Bifurcation Analysis Of A Nonlocal Holling-Tanner Predator-prey Model

All creatures in nature are interrelated and mutually conditioned.Interspecies relationships refers to the relationship formed by the interaction between different species.The interactions between the two populations are usually characterized by competition,predation,symbiosis,and parasitism,etc.Particularly,the relationship between the predator and prey is the most common one in nature.Therefore,the research on the relationship between the predator and prey will help us to understand the dynamics of the ecosystem.A diffusive predator-prey system with nonlocal competition and Neumann boundary condition is considered in this paper.We derive some results on the existence of Hopf bifurcation,the stability of periodic solutions,and the direction of the bifurcation.The main contents are as follows:We first study the existence of Hopf bifurcation of the system.Through analyzing the eigenvalues,we obtain the eigenvalues distribution of the characteristic equations corresponding to the constant positive equilibria of the system.Then we obtain the stability of the equilibria and the sufficient condition for occurrence of Hopf bifurcation near that point.Then,we investigate the stability of periodic solutions and the direction of the bifurcation.For spatial nonhomogeneous periodic solutions,we first transform the system to be solved into an equivalent system by variable substitution.Based on the center manifold theory and the normal form theory,the stability of the spatial nonhomogeneous periodic solutions and the direction of the bifurcation are obtained.In fact,the calculation of the bifurcation direction is more complex for spatial homogeneous periodic solutions,but the stability of the periodic solutions and the direction of the bifurcation can be similarly obtained by using the center manifold theory and the normal form theory.It is shown that the Hopf bifurcation is more likely to occur when there is a nonlocal competition among the prey.At this time,the periodic solutions near the bifurcation values can be spatially nonhomogeneous and orbitally asymptotically stable.In addition,it can be found by numerical simulation that the periodic solutions concentrate more on the boundary of the region when the spatial variables become larger.Therefore,when there is nonlocal competition among the prey,the system will have more dynamic properties.