Research On A Predator-Prey System In Spatially Heterogeneous Environment

Biomathematics is a combination of mathematics and biology,and has established and perfected its theoretical system in a wide range of applications,and has developed many unique mathematical methods adapted to the characteristics of biology.At present,biomathematics is a relatively complete and relatively independent subject,which con-tains many important branches.Among them,population dynamics is by far the most extensive and in-depth application of mathematics in ecology,and the most systematic and mature branch.Because predator-prey models usually have rich population dynam-ics,predator-prey models have always been an important subject for many mathematicians and biologists,spatial inhomogeneity and functional response functions usually have im-portant effects on predator-prey models,so this paper mainly considers a predator-prey model with Beddington-DeAngelis functional response function in inhomogeneous space and the structure and stability of the positive steady-state solution are given,and the in-fluence of spatial inhomogeneous and functional response functions on the predator-prey system is obtained.The second chapter first gives the priori estimate and basic dynamic behavior of the positive steady-state solution.Then,considers ? as the bifurcation parameter,the global divergence from the semitrivial steady-state solution curve is determined for using the di-vergence theory and the priori estimate,etc.And then the existence and nonexistence of positive steady-state solution is given,we further determine the coexistence area of preda-tor population and prey polulation.The results show that different spatial inhomogeneities and different functional response functions have different effects on the population and can produce different critical values.The third chapter gives the influence of m in the Beddington-DeAngelis functional response function on the coexistence area.First,the necessary and sufficient conditions for the existence of a positive steady-state solution when k is sufficiently large are given,and the uniqueness and global asymptotic stability of the positive steady-state solution are given.Then,a sufficient condition for the existence of the positive steady-state solution is given when m is sufficiently large,and the uniqueness and multiple existence of the positive steady-state solution are discussed.When the positive solution is unique,the global asymptotic stability of the positive solution is obtained.The results show that when-c/m<??0,the size of the coexistence area has nothing to do with k,and decreases with the increase of m;and when 0<?<?1D(?0),the coexistence area increases with the increase of k or m.