Bifurcation Analysis Of A Predator-Prey System With Fear Effect

Fear is an innate psychological activity of any species.In recent years,many scholars have confirmed that fear has a certain impact through theory and experiment on the reproductive rate of biological population,habitat use,foraging behavior and so on.It is well known that the presence of large carnivores help regulate biodiversity.They are at the top of the food chain and the food web,they can directly kill or cause fear of the prey and indirectly control the number of the prey,thereby maintaining the balance of the ecosystem.However,with the continuous decrease of large carnivores causing a series of ecological problems,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 can explain various ecological phenomena well.On the basis of the traditional predator-prey system,based on the influence of fear on the foraging behavior of predators,this article will establish a predator-prey system in which the predators are subject to fear,taking fear as a variable of continuous change.In the second chapter,we consider a class of Holling-Ⅲ predator-prey system with fear effects.Firstly,the existence and local stability of equilibrium point are discussed.Then,by using bifurcation theory,we obtain that the system will experience saddle node bifurcation,Hopf bifurcation,Bogdanov-Takens bifurcation and so on.it is found that changing the fear parameters can produce new phenomena in the system through numerical simulation,and with the increase of the fear level,the periodic oscillation of the system will gradually decrease,enhancing the stability of the system.If the fear level is too large,the dynamic behavior of the system will change,and eventually lead to the extinction of the population of medium predators.This suggests that fear can effectively control the density of medium predators and prevent the excessive growth of species below the food chain,thus maintaining the balance of the ecosystem.In the third chapter,considering the digestive time delay of predators,a predator-prey system with time delay and fear effect is established.Taking the time delay as a bifurcation parameter,the conditions of Hopf bifurcation at the positive equilibrium point of the system are discussed.Next,we analyze that the direction of Hopf bifurcation and the stability of periodic solutions of bifurcation by using the central manifold theorem and the canonical form theory.Finally,through numerical simulation,it is found that with the increase of time delay,the stability of the system is destroyed,and under a large time delay,the system presents some special oscillation behavior,and the oscillation period will increase,which is easy to lead to the extinction of predators and prey,and is not conducive to maintaining the balance of the ecosystem.