Bifurcation Analysis Of A Predator-prey Model With Time Delay And Allee Effect

The study of predator-prey model is a hot topic in theoretical ecology and applied mathematics.Among them,the research on the effects of various ecological effects on the dynamics of population model is particularly rich.In this paper,the Allee effect is introduced into the predator-prey model,and the effect of time delay on the dynamic behavior of the model is considered.The paper is divided into three parts:In the first place,a predator-prey model with strong(weak)Allee effect is constructed,and the boundedness,equilibrium point,stability and Hopf bifurcation of the model are analyzed;There is one more point,based on the predator-prey model with weak Allee effect,the search delay and digestion delay are introduced to analyze the effect of delay on the stability of the model;The last but not the least,considering the spatial structure of the original model,a predator-prey model with weak Allee effect and search delay is constructed to analyze the Turing instability caused by weak Allee effect and delay respectively,and the influence of weak Allee effect and search delay on the pattern formation is studied.The results show that the Allee effect enriches the dynamic behavior of the predatorprey model;the introduction of time delay destroys the stability of the coexistence equilibrium point,and the continuous oscillation of the two populations is observed;the increase of weak Allee effect results in the decrease of patch length and density,and the increase of time delay results in the change of patch shape from coexistence of spotted and stripe pattern without time delay to stripe pattern.So we find that weak Allee effect and search delay play an important role in the formation of population morphology.