Dynamical Analysis Of Two Kinds Of Models With Impulsive State Feedback Control

In view of the problem of pest management,in this thesis,two types of ecological models with impulsive state feedback control are established.Using the theory and method of semi-continuous dynamical system and computer simulation,the dynamic properties of the ecological model under impulsive state feedback control are analyzed including the existence and stability of order-1 periodic solution,the existence of order-1 homoclinic cycle and the existence of order-1 homoclinic bifurcations,etc.Our research provides a theoretical basis for practical pest management.This thesis is divided into four chapters.In Chapter 1,we mainly give the research background and development status of this topic,the basic theoretical knowledge supporting this research and the main work and innovations of this thesis.In Chapter 2,based on the ecological significance of the model,we construct a kind of integrated pest management model with nonlinear impulsive state feedback control in food limited environment.Using the theory and method of differential equation,the dynamic properties of the model are analyzed.First,for a system without impulse effects,the equilibrium type of the system is obtained and the local stability of the equilibria is analyzed by using the eigenvalue method,and the globally asymptotic stability of the system's positive equilibrium is proved by constructing the Dulac function.Second,we focus on the analysis of the dynamic properties of the system under the nonlinear impulsive state feedback control.We construct a semi-continuous dynamic system,and according to the relative position of impulsive set and the phase set,we prove the existence of the order-1 periodic solution of the system in three cases by using the successor function,and we also prove the orbit asymptotic stability of the order-1 periodic solution of the system by using Analogue of Poincare criterion and obtain the sufficient conditions for the orbit asymptotic stability of the order-1 periodic solution.Finally,we give some numerical simulations and compare the properties of the solution of the system without and with impulsive effect.The results show that the impulsive state feedback control system can effectively simulate the integrated pest management.Under the impulsive state feedback control,the integrated pest management can be effectively implemented to maximize the protection of the ecological environment and maintain ecological balance.In Chapter 3,based on a kind of ecological model with undercrowding effect and Holling ? type functional response,we consider the effect of impulsive state feedback on the dynamics of the model.Using the theory and method of semi-continuous dynamic system,we analyze the dynamic properties of the ecological model under the control of impulsive state feedback.First,the existence of order-1 homoclinic cycle is proved by using the monotonicity of impulsive function,and then by choosing q₁ as the bifurcation parameter as the control parameter,the existence,uniqueness and asymptotic stability of the order-1 homoclinic bifurcations are considered.It is shown that under the control of impulsive state feedback,the system has rich dynamic properties,for example,the system has unique order-1 homoclinic cycle,and by selecting appropriate bifurcation parameter,when the bifurcation parameter changes,the order-1 homoclinic cycle disappears,and the system bifurcates out an orbit asymptotically stable order-1 periodic solution.In Chapter 4,we summary the whole thesis,and prospect the future research.