Study On Reaction Diffusion Model With Three Species

At present,the development of Biomathematics is very rapid.Using PDEs to study biodynamics has become a very active research direction in the field of nonlinear PDEs.Among them,the reaction-diffusion equation has been widely and deeply applied in the biomathematics.Through the establishment and analysis of the corresponding reactiondiffusion model,scholars can understand and study the diffusion of species in the spatial heterogeneous environment and the corresponding population dynamics problems,and have made a lot of important achievements and progress.In this paper,the reaction-diffusion model of general three competitive species is explored.Based on the environment of spatial heterogeneity and time constancy,the effects of diffusion rate and interspecific competition on population survival and evolution are mainly studied.The existence and stability of semi trivial equilibrium solution and coexistence equilibrium solution are discussed by using the stability discrimination method of equilibrium solution of eigenvalue and eigenfunction.In the first chapter,the research background and significance of reaction-diffusion model are briefly introduced,and the research and development status of reaction-diffusion model are described.In the second chapter,the research model is described in detail,the basic concepts and related theorems of semi trivial equilibrium solution,coexistence state,upper and lower solutions are introduced,and some basic properties of the main eigenvalues are summarized.In the third chapter,when the intrinsic growth rate is the same and the diffusion rate is the same,but the interspecific competition coefficients of the three species are different,on the one hand,the stability of a kind of semi trivial equilibrium solution is clearly described by using the principal eigenvalue theory and other methods.On the other hand,the upper and lower solution methods are used to explore the coexistence state of the system,and the Gagliardo-Nirenberg inequality is used to estimate the energy,and the uniform boundedness of the global solution of the system is obtained.In the fourth chapter,when the three species are in the same heterogeneous environment,the stochastic diffusivity is different,and the interspecific competition coefficientis not the same,the influence of intraspecies competition is not considered,that is,the intraspecies competition coefficient is 1,the stability of the two kinds of semi trivial equilibrium solutions of the system is studied by using the stability discrimination method of the equilibrium solutions of eigenvalue and eigenfunction.At last,we summarize the main research results of this paper,and put forward the the future research work.