On The Dynamic Of Nonautonomous Lotka-Volterra Competitive Systems With Feedback Controls

As we know, ecosystem in the real world continuously disturbed by unpredictableforces which can result in changes in the biological parameters such as survival rates.Of practical interest in ecosystem is the question of whether or not an ecosystem canwithstand those unpredictable forces which persist for a finite period of time. In the lan-guage of control variables, we call the disturbance functions as control variables. However,on the other hand, a lot of experiment illustrate that it may realize the permanence ofthe species by feedback controls. These will have very important theoretical and practicalmeaning to keep the population diversity and to sustainable development of ecosystem. Inthe paper, we will investigate the dynamical behaviors of nonautonomous Lotka-Volterracompetitive systems with feedback controls from there facts. they include boundedness,permanence, global attractive, periodicity and almost periodicity.In the second section, the permanence of nonautonomous n-species Lotka-Volterracompetitive systems with feedback controls is studied. firstly, the boundedness of allpositive solutions of the system discussed using the di?erential inequalities method andcomparison theorem. Then, some new integrational criteria on the permanence for allpositive solutions are established by analysis technique.The corresponding results givenby Chen in [8] are improved. Finally, some corresponding results are given for periodicsystem.In the third section, the global attractivity of nonautonomous n-species Lotka-Volterracompetitive systems with feedback controls is discussed. The proof contributes to theworks of the global attractivity of nonautonomous n-species Lotka-Volterra competitivesystems with delays and feedback controls in [11]. new su?cient conditions of whichall solution of the system is global attractive are established by constructing Liapunovfunctionals and Barbalat's lemma.In the forth section, the almost periodic solution of nonautonomous n-species Lotka- Volterra competitive systems with feedback controls is studied. We mainly discuss the hullequation of the feedback controls. For almost periodic system, it will exist the only globalasymptotic stability almost solution. And some examples to illustrate the conclusionsobtained in this paper are given as the application of theorems.