Analysis And Control For Discrete Hierarchical Model Of Two-species Competing System

In real ecological environments,every biological population often coexists with other populations,and there are competitive relationships between the populations,they compete for resources such as food,water,and territory.In addition,there are differences in hierarchical status among individuals within a population,which has an important impact on individuals’ acquisition of survival resources,and in turn has an impact on the evolution of the populations.By considering the above two factors comprehensively,a class of two-species competition model with discrete hierarchical structure is established in this thesis,to describe the evolution process of the population system.The model is a class of nonlinear high-dimensional system of difference equations.In the thesis,the eigenvalue theory of non-negative matrices,discrete system dynamics and control theory are used to study the dynamic behaviors and control problems of the model.The research results are included in the chapter 2 and Chapter 3.The dynamic properties of the model are analyzed in the chapter 2.The population system model is proposed and the meaning of the variables and parameters in the model are explained in section 1.The boundedness of the model solution(the state of the population system)is proved and the existence of a positive equilibrium is discussed in section 2.In section 3,the eigenvalue theory of primitive matrices is used to prove the local stability of the zero equilibrium;and according to the Gersgorin discs theorem,the local asymptotic stability of the non-negative equilibrium is given;finally,the global asymptotic stability of zero-equilibrium and non-negative equilibrium are established by constructing the Lyapunov functions.The validity of the theoretical results are verified by numerical simulations conducted by MATLAB software in section 4.Some control problems of the model are studied in chapter 3.In section 1,the state model of the control system is introduced briefly,then the controllability and stabilizability of the system are analyzed,and the feedback matrix of system is also given.Section2 mainly focuses on the optimal harvesting problem of the system.The existence of the optimal control strategy is proved;then based on the Pontryagin principle of discrete systems,the necessary conditions to be satisfied by the optimal harvesting strategy of the control system and the specific description of the optimal harvesting strategy are derived;furthermore,the optimal harvesting strategy formula are specified in three special cases.In section 3,some numerical simulations are carried out by using LINGO software,to investigate the effects of changes in parameters,such as individual economic value,on the optimal strategy and maximum economic benefit.