Dynamic Analysis Of Competitive Systems With State Feedback Control

The development,utilization and optimal management of renewable resources can promote the coordinated and sustainable development of population,resources and environment,and help us realize the overall progress of society and people.There is a competitive relationship in the progress of long-term development and evolution of the nature.The management of the competitive population is beneficial to the sustainable development and utilization of renewable resources,but different management measures will have different effects on the renewable biological resources.In this thesis,several competitive models with different management modes are constructed based on the mixed culture model of fish and crab in fishery.The effects of competition between two species on population density are studied by using the qualitative and stability theory of ordinary differential equation,and the effects of harvesting or releasing competing species to protect vulnerable species are discussed by using the theory of impulsive differential equation.This thesis is divided into the following four parts.In the first chapter,we introduc the research status,existing problems and related mathematical models of fish and crab mixed culture in population ecology and fishery,and gives some basic theoretical knowledge of ordinary differential equation and impulse differential equation.In the second chapter,with the background of breeding crayfish and river crab,a species competition model is established.Based on the nonlinear dynamical system theory,the qualitative and stability are nanlyzed,the positive,the boundedness of the model,as well as the existence and stability of equilibrium condition are obtained,and the theoretical results are verified by numerical simulation finally.The results showed that when there is no positive equilibrium point,crayfish or river crab tended to become extinct.When there is only a positive equilibrium point in the model,two species may coexist or one may go extinct.Bistability occurs when there are two positive equilibrium points.In the third chapter,based on the analysis of chapter 2,in the case of continuous model without intervention,there may be a species extinction in the competition process.In order to protect vulnerable species,prevent species extinction,and achieve the purpose of species coexistence and sustainable utilization,this chapter constructed a competition model with state feedback control.In this thesis,the existence and stability of the periodic solution of the state feedback control competition model are studied by using methods such as Poincaré mapping and analogue of the Poincaré criterion.The results show that the state feedback control is effective for the coexistence of two species,but there may be an imbalance between the two species.In the last chapter,according to the analysis of chapter 2,the occurrence of bistability is related to the initial value,and the fate of the two competing species will be changed.Under certain conditions,even if it is controlled by a pulse,there may be a phenomenon that one is relatively too much and the other is relatively too little when two competing species coexist.In this chapter,we consider a class of semi-continuous dynamics models with bilateral control and study the existence of periodic solutions.The numerical results show that under bilateral control,vulnerable species can be well protected and resources can be sustainably utilized.