Bifurcation Analysis In A Class Of Predator-prey Model With Time Delay And Additive Allee Effect

From perspective of population ecology,the relationship between predator and prey and individual’s share of resources such as food,space and so on are related to population density.Under the interaction of diffusion,additive Allee effect and time delay,it is important to study the dynamic change of the population density of predator and prey to protect species diversity.In this paper,the bifurcation analysis of a class of predator-prey model with time delay is presented,and the following two kinds of cases are mainly discussed.The first case is a predator-prey system without diffusion term.Firstly,the condition and number of the boundary equilibrium point and the positive equilibrium solution are given.Secondly,by analyzing the distribution of the roots of the characteristic equation,we have obtained the condition of the existence of pure imaginary roots and the sufficient condition for stability of the fixed point when time delay is zero.The implicit function theorem is used to prove that the real parts of the characteristic roots corresponding to the pure imaginary roots satisfy the transversal condition,and the critical value of the system experiencing Hopf bifurcation is solved.Thirdly,by using the infinitesimal generator,the predator-prey model is written as an abstract ordinary differential form,and the reduction equation of the model confined to the central manifold is obtained by using the form adjoint theory and infinite dimensional system’s normal form theory.Finally,the predator-prey system is numerically simulated to explain the theoretical analysis results.The second case is a diffusive predator-prey system that satisfies the Neumann boundary conditions.Firstly,the characteristic equation of the model is solved by Fourier series in the real-valued Soblev space.Secondly,the distribution of the roots of the characteristic equations is analyzed as time delay varies,and sufficient conditions for stability of steady-state solution and the existence of Hopf bifurcation are given respectively.Thirdly,the infinitesimal generator and form adjoint operator are defined by the Riesz representation theorem,and the eigenvector is obtained,then the normal form of the model’s reduction equation is deduced by using the spectral decomposition theory and partial functional differential equations’ central manifold theory and normal form method.Then we can determine the direction of Hopf bifurcation and the stability of bifurcation periodic solution.Finally,the predator-prey system with homogeneous Neumann boundary conditions is numerically simulated to explain the theoretical analysis results.