The Global Existence And The Qualitative Analysis Of Reaction-diffusion Population Systems

Research on the global existence and the qualitative analysis of solutions to reaction-diffusion population systems is a hot topic recently in study of biomath-ematics. In this dissertation, we investigate the existence of solutions to three kinds of strongly coupled reaction-diffusion predator-prey systems and a kind of reaction-diffusion elliptic competition system, and consider the qualitative analy-sis of solutions to four kinds of reaction-diffusion predator-prey systems. Employ-ing the energy estimates and Gagliado-Nirenberg type inequalities, we establish the global existence and uniform boundedness of solutions for a general strongly coupled predator-prey system; By applying the contraction mapping principle, the Holder continuity, the parabolic Schauder estimates and parabolic Ln esti-mates, we prove that there exist global classical solutions to a kind of the food chain strongly coupled model in three spices including two-preys-taxis and a kind of strongly coupled predator-prey model with self-diffusion and prey-taxis; Ap-plying the method of upper and lower solutions developed by Pao, we obtain the sufficient conditions for the existence of positive solutions to a strongly cou-pled elliptic competition system; Using the method of eigenvalue and Lyapunov function, we investigate the local stability and global stability to four kinds of reaction-diffusion predator-prey systems; Using the homotopy invariance of topo-logical degree, we prove that there exist non-constant steady states to three kinds of the reaction-diffusion predator-prey systems. This dissertation consists of four chapters.Chapter2is devoted to proving the global classical solution of the strongly coupled reaction-diffusion predator-prey system.In section2.1, we are concerned with a cross-diffusion system arising in a predator-prey population model with Holling type-II functional response in a bounded domain with no flux boundary condition. Employing the energy esti-mates and Gagliado-Nirenberg type inequalities to establish W21-bounds uniform in time, we prove the global existence and uniform boundedness of solutions for a strongly coupled system if the diffusion coefficients satisfy certain conditions. The global asymptotic stability of the constant positive steady state is also in-vestigated.In section2.2, we are concerned with the food chain model in three spices including two-preys-taxis and Holling type-II functional response under no flux boundary condition. The central point of this system is that the spatial-temporal variations of the predators'velocity are directed by the preys'gradient. By applying the contraction mapping principle, the parabolic Schauder estimates and parabolic LP estimates, we prove that there exists a unique global classical solution of this system.In section2.3, we are concerned with predator-prey model with self-diffusion and prey-taxis incorporating Holling type II functional response under homoge-neous Neumann boundary condition. In addition to random diffusion the preys and predators have self-diffusion and prey-taxis. By applying the contraction mapping principle, the Holder continuity, the parabolic Schauder estimates and parabolic LP estimates, we prove that there exists a unique global classical solu-tion of this system.Chapter3is devoted to investigating the stability, bifurcation and the ex-istence of non-constant steady states to the reaction-diffusion predator-prey sys-tems.In section3.1, we are concerned with a diffusive predator-prey model with a constant prey refuge which provides a condition for protecting of prey from pre-dation under homogeneous Neumann boundary condition. The stability of equi-librium points and Hopf bifurcation are investigated. We obtain that the positive constant solution is globally asymptotically stable when the constant refuge is sufficiently small and the semi-trivial equilibrium point is globally asymptotically stable when the constant refuge is between two positive constants. Furthermore, we prove that this system has the periodic bifurcation.In section3.2, we are concerned with a predator-prey model with medium interaction incorporating Holling type II functional response under homogeneous Neumann boundary condition. The stability of equilibrium points and existence of non-constant steady states are investigated. We obtain that if the carrying capacity of prey is bigger than positive integer one, then the unique positive constant solution is globally asymptotically stable when interaction coefficient is between two smaller positive constants, and between other two medium positive constants there exists at least one non-constant steady-state under other suitable conditions; and if the carrying capacity of prey is smaller than positive integer one, then the unique positive constant solution is globally asymptotically stable. Moreover, we find that the semi-trivial solution is globally asymptotically stable when the interaction coefficient is sufficiently small. Furthermore, we derive that the periodic solutions bifurcate from the positive constant solution.In section3.3, we are concerned with a cross-diffusion system arising in a predator-prey population model in a bounded domain with no flux boundary condition. We find that cross-diffusion coefficients create coexistence states for this system, that is, there exist non-constant steady states for this cross-diffusion system under certain conditions while without cross-diffusion there doesn't exist non-constant steady state for this corresponding system which is globally asymp-totically stable for the only one positive constant solution.In section3.4, we are concerned with a cross-diffusion system arising in a Leslie predator-prey population model in a bounded domain with no flux bound-ary condition. We investigate sufficient condition for the existence and non-existence of non-constant positive solution. We obtain that if natural diffusion coefficient of predator is large enough and cross-diffusion coefficients are fixed, then under some conditions there exists non-constant positive solution. Further-more, we show that if natural diffusion coefficients of predator and prey are both large enough, and cross-diffusion coefficients are small enough, then there exists no non-constant positive solution.In chapter4, we consider a elliptic competition system with cross-diffusion under homogeneous Dirichlet boundary condition. Existence and non-existence of positive solutions are investigated. Applying the method of upper and lower solutions developed by Pao, we obtain the sufficient conditions for the existence of positive solutions when the cross diffusions coefficient is sufficiently small. Moreover, we also find sufficient conditions for non-existence of positive solutions.