Analysis Of Several Dynamic Model In A Predator-Prey System With The Impact Of Fear

Fear,a inherent psychological reaction of creature,is good to increase alertness and reaction ability.Although it can help animal avoid danger,it also reduce the growth of biological by affecting the reproduction rate,physiological status,habitat use,and foraging behavior,etc.It is a good advice for eliminate the ecological imbalance caused by human hunting large carnivores.Population dynamics model,setting up using mathematical theory,can well explain various biological phenomena.The model of the fear affect on biological growth population has been gradually perfect.However,there are still shortcomings in the research about the influence of the fear on biological foraging behavior.Therefore,this paper established the predator-prey population participation rate function under the influence of fear.And the predator-prey model with the fear influence on biological foraging behavior is studied.Firstly,a predator-prey model with Logistic growth and Holling II functional response function is established in this paper.And it is found that the system undergo a series of complex dynamic behavior: Saddle node bifurcation,Hopf bifurcation,Bogdanov-Takens bifurcation,generalized Hopf bifurcation and saddle-node bifurcation of limit cycle,etc.After adjusting the fear factor,we found the fear on medium-sized predator can prevent lower trophic level species extinction and maintain the stability of ecosystem.In addition,with the increase of degree of fear,periodic oscillation of the system become smaller gradually.And it can increase the stability of the system.However,the high level fear can lead to the excessive growth of lower trophic level species and the extinction of the middle nutrition level.Secondly,we established a predation-prey model with time delay.Analysing the dynamic characteristic,we found that the time delay does not change the stability of the trivial solution and semi-trivial solution.But it can change the stability of the positive equilibrium under certain condi-tions.In the meantime,the system undergo a series of Hopf bifurcation.After the numerical simulation about time delay impact on system,it is found that the system undergoes a supercritical or subcritical Hopf bifurcation.And the period of oscillation will also increase with the increase of time delay.Especially the appear of large limit cycle can lead to the extinction of the predator and prey species populations.This could break down the steady development of the ecosystem.