Analysis Of A Predator-Prey Model With Strong And Weak Allee Effects

Aiming at the predator-prey problem with Allee effect affecting the survival of endangered species,a predator-prey model with Allee effect is established based on the essential difference between the strong Allee effect and the weak Allee effect,under weak Allee effect and strong Allee effect,the global behavior of the system is analyzed by theoretical proofs,and verified by numerical simulations.It provides the theoretical basis for formulating effective strategies to protect endangered species,and further enriches the research topics of Allee effect in predator-prey model.On the one hand,the weak Allee effect of prey is studied.According to the existence of equilibrium points,the index of biological viability is proposed,and the types of boundary equilibrium point and positive equilibrium point are proved by eigenvalue theory and formal series.The existence and uniqueness of limit cycles are proved by combining Poincare-Bendixson theorem with Zhang Zhifen theorem.Furthermore,the global asymptotic stability of positive equilibrium point is obtained.The conclusion is realized by using Python software.It is concluded that the weak Allee effect makes the population spend more time to reach the equilibrium state,and increases with the increase of the degree of weak Allee,it also gives the reasonable explanation for the species with sparse population density but periodic population change in nature,which generalizes the results of Fu Xiaoyue et al.(2012),and enriches the research and conclusion of weak Allee effect in predator-prey model.On the other hand,the strong Allee effect of prey is studied.Using the index of biological viability,the existence and types of equilibrium points are obtained by using eigenvalue theory and other methods.The existence of heteroclinic orbit and singular closed orbit connecting saddle point and node of the system are proved by finding tangent free lines and Bendixson-Dulac criterion.The validity of the results is verified by numerical simulation.It is concluded that it takes more time for the strong Allee effect to make the population reach equilibrium than the weak Allee effect,and it increases with the increase of the strong Allee effect.At the same time,two critical cases of the system are confirmed.It provides the strong basis for maintaining the survival of species,supplements the conclusion of Wang Wanxiong et al.(2013),enriches the research and conclusion of predator-prey model under strong Allee effect,and obtains a new discovery on singular closed orbit.