The Dynamic Mechanism On The State-dapendent Impulse Of A Predator-prey System With Allee Effect

In this paper,the dynamics behavior of a predator-prey system with state-dependent impulsive control strategy and Allee effect is studied theoretically and numerically,and some meaningful results are obtained.In Chapter 1,the backgrounds and research status on mathematical system and impulsive control strategy are introduced.Some basic concepts and lemma are also introduced,which provides a theoretical basis for the subsequent chapters.In Chapter 2,a ecological predator-prey dynamic system with Allee effect is constructed.How some key factors influence dynamic evolutionary mechanism of steady conversion and bifurcation behavior in predator-prey system is explored.According to the stability analysis theory of dynamics,the equilibria and bifurcation behaviors are analyzed.Which are proved that the existence and stability of the equilibria,as well as the occurrence of bifurcation behaviors will be controlled by a certain parameter.The corresponding numerical simulation ensures reliability of the theoretical results.In the following chapters,predator-prey systems in the presence of state-dependent impulsive control strategy are established on the predator-prey dynamic system that constructed in this chapter,then,this chapter is the basis of the following research.In Chapter 3,based on the results obtained in chapter 2,a predator-prey system with state-dependent impulsive control strategy is established,the Homoclinic and Heteroclinic bifurcations of this predator-prey system are studied.Firstly,the existence of Homoclinic and Heteroclinic cycles are determined by the geometric method in semi-continuous dynamic systems.Then,through the theory of rotated vector field,it is proved that the system exhibits Homoclinic and Heteroclinic bifurcations.And the existence and uniqueness of order-1periodic solution that arising through bifurcation also been obtained.Finally,for all these theoretical results,the corresponding numerical examples are provided respectively.In Chapter 4,according to the above results,another state-dependent impulsive control strategy is introduced on the predator-prey dynamic system in chapter 2.The dynamical behavior of predator-prey system with state-dependent impulsive control strategy is researched.Firstly,it is proved that system own an orbitally asymptotically stable semi-trivial periodic solution when ?(28)0 by means of the analogue of the Poincaré criterion.Then,in the case of ?(29)0,the existence and stability of positive periodic solution are discussed.Finally,those results are verified through numerical simulations.