Dynamic Behaviors Of Three Types Of Stage-structured Population System

This paper consists of three parts:Firstly,a ratio-dependent predator-prey system with a constant proportion of prey refuge and stage-structure for prey species is proposed and studied.By applying the comparison principle of differential equations,a set of sufficient conditions which ensure the permanence of the system are obtained.After that,by using of variational matrix,the local stability of the equilibria is discussed,we show that the stage structure and refuge play an important effect on the local stability of the positive equilibrium.Furthermore,we discuss the effect of refuge on the prey and predator densities.Finally,by introducing a new lemma and applying the iterative method,we obtain some sufficient conditions which ensure the global attractivity of the equilibria of the system.Several examples show the feasibility of our results.Secondly,we introduce and study a stage-structured plant-pollinator mutualism model with Beddington-DeAngelis functional response.By applying the comparison principle of differential equations,we obtain some sufficient conditions which ensure the permanence of the system.After that,by analyzing the variational matrix,the local stability of the equilibria is investigated.Introducing a new lemma and applying the iterative method,under some suitable assumption,we prove the global attractivity of the system.Our results show the mutual interference concerning pollinators has no effect on permanence and the extinction of system.Thirdly,a stage-structured predator-prey model with Allee effect on predators pecies and harvesting on prey species is introduced and studied.The local stability of the positive equilibrium is discussed by analyzing the variational matrix.Furthermore,by introducing two new lemmas and applying the iterative method,some suitable sufficient conditions which ensure the global attractivity of the equilibrium of the system are obtained.Our results show that the Allee effect has no effect on the local stability of the positive equilibrium and the global attractivity of the equilibria,such a result is different from the known results.