Dynamics Of A Predator-prey Model With Time Delay And Fear Effect

The study of dynamics for predator-prey system has always been an important subject in biological mathematics.Motivated by the ODE model of predator-prey with fear effect,considering the age structure of prey,we established a predator-prey model with time delay and fear effect in this paper.And the long-time dynamical behaviors of this system was investigated.Firstly,the existence,uniqueness,positivity and eventually uniformly boundness of the solutions of our model were proved by the related theories of functional differential equations and monotone dynamical systems.Meanwhile,the uniformly persistence of the solutions was analyzed.Then,under the condition that there is no positive equilibrium,we proved the global stability of the trivial equilibrium and the predator-free equilibrium by constructing the Lyapunov function and using the comparison principle and the theory of asymptotically autonomous dynamical systems,respectively.Next,it was proved that there exists the unique positive equilibrium when the predatorfree equilibrium loses its stability.Then,the existence condition of Hopf bifurcation was analyzed by the geometric discriminant criterion of Beretta and Kuang,and the stability problem of Hopf bifurcating periodic solution was studied by the central manifold theorem and the normal form method.Finally,we performed some numerical simulations and the simulation results are consistent with our theoretical results obtained in the previous parts.Moreover,we also observed that the periodic solution of large amplitude of the ordinary differential model with non-delay will disappear with the increase of time delay.Combined with the analysis of our mathematical results,it can be found that its disappearance occurs at the same time as the stability switch.When the time delay continues to increase,the positive equilibrium will become globally stable.