay and diffusion,we investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations.Local and global stability of the positive constant steady state for the non-delayed system is studied.When considering the effect of mature time delay,we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points.We obtain the complicated dynamics phenomena,including periodic solutions,quasi-periodic solutions on a three-dimensional torus,the coexistence of two stable nonconstant steady states,the coexistence of two spatially inhomogeneous periodic solutions,and strange attractors.