Qualitative Analysis Of Two Types Of Reaction-diffusion Population Model

The development of modern science and technology is inseparable from mathematics,and the accuracy of mathematics depends on various mathematical models.Moreover,a large number of mathematical models can be classified as reaction-diffusion equation.In recent decades,with the aid of the background of the interactions between species,the mathematicians and biologists have established and studied some population diffusion models.In this paper,nonlinear analysis and partial differential equation theory is used to qualitatively analyze two kinds of population diffusion models.The main contents of the study include the local and global existence,stability and uniquness of the positive solutions of the model.The main mathematical theories and methods used include the comparison principle,the implicit function theorem and the fixed point index theory.The main contents of this paper are as follows:In Chapter 1.The background of population dynamics and the theoretical framework of reaction-diffusion equation model are introduced.After that,it enumerates and states the biological background,the original model,biological significance of the basic parameters of the model and present situation at home and abroad respectively with fear effect,Beddington-DeAngelis functional response,prey-taxis and Leslie-Gower types.Finally,the main works of this thesis are introduced.In Chapter 2.We considered a predator-prey model with fear effect and B-D functional response.First,a priori estimate is obtained by using the properties of eigenvalue problems of elliptic equations and the comparison principle of the positive solution of the model.Secondly,take advantage of using the degree theory of elliptic equation,the existence of the solution of the model is built by calculating the index of the semi-trivial solution.Finally,the local and global bifurcation theory is used to study the local and global bifurcation structure of the model's semi-trivial solutions.Furthermore,the stability and uniqueness of the semi-trivial solutions are obtained.In Chapter 3.A three-species model with two prey-taxis and Leslie-Gower type is studied.Firstly,the priori estimate and local existence of classical solutions of the model is obtained by using common algebraic tools and classical parabolic equation theory.Secondly,using the operator semigroup theory under Neumann boundary conditions,the boundedness of the positive solutions of the model is established in Lk(?)(k?2)space.Then we prove the global existence and boundedness of the positive solutions.Finally,the method of constructing Lyapunov function and Routh-Hurwitz criterion is used to obtain that the positive solution of the model is local and globally asymptotically stable when the coefficients under some certain conditions.In Chapter 4.We analyzes the biological significance of the two kinds of population diffusion models,the deficiencies and proposes the future work of the article.