Dynamic Analysis Of A Predator-prey Model With Fear Effect And Economic Benefit

The interaction between predator and prey is a central topic in ecology and evolutionary biology.The long-standing view is that,predators only have a direct impact on preys by directly killing prey population.However,after a large number of experiments,it is found that the indirect effect(fear effect)on the prey population is sometimes even stronger than the direct effect,so the influence of fear effect on predator-prey system can not be ignored.In addition,from the perspective of human needs,it is of great significance to harvest species reasonably to achieve sustainable economic growth.The harvest of species is commonly used in fishery,forestry and agriculture,so as to achieve a balance between protecting resources and obtaining greater profits.In general,the populations which are harvested often sold for economic benefits.Based on the above facts,this paper studies a predator-prey model with fear effect and economic benefit.In this paper,the ordinary differential equation corresponding to the model is analyzed and we obtain the positivity and boundedness of the solutions.We study the diffusion model and give su cient conditions to ensure the existence of constant equilibria.The boundedness of solutions and the extinction of prey species are obtained by using upper and lower solution method and comparison principle.The local asymptotically stability of equilibrium solutions is discussed by using eigenvalue theory,and the global asymptotically stability of semi-trivial solution and positive constant equilibrium are proved by constructing Lyaponov functions.Furthermore,by using the maximum principle and comparison principle,the prior estimations of positive steady state solution are obtained,and the nonexistence of non-constant steady state solution is obtained by using Poincare inequality.Finally,the occurrence of the Hopf bifurcation and steady state bifurcation from the positive equilibrium are studied with the economic benefit as the bifurcation parameter.In addition,the theoretical results are also verified by Matlab simulation.