Stability Of Two Types Of Predator-prey Models With Allee Effects And Beddington-DeAngelis Response

This dissertation mainly studies the steady-state solution of the cross-diffusion predator-prey models with Allee effect and Beddington-DeAngelis response,and discusses the existence,boundedness,stability of equilibrium point and the existence of the non-constant steady-state solution.The content of this dissertation is as follows:Chapter 1 first introduces the biological significance of the predator-prey system,related research background and research progress.Secondly,this chapter introduces the influence of the Allee effect on the predator-prey system,and lists the preliminary knowledge such as the maximum value principle and Harnack inequality used in this dissertation,and briefly introduces the main work of this dissertation.Chapter 2 studies the Beddington-DeAngelis type cross-diffusion predator-prey model with strong Allee effect.This chapter first uses the comparative principle to obtain the boundedness of the positive solution,secondly uses the linearization method to prove the stability of the equilibrium point,and finally discusses the conditions for the existence of the non-constant steady-state solution by using the Leray-Schauder degree theory.Chapter 3 studies the Beddington-DeAngelis type cross-diffusion predator-prey model with weak Allee effect.This chapter analyzes the stability of the equilibrium point of the model,gives a priori estimation of the solution,and further proves the existence of the non-constant steady-state solution.It is found that a sufficiently large self-diffusion coefficient hinders the existence of a non-constant steady-state solution,and a sufficiently large cross-diffusion coefficient is conducive to the existence of a non-constant steady-state solution.Finally,the paper summarizes and prospects,summarizes some conclusions on steady-state solutions of the two types of models,and puts forward questions that require further research.For example,whether there are periodic solutions in these two types of models,and the stability of the solutions of the above two types of models in the non-uniform space mode.