Bifurcation Research Of A Ratio-dependent Diffusion Predator-prey Model

Predator-prey relationship is one of the important relationships among populations.Studying this biological relationship can help us better understand the related species,control the number and change of population,and is of great significance in protecting species as well as maintaining ecological balance.Ratio-dependent predator-prey model is a biological model based on the assumption that the individual growth rate of predator and prey is the ratio of the number of predators to the number of predators.Compared with the general predator-prey model,it can produce more abundant and reasonable dynamic properties.The results are also supported by many experimental data,which is a kind of biological model that is very popular among mathematicians and biologists and has great research significance.In this paper,we study the bifurcation problem of a ratio-dependent predator-prey model with diffusion term under homogeneous Dirichlet boundary conditions.In this problem,both u and v are zero on the boundary and the non-linear term contains u/(u+v),which is equivalent to a zero-to-zero form,leading to a difficulty of dealing with this problem.In order to overcome it,we linearize the system at its nontrivial equilibrium solution and transform it into a linear system.According to the relationship between parameter c and 1,the position of bifurcation and the existence of positive solutions in three cases are considered respectively.Then,in each case,a prior estimate of the positive solution of the system is given by using the maximum principle of the second order elliptic partial differential equation.Then the system is transformed into a linear system at the equilibrium solution by translational transformation.According to the related theory,the transformed linear system can be transformed into a compact operator,and the parameters of the bifurcation are determined by studying whether the compact operator has zero eigenvalues.Then,the existence of global bifurcations is studied based on local bifurcations by using the fixed point index theory to analyze whether the compact operator constructed has eigenvalues greater than 1.Finally,it is proved that there are positive solutions in all three cases.And the branch is bounded if parameter c less than or equal to 1,the branch tends to be infinite if parameter c greater than 1.