Dynamics Of The Stage-structured Predator-prey Model And The Global Behavior Of A Monotonic Syste

This paper focuses on exploring the long-term behavior of the stage-structured predatorprey model with Beddington-DeAngelis functional response,the general stage-structured predator-prey model with predator interference,and a system of two specie-groups competition with an inhibitor,which includes the following three aspects:Ⅰ.The existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response will be investigated.We first derive an existence criterion of periodic orbit in terms of inequalities of system’s nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation and hypersurface theories.We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value,or between its Hopf bifurcating values.Combining theoretic analysis and numerical simulations,we investigate global bifurcations of limit cycles for a varying parameter system,which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles.Similar results are obtained for the system without mutual interference.Ⅱ.We study the global asymptotic stability of a general stage-structured predator-prey model by using the Lyapunov method,and explore how the predator interference affects and changes the model’s dynamic behavior.We first consider the equilibrium of the system and provide the necessary and sufficient condition for the existence of the positive equilibrium,and investigate the global stability of the boundary equilibrium.Then,we present the necessary and sufficient condition for the uniform persistence of the system.A criterion is derived by Lyapunov method for the unique positive equilibrium to be globally stable.Applying it to the model whose predator rate is of either Beddington-DeAngelis type or Crowley-Martin type,its unique positive equilibrium is globally asymptotically stable when the intensity of mutual interference is suitably strong,which rigorously proves that the presence of mutual interference by predators can globally stabilize the positive equilibrium and eliminate limit cycles,as has been demonstrated by[51]in the model without considering stage-structure.Finally,numerical simulation and comparison can show the superiority of our global asymptotic stability criterion.Ⅲ.Two specie-groups competition model with an inhibitor is established,and conducting a complete classification of its global dynamics.Firstly,we study the boundary dynamics of the system and discuss the stability of the boundary equilibrium.Subsequently,we provided the necessary and sufficient conditions for the existence of two and unique nontrivial positive equilibrium,and prove their hyperbolicity.Based on these results,we give the classification for the global dynamics of the system by applying the connecting orbit theory and the properties of type-K monotone systems.This classification is discussed according to the cases of two,unique and no positive equilibrium,thus providing a complete classification of the global dynamics of the system according to the relationship between the coefficients.