Qualitative Analysis Of A Predator Model With Toxins And Allee Effect

The predator-prey model has been favored by many scholars in recent decades.With the deepening of the research,experts have discovered that the interaction among the populations,and the interaction between population and environment,whose effects are significant for the population system.Based on the previous theories and techniques,this paper mainly studies the properties of solutions for two kinds of predator-prey models.One is predator-prey model with toxin effect the other is the modified Leslie-Gower model with Allee effect in predatorThe main contents in this paper are follows:In section 1,the realistic background of the above two models are introduced and the prepared knowledges are listed.In section 2,we investigate the prey-predator model with effect of toxin.Using bifurcation theory,the bifurcations from the semi-trivial solutions are acquired by looking on a and d as bifurcation parameters,and then the local branch is extended to the global branch.Furthermore,the stability of bifurcation solutions is obtained.In section 3.an improved Leslie-Gower model with Allee effect is discussed.Firstly,the priori estimate of positive solutions and the asymptotic stability of posi-tive constant solution are obtained.Secondly,the non-existence of the non-constant positive solutions is verified.Finally,sufficient condition for the existence of non-constant positive solutions is derived.