Dynamic Behavior And Optimal Harvesting Strategy Of A Predator-prey System With Double Delay Effect And Holling Type ? Functional Response

In theoretical ecology,both mathematicians and ecologists are always focusing on the research on predator-prey system.Researchers have made a lot of explorations on predator-prey system with functional response,the harvesting effect,the time delays,the habitat complexity and some other effects.Based on the published researches,a predator-prey system with harvesting effect and time delays is presented in this paper.Holling type ? functional response(i.e.,the prey shows group defense ability)and linear harvesting function are used.Both production delay and dimension delay are considered.Firstly,without considering the time delays,we study the uniform boundedness of solutions and the existence and stability of equilibrium points of the presented system.Secondly,we proved that only when the both time delays coexist,the system has stable positive equilibrium points and Hopf bifurcations.Finally,Pontryagin maximum principle is used to explore the optimal harvest strategies.Our results show that in the presented system without time delays,as long as the initial values are in R2+ and the intrinsic growth rate is greater than the predation rate,the solutions of the system are all uniformly bounded,and there are two boundary equilibrium points and at most two positive equilibrium points.The equilibrium point E0 is a saddle and the stability of the axial equilibrium point El is depended on the choices of the involved parameters.Furthermore,if the trace of the Jacobian matrix is less than zero and the harvesting strength satisfies the inequality max{U2,W1}<E<min{W2,U1},then the positive equilibrium point E1*is globally asymptotically stable.Otherwise,E1*is unstable.If E2*exists,it must be unstable.In the system with time delays,we obtained the conditions for the persistence of Hopf bifurcations and the global asymptotic stability of the positive equilibrium points E*under three cases(i.e.,consider only predator-prey delay,only digestion delay or both delays).Finally,the numerical simulations are implemented,and the results are consistent with our theoretical analyses.