Bifurcation Analysis Of General Autonomous Impulsive Semi-dynamical System With Applications To Pest And Disease Control

In the process of control and treatment of pests,infectious diseases and even tumors,it is often based on the number of pest populations,the size of susceptible populations and even the size of tumors to decide whether to adopt integrated pest control strategies,vaccination of infectious diseases,radiotherapy or chemotherapy of tumors,etc.The above threshold control strategies based on the number of pest populations,the size of susceptible populations and the size of tumors have been widely used in many fields such as biology,life sciences and agronomy.The mathe-matical models developed and established have a very important role in predicting and evaluating the effectiveness and sensitivity of threshold control strategies.How-ever,the mathematical model based on threshold control strategy is not continuous or even smooth,which brings challenges to the theoretical and numerical analyses of the systems.If the above feedback control strategy based on system state or threshold level is applied instantaneously,the corresponding state-dependent feedback control im-pulse system can provide a natural description of the above phenomena.In recent years,the qualitative theory of special state-dependent feedback control systems has been studied comprehensively.It mainly includes the existence and stability of the order k periodic solution of the system and the analysis of the complex dynamic behavior of the system by numerical techniques,and revealing the corresponding biological conclusions.However,the qualitative behavior of general planar state-dependent feedback control systems is rarely studied and there are some difficulties,especially the corresponding bifurcation analysis.Therefore,a class of generalized Kolmogorov systems with linear state-dependent feedback control is proposed in this paper.Various bifurcations of the boundary order-1 periodic solutions on key parameters are studied comprehensively and systematically,which has important theoretical and practical application value.In this paper,the critical conditions for the existence and stability of semi-trivial periodic solutions of the system axe obtained.Then,combining the properties of impulsive and differential dynamical systems,the one-parameter family of a dis-crete map,i.e.the Poincare map,is defined,which is determined by the difference equation of the iteration relation of the impulsive point sequence of the proposed system.Further,bifurcation theorems related to the discrete map are used to ad-dress the bifurcations of a semi-trivial periodic solution of the proposed impulsive semi-dynamical system.By selecting threshold level,control parameters and system parameters,the transcritical bifurcations,pitchfork bifurcations and backward bi-furcations related to semi-trivial periodic solutions and their corresponding critical conditions are obtained,as well as the flip bifurcations of order-1 periodic solutions within the system.A new method is developed to study the existence and stability of order-2 periodic solutions of systems.As applications,the main results are ap-plied to analyze the models concerning integrated pest management and infectious disease control,and the corresponding theoretical results and application fields are extended.