Stability And Hopf Bifurcation Of Predator-prey Models With Fear Effects

In nature,fear effect can be seen everywhere.Fear can not only help organisms improve their alertness,but also help them avoid danger.Fear is also an innate psychological reaction of organisms.At the same time,fear will also have a certain impact on the reproductive ability,foraging behavior,physiological state and so on.This provides a good solution to the ecological imbalance caused by the disappearance of large carnivores.In recent years,due to the excessive control of human beings,some large carnivores are facing the extinction problem.The loss of the top of the food chain will destroy the balance of the ecosystem.How to adjust the ecological balance in a short time is a major problem facing human beings.Using theoretical knowledge to establish the corresponding population dynamics model,we can get a good explanation of this phenomenon.At present,the population model of the effect of fear on the growth rate of organisms has been relatively perfect.Therefore,this paper establishes a population model to study the effect of fear on the foraging behavior of organisms.In the paper,based on the traditional predator-prey model and the effect of fear on the predation rate of predators,the fear effect was taken as continuously changinng variable.The dynamic behavior of a class of Holling II Lotka-Volterra model with fear effect is studied.For the local model,the existence and stability of the non-negative equilibrium are studied,the conditions for the existence of the positive equilibrium are given,and the stability of the positive equilibrium is analyzed.Secondly,a series of complex dynamic behavior states,such as Hopf bifurcation and saddle-junction bifurcation,were found by analysis,and their behavior was studied in detail.The results show that when the fear passes a threshold,a supercritical limit cycle with asymptotically stable orbit will appear in the model by adjusting the fear factor.Fear of medium-sized predators can prevent the extinction of low-trophic species and thus maintain the stability of the ecosystem.In addition,with the increase of the fear level,the periodic shock of the system will become smaller and the stability of the system will be strengthened,but there is an optimal area.If the fear level is too large,it will lead to the excessive increase of the low trophic level,and even threaten the survival of the middle trophic level.For the reaction-diffusion model,the Turing instability caused by diffusion of positive equilibrium and bifurcated periodic solutions,the existence and direction of Hopf bifurcated solutions and the stability of bifurcated periodic solutions are investigated.The results show that fear effect plays an important role in determining the stability and branching behavior of the model,it can induce the Turing instability phenomenon,which is fundamentally different form the situation without fear effect.The fear effect can reduce the density of predators at the positive equilibrium point,but it will not cause the extinction of predators.The fear effect can also increase the stability of the system by eliminating the existence of periodic solutions.The above results show the complex dynamic behavior of the model.