Kinetic Analysis Of Three Types Of Reaction Diffusion Models

In recent decades,all kinds of mathematical models have received increasing attention from ecologists and mathematicians.In particular,the spatially inhomo-geneous terms is very important issue,for example,the diffusion term,or cross-diffusion term,which has been extensively studied by many scholars.With the development of the nonlinear function analysis and the theory of nonlinear partial d-ifferential equations,especially,the development of the reaction-diffusion-equations,qualitative research work on the ecological model has stepped into a new stage,more results of practical application value have been made.In this paper,we are mainly used the theories of nonlinear function analysis and nonlinear partial differential equations,especially the reaction-diffusion equa-tions and the elliptic partial differential equations of second order to study the dy-namical behaviors of the three biological model with reaction and diffusion,such as the existence,uniqueness,multiplicity,stability,asymptotic behavior and the glob-ally behaviors of positive solutions of system.The methods involve the super-sub solutions method,the maximum principle,the local and global bifurcation theo-ry,the Leray-Schauder degree theory,the stability theory,the regularity theory,Lyapunov-Schmidt reduction,the perturbation technique and the theory of mono-tone dynamical systems.The main structure and contents of this paper are as follows:In chapter 1,we introduce the background and research situation of Lotka-Volterra model and chemostat model.Some basic results will be used in the forth-coming chapter.In chapter 2,a two-species cooperative model with diffusion and under ho-mogeneous Dirichlet boundary conditions is investigated.By virtue of the global bifurcation theorem,we show that the global bifurcation solutions can be extended to oo by increasing parameter a to oo,and for large a or large a,the system has a unique positive solution,which is globally asymptotically stable of the system by the theory of monotone dynamical systems.Finally,the existence and stability of positive solutions of system can be established by the Lyapunov-Schmidt technique.In chapter 3,a predator-prey model with cross-diffusion under homogeneous Dirichlet boundary conditions is studied.Firstly,the stability of the trivial solution and the semi-trivial solutions are considered by the spectral analysis,and we can give some sufficient conditions for the existence of positive solutions of system by the Leray-Schauder degree theory.Sceondly,we deriver the multiplicity results when some parameters are suitably large.Moreover,the existence and stability of positive solutions of systems are considered by the bifurcation theorem and spectral analysis.Furthermore,we consider the effects of cross-diffusion for the positive solutions of system with the spatial dimension 1 ? N ? 5.Finally,the uniqueness of positive solution is studied when the spatial dimension is one.In chapter 4,the two similar competition species in a competition unstirred chemostat model with diffusion is studied.Firstly,by the Taylor expand and the spectral analysis,we give the stability of the semi-trivial solutions of system.Second-ly,by the maximum principle,the LP estimates and Sobolev embedding theorems,we show that all positive solutions of system converge to the degenerate solutions when the perturbation parameter tends to 0,and hence by the Lyapunov-Schmidt reduction,the existence and nonexistence of positive solutions of system are estab-lished when the perturbation parameter is suitably small.Moreover,we consider the stability of positive solutions,and hence the one of the semi-trivial solutions,or the unique positive solution of system is the globally attractor can be established under some conditions by the theory of monotone dynamical systems.