Study On Dynamics Of A Predator-prey System With Allee Effect In Predator

In the predator-prey system,the complexity of the dynamics of the predatorprey system is closely related to the different choices of the per growth rate of the population.In the case of scarce resources,when the population density is relatively small,the increase of the population density is conducive to reciprocal capture,cooperative foraging,and reproduction.In this case,the per growth rate of the population increases with the increase of population density,which is called the Allee effect in biology.For the predator-prey system,Allee effect is often encountered in the prey population and has been widely investigated.In real life,the predator population is relatively small compared with the prey population,and the increase of the predator density will significantly improve the reciprocal foraging efficiency and predation ability,and then increasing the chance of predator's reproduction.Therefore,the phenomenon of Allee effect is more common in the predator population.However,in the existing literature,there are few studies on predator population with Allee effect compared with those on prey population with Allee effect,and most of the studies are based on the results of numerical analysis.This paper mainly studies from a qualitative perspective the influence of the Allee effect intensity on the existence and stability of the coexistence equilibria of a predator-prey system with an Allee effect in predator.And the influence of key parameters such as the incubation delay of the prey population,the strength of the Allee effect,and the diffusion coefficient on the stability of the coexistence equilibria.The research of this paper is mainly divided into four chapters:In chapter 1,we mainly introduce the research background of the predator-prey system with the Allee effect in predator and the model studied in the article.In chapter 2,we study the existence and stability of the positive equilibria of the predator-prey system with the Allee effect and intra-species competition.Firstly,for the system without delay,the influence of the strength of the Allee effect on the existence and stability of the positive equilibria is studied,it is shown that when the strength of the Allee effect is less than a certain critical value,the system has two positive equilibria,and when the strength of the Allee effect is greater than this critical value,the positive equilibria of the system disappear.Secondly,under the assumption that the positive equilibrium with a large predator biomass is stable,we study the influence of the incubation delay of the prey population on the stability of the positive equilibrium.The results show that under certain conditions,the increase in delay will cause the instability of the coexistence equilibrium,and lead to stability switches,as well as the Hopf bifurcation and periodic vibration induced by the delay.In chapter 3,we study the Holling-? type diffusion predator-prey model,in which the predator has the Allee effect.First of all,without diffusion,we study the influence of the Allee effect on the number and stability of coexistence equilibrium is discussed.We give a critical value of ?*for Allee strength which is dependent on other parameters of the system.When 0<?<?*,the system has two positive equilibria.The stability of the positive equilibrium with smaller biomass of predator is not affected by the strength of Allee effect and is always unstable.We focus on how the strength of the Allee effect affects the stability of the coexistence equilibrium corresponding to high predator biomass.Our theoretical results show that there are three kinds of possible cases about the stability:(?)when 0<?<?*,the strength of the Allee effect does not affect the stability of this coexistence equilibrium;(?)there exists a unique Hopf bifurcation value ?H*,such that this coexistence equilibrium is asymptotically stable for 0<?<?H*and unstable for ?H*<?<?*;(?)there exists two Hopf bifurcation values ?H(1)and ?H(2),such that the stability switches induced by Allee effect occur,i.e.,this coexistence equilibrium is asymptotically stable for 0 ??<?H(1)or ?H(2)<?<?*,and unstable for ?H(1)<?<?H(2).Secondly,under the assumption that the positive equilibrium with higher predator biomass is stable,the influence of the diffusion term on the stability of the positive equilibrium and the pattern induced by diffusion are studied.The research results show that when the predator population spreads more slowly than the prey population,due to the existence of the Allee effect in predator,the system will experience diffusion-driven Turing instability at this time.In chapter 4,we summarize the results of the paper and look forward to the future work.