The Study On The Dynamical Behaviors In Species And Epidemic Model With State Dependent Impulse Control

In recent years, state dependent impulse differential equations has been widespread attention to domestic and international scholars, especially with the study on the dy-namical models with state dependent impulsive control strategies. This dissertation has conducted the research focusing on the population dynamical models and epidemic dynamical model with state dependent impulse control, the effects of state dependent impulsive control on the extinction and persistence of population, the extinct and per-manence of epidemic are investigated. Its main contents can be summarized as follows:1. In the first part, we mainly discuss the dynamics behaviors of a two-species com-petitive models with state dependent impulsive effects. By the Poincare map, analogue of Poincare criterion and qualitative analysis method, some sufficient conditions on the existence and orbitally asymptotical stability of the semi-trivial periodic solution are obtained. Further, by the bifurcation theorem, the positive periodic solution bifurcates from the semi-trivial solution through a fold bifurcation are obtained. Finally, a discus-sion on dynamics behaviors of two-species competitive model with multi-state dependent impulsive effects, some general criteria for the existence and stability of two semi-trivial solutions and two positive periodic solutions are established. Numerical simulations are carried out to demonstrate the feasibility of the main results.2. In the second part, two-species competitive model with state dependent impulsive replenish the endangered species is proposed, the existence and orbitally asymptotical stability of positive order-1 periodic solution and semi-trivial periodic solution are ob-tained. Numerical simulations are carried out to illustrate the feasibility of our main results and the suitability of state dependent impulsive control strategies.3. In the third part, we propose a novel SIR epidemic dynamical control mod-el, where the state dependent impulsive vaccination and medication treatment control strategies are being introduced to prevent the spread of disease at different control thresh-old values. Some sufficient conditions of two positive order-1 periodic solutions for this control model are constructed. Numerical simulation time phase diagram was given, under the control strategy is verified enables to effectively control the spread of the disease.