Analysis Of A Cross-Diffusive Holling-Tanner Model With A Prey Refuge

As one of the important predation models,the Holling-Tanner prey predator model can accurately describe the interspecies interaction in ecosystems such as bobcats and rabbits.Based on previous studies,this paper established a cross-diffusive Holling-Tanner model with a prey refuge,analyzed the properties of the equilibrium point of the non-diffusion system and cross-diffusive system,and got some meaningful conclusions.For the non-diffusion system,on the basis of Hurwitz criterion and constructing Lyapunov function method,we get the sufficient condition that the positive equilibrium point is a global asymptotically stable node or focal point.According to the Bendixson-Dulac criterion,we proved that there is not exist the limit cycle.The results of the study on the positive equilibrium point of the ODE system are added to Jiang Song's "The existence of the existence of a kind of diffusion holling-tanner model".Considering the positive equilibrium point as a continuous function of the coefficient of refuge,we obtained the analysis of the coefficient of refuge will affect the density of the prey and predator population,but will not affect the stability of the system.The conclusions of the influence of coefficient of refuge on the stability of various predation systems are generalized,and the feasibility of the conclusion is verified through the numerical simulation.For the diffusion system,mainly based on the reaction diffusion system stability theorem discusses the diffusion effect on the stability of the system,we get the following conclusion:When only free diffusion occurs,there will be no Turing instability;When the cross-diffusion is introduced,the instability phenomenon will be generated at the original stable equilibrium point under certain conditions.The model proposed in this paper is the further development of the model of Li Yanling's "A kind of the existence of a kind of intersecting predator-prey model with very few positive solutions".The Turing instability of model was studied.Therefore,it is better to explain and control the phenomenon that the two populations can coexist peacefully in the ecological environment,and enrich the research results of this kind of model.