High Codimensional Bifurcations Analysis Of Some Biomathematics Systems

Through the study of biomathematics system,we can reveal the essence of complex ecological phenomena in nature,and to guide the human beings for reasonable protection and utilization of the ecosystem.Therefore,it is of great practical significance to research the dynamics behavior of the system.In this paper,we will use the qualitative theory of differential equations,bifurcation theory,the center manifold theorem,spectral theory,perturbation theory and normal form theory to study several biomathematics systems.In chapter 3,a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting is studied.On the basis of the existing literature,we focused on the high codimensional bifurcation phenomena of the system near its unique interior equilibrium.We find that the unique interior equilibrium can bea saddle-node of codimension 1 and a Bogdanov-takens type cuspsof codimension 2 and 3.We also prove analytically that the system undergoes Bogdanov-Takens bifurcations of codimension 2 and 3.In order to explore the changes of dynamics in the same biological system by the same harvesting type fordifferent populations,a Leslie-Gower predator-prey model with Michaelis-Menten-type predator harvesting is considered in chapter 4.The results show that the system has rich dynamics,the equilibria can be topological saddles,nodes,foci or centers,saddle-nodes,nonhyperbolic node of codimension 2,cusps of codimension 2 and 3.Complex bifurcations is also appeared in the system,such as saddle-node bifurcation,transcritical bifurcation,pitchfork bifurcation,Hopf bifurcations,homoclinic bifurcation,Bogdanov-Takens bifurcations of codimension 2 or 3,and so on.At last,several numerical simulations are carried out,andthe reasonable biological explanations for these complex bifurcations also are provided.To study the spatio-temporal dynamics of a reaction-diffusion system with dissipative term,the Turing-Hopf bifurcation of codimension 2 in general Brusselator reaction-diffusion model is investigated in chapter 5.We first use the spectral theory of Laplaceoperator to transform the partial differential equation into a system composed of countable duality differential equations,and the transversal condition of the Turing instability and the Turing-Hopf bifurcation of codimension 2 is obtained by solving the eigenvalue problem of the linearized matrix at the constant steady-state solution.Then the normal form on the center manifold of the perturbation system is analyzed by the center manifold theorem.It is proved that the system undergoes a Turing-Hopf bifurcation of codimenion 2 under suitable conditions.Finally,we analyzed the Turing-Hopf bifurcation of codimenion 2 for a specific example,and the six complex dynamicalbehaviors associated with bifurcation are carried out by numerical simulations to illustrate the validity of the theoretical results.