Dynamic Studies On Some Predater-pray Models With Taxis

Many processes in medicine,engineering,physics,chemistry,and biology can be described mathematically using nonlinear reaction-diffusion equations or nonlinear pulse equations.In space predation,in addition to the random movements of predators and preys,predators can gather also in areas where prey’s density is high(or prey moving in the opposite direction of predator’s clusters),a phenomenon known as prey chemotaxis(or predator chemotaxis).Compared with chemical chemotaxis,the research on the chemotaxis of predator-prey reaction-diffusion systems has a relative short history.Therefore,it is necessary and meaningful to study the predator-prey models with chemotaxis.In this dissertation,we have studied the dynamic properties of reaction-diffusion systems with chemotaxis and with two to four populations of preys/predators.We have proved,under suitable conditions,the global existence and boundedness of the solutions of these models.Below we describe them in more details:Firstly,the predator-prey reaction-diffusion systems with prey-chemotaxis for threepopulations,setting in a smooth bounded domain and with homogeneous Neumann boundary conditions,where two predators compete for one prey have been studied.Under the random diffusion and prey chemotaxis,we describe the dissipation structure and pattern formation of such systems.In fact,the introduction of chemotaxis makes the dynamic properties of such predator-prey reaction-diffusion system being more complicated.We use the theories from operator semigroups and the quasi-linear parabolic equations,as well as a beautiful bootstrap method,to prove,under very loose conditions,the global existence and uniform boundedness of solutions to such system.Our result has improved many known ones.Secondly,some special cases of the general models mentioned as above,where both predators are cooperative and are attracted by the prey are considered.Our results show that the prey chemotaxis has an important effect on the dynamic properties of the system:when the sensitivity coefficient of the prey is small,the stability of the positive equilibrium solution of the system is not destroyed;but the positive equilibrium solution becomes being unstable and that the system adapts extraordinary spatio-temporal patterns,when the prey chemotaxis sensitivity coefficient growing.It is noted that the stability of constant positive equilibrium solution in mathematical model indicates that the population is homogeneous distributed,while the appearance of non-constant spatio-temporal patterns indicates the rich dynamic properties of the system.Finally,based on the previous analysis,we study the bifurcation of such systems.We take the prey chemotaxis sensitivity coefficients of prey(or predator)as the parameters,and apply the Grandall-Rabinowitz local bifurcation theorem.We analyze the bifurcation near positive constant steady-states of the system,and obtain concrete conditions for the emergence of bifurcation.This reveals the rich kinetic properties of such three population systems with two preys.At the same time,we study the positive solutions of general second order m-point boundary value problems for nonlinear singular impulsive dynamic equations on time scales.The existence and uniqueness of the positive solution are obtained by using the theorem of mixed monotone fixed point on the cone.The nonlinear items of the equations may be singular.We illustrate the results by examples,showing that these results can enrich the kinetic behavior of the chemotactic predation-prey systems,and provide a theoretical basis for some known numerical results that the prey chemotaxis could reduce the pattern formation in predatory-prey systems.