Dynamics Study Of A Predator-prey System With Fear

In recent years,due to the lack of deeper cognitions on the importance of large carnivores,a large number of large carnivores have been disappearing or even extinct.The absence of food chain tops will damage the balance of ecosystems.It is an imminent and major issue that how to reasonably adjust the balance of the ecosystem in a short time,then people and nature live together in harmony.If humans no longer destroy animal habitats and work together to protect the population,the number of endangered populations is still expected to increase slightly.However,the number of populations cannot be restored in a short time and it will take a long time.A previous pseudo natural enemy experiment in Canada showed that it was possible to ease this situation for a short period of time by creating pseudo natural enemies.In the experiment,the dog barking in the tape was used to simulate a scene of fear for the raccoon,so that the raccoon who had no survival pressure felt the enemy again,this will help protect the diversity of the population under the food chain and maintain the balance of the ecosystem.So far,many biologists have confirmed through experiments and simulations that fear plays a decisive role in the ecosystem.However,there is not much research on the effects of fear on the predator-prey system through mathematical models.In this paper,based on the traditional predator-prey model and considering the influence of fear on prey rate,a predator-prey model considering the fear of predator is established by taking fear as a continuous variable.In the text,fear mainly affects the predator-prey ratio.Through analysis,when kcp leqd,the system has only extinction equilibrium point and boundary equilibrium point,which means that the medium-sized predator will become extinct.When Kcp > d,according to the Hurwitz criterion,the system has a unique positive equilibrium point,it is locally asymptotically stable as long as it exists.The global behavior of the system equilibrium point is further analyzed.When Kcp ? d,the boundary equilibrium point is globally asymptotically stable.When Kcp > d,the boundary equilibrium point is unstable.Under certain conditions,we use the second additive compound matrix to prove the global asymptotic stability of the positive equilibrium point.It is verified by numerical simulation that the time for the system to stabilize is not related to the selection of the initial value,but to the magnitude of the fear,and the time for the system to stabilize is shorter with the increase of the fear.Secondly,a predator-prey model with Holling II functional response considering the response of fear is established by introducing fear influence factor function into the model by using the qualitative and stability theory of differential equations,it is proved that there are at most three equilibrium points in the system.In addition,there are abundant dynamic forms in the system.With the different number of equilibrium points,the stable States of the system are different.Under certain conditions,if there is only one positive equilibrium point,the system will have Hopf bifurcation.Numerical simulation shows that when there are three equilibrium points,the behavior of the equilibrium point is different.