Qualitative Theoretical Research On State-dependent Feedback Control Ecosystems

It is well-known that many important applications in life sciences,agriculture,and medicine can be described and characterized by impulsive semi-dynamical sys-tems.In some applications such as impulsive vaccinations and drug administrations for disease prevention and treatment,the impulsive control is implemented at fixed moments to reflect how human actions are taken at fixed periods.In other appli-cations,however,state-dependent control strategies have to be used,for example,in an integrated pest management(IPM)strategy,actions are taken only when the population of pests reaches an economic threshold;The control of diabe.tic patients'blood glucose is also based on the monitoring of blood glucose concentration and the implementation of control measures of the insulin injection.Whether it's the economic threshold of pest control or the blood glucose level of diabetes treat-ment,the feedback control is adopted depending on the state variable reaching a certain threshold level,and state-dependent impulsive differential equation is often used to describe it.In recent years,the modeling and theoretical analysis of state-dependent impulsive differential equation have been developed rapidly,however,the non-smoothness of the system brings great challenges to the complete analysis of such systems and restricts their applications.Therefore,this paper aims to develop a qualitative method for analyzing the global dynamic behavior of state-dependent feedback control impulse system and to promote its theoretical and applied devel-opment.Based on this,this paper selects a state-dependent impulse system that is widely used in the field of integrated pest management,HIV and tumor immunotherapy,and develops novel analytical techniques to analyze the qualitative behavior of the impulse dynamical system.In Chapter 2,we firstly give the exact domains of the impulsive set and the phase set of the model,and determine the analytical formula of the Poincaré map defined by the impulsive points in the phase set.The main property related to the differential systems are studied by using the first integral and Lambert W functions and it's property;Secondly,we determine the key parameters of the model(e.g.the instant killing rate,threshold level and releasing constant),according to the relationship of parameter space,which can be divided into three cases and the exact domain of the model's impulse set and phase set are obtained Finally,we examine the sign of three key parameters and two important variables and the relation with the domain of Poincaré map,using Lambert W functions gives the analytical expression of Poincaré map,all those provide the useful tools for the qualitative analysis of system in the following chaptersIn Chapter 3,we first discuss some important equalities and inequalities and their relations by using the key parameters of state-dependent feedback control(releasing constant).Then,we employ the analytical formula of the Poincaré map to investigate the existence and stability of the fixed point of the Poincaré map in the phase set,which can help us to obtain the existence and stability of the order-1 limit cycles of the original impulse system with state-dependent feedback control In particular,we completely give the existence,local and global stability of the system order-1 limit cycle;as well as the sharp sufficient conditions for the global stability of the boundary order-1 limit cycle;We examine the flip bifurcation related to the existence of an order-2 limit cycle.We show that the existence of an order-2 limit cycle implies the existence of an order-1 limit cycle,these lay a foundation for qualitative analysis of state-dependent impulse systemIn Chapter 4,to address the global dynamic behavior of the model related to state-dependent feedback control completely,firstly,we derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite state-dependent feedback control actions.Secondly,we also prove that there does not exist a limit cycle with order-k(k?3),and solve a hard problem of qualitative analysis in this aspect.Finally,we investigate multiple attractors and their basins of attraction,as well as the interior structure of a horseshoe-like attractor,and discuss implications of the global dynamics for IPM strategy in detailBased on the study in Chapter 4,it is known that the solution of the system may experience a finite number of impulsive effects or even no impulsive effect,that is to say,the domain and range of Poincaré map can be very complex,in particular,it may arise non-smooth even discontinuity,this makes it great difficult to study the qualitative behavior of the impulsive system by using Poincaré map.Based on this,in Chapter 5,we give a realistic exarmple of the discontinuity of Poincaré map,by the continuity of the function describing the times of meeting impulsive set(equivalent to the continuity of Poincaré map),we give a general sufficient condition to guarantee the continuity of Poincaré map,it provides an important theoretical support to comprehensively and systematically analyze the global qualitative behavior of the state-dependent impulsive system.