Study On Spatiotemporal Pattern Dynamics Of Several Population Models

Pattern dynamics mainly studies the formation and evolution of the spatiotemporal ordered structure when the system is far from the thermodynamic equilibrium state.The bifurcation theory is an important tool to investigate the pattern formation of partial differential equations.In recent years,the study of pattern dynamics mainly focuses on the dynamic behavior of the system near the high codimension bifurcation and advanced bifurcation.In this paper,based on the population models,the spatiotemporal patterns generated by the Turing-Hopf bifurcation of spatially homogeneous steady states and the Hopf bifurcation of spatially inhomogeneous steady states are studied by using the center manifold theorem,normal form theory and implicit function theorem.The main content of this thesis are:1.Based on the abstract normal form theory proposed by T.Faria et al.,a specific formula for calculating the normal form of a general reaction-diffusion equation with a discrete time delay near the Turing-Hopf bifurcation.The coefficients of the formula can be expressed explicitly by the coefficients of the original equation.By analyzing the third-order truncated normal form and combining with the center manifold convergence theorem,the possible stable attractors near the Turing-Hopf bifurcation are obtained.They are respectively spatially homogeneous steady state,spatially inhomogeneous steady state,spatially homogeneous periodic solution,spatially inhomogeneous periodic solution and spatially inhomogeneous quasi-periodic solution.We theoretically prove that the TuringHopf bifurcation can lead to the generation of spatiotemporal ordered structures.2.The bifurcation of a class of Holling-Tanner predator-prey model is studied.By choosing the space length l and the birth ratio of predator to prey as parameters,we discuss the existence conditions of multiple bifurcations.According to the normal form method,the third-order truncated normal form of Holling-Tanner model near the Turing-Hopf bifurcation is obtained.By analyzing the corresponding amplitude system with the unfolding of type VIIa,we reveal the dynamics of the original system near the Turing-Hopf bifurcation,such as one pair of stable spatially inhomogeneous periodic solutions coexist,one pair of stable spatially inhomogeneous quasi-periodic coexist and one stable spatially homogeneous steady state coexist with one pair of stable spatially inhomogeneous quasiperiodic solutions.3.The bifurcation of a time delayed Holling-Tanner predator-prey model is studied.The time delay reflects the hysteresis phenomenon caused by intraspecific competition.Based on the normal form theory and by discussing the corresponding amplitude system with the unfolding of type IVa,we obtain many interesting dynamical behavior of the system near the Turing-Hopf bifurcation.For example,two stable spatially inhomogeneous steady states can coexist in some parameter regions,but due to the effect of time delay,these two spatially inhomogeneous steady states can lose their stability via the Hopf bifurcation and eventually lead to the generation of two stable spatially inhomogeneous periodic solutions.4.A memory-based diffusion population model with nonlocal maturation delay is studied.By using the Lyapunov-Schimidt reduction,we obtain the existence of the spatially inhomogeneous positive steady state.By combining the prior estimation of eigenfunctions,the implicit function theorem and the geometric method that deal with the eigenvalue problem with two delays,the characteristic equation of the system at the positive steady solution is discussed.The characteristic equation is a class of partial differential equations with two delays.We study the existence conditions of the zero real part eigenvalues,and then give the stability of the positive steady state and the parameter conditions for the system undergoes the Hopf bifurcation at the positive steady state.