Qualitative Study On Two Classes Of Differential Systems With State-dependent Impulses

Pulse control is widely utilized in biology,econmy,engineering technology and other fields,such as the treatment of infectious diseases,pest management,fishing and so on.In the past twenty years,the impulsive differential equations have been studied deeply,especially the state-dependent impulsive equations.Such issues become more complicated because of the discontinuity of the system.In this paper,based on the existing research,the dynamic properties of two classes of state-dependent impulse differential systems are studied.In the second chapter,a class of general predator-prey models with state-dependent impulses is studied.Based on geometric analysis,Poincaré maps and successor functions,three typical Bendixson domains are constructed,and some sufficient conditions for the existence of order-1 periodic solutions are obtained.At the same time,we make some discussion on the existing domains according to the system parameters.Furthermore,the asymptotic stability of the order-1 periodic solution is obtained by exploiting the Analogue of Poincare′ Criterion.Finally,to illustrate the theoretical results,an example is given and some numerical simulations are carried out.In the third chapter,a class of SIR models with nonlinear statedependent impulses is studied.Comprehensive interventions including vaccination,isolation and treatment are carried out when the susceptible population reaches a threshold.Based on the Poincaré map and the basic reproduction number,the existence and the stability of the disease-free periodic solution of the system are established.In addition,by using the theory of the bifurcation of the single-parameter mappings related to Poincaré maps,it is found that there are some transcritical bifurcation and Pitchfork bifurcation occur.