Research On The Persistence,Extinction And Stability Of Three Ecosystem With Feedback Controls

This paper consists of three parts:Firstly,a two species Lotka-Volterra predator-prey system with single feedback control is investigated.By constructing suitable Lyapunov functions,sufficient conditions which guarantee the global stability of the positive equilibrium and boundary equilibrium are obtained,respectively.Our results show that under suitable conditions,the single feedback control variable only change the position of positive equilibrium and boundary equilibrium,and retain its global stability property.Furthermore,if the predator species is extinction in the Lotka-Volettra system,then the feedback control variable has no influence on the dynamic behaviors of the predator species,i.e.,the predator species will still be driven to extinction.Secondly,a new kind of Lotka-Volterra competitive system with the effect of toxic substances and feedback controls is proposed and studied,where every feedback control variables could have direct influence on any species of the system.By constructing a suitable discrete Lyapunov type extinction function,we obtain the sufficient conditions which guarantee that one of the two species will be driven to extinction;By giving a detail analysis of the right-hand side of the system,sufficient conditions which ensure the permanence of the rest species is obtained.We also obtain the sufficient conditions which ensure the permanence and global stability of the system.Thirdly,under the assumption that the intrinsic growth rate maybe negative,we revisit the dynamic behaviors of the Lotka-Volterra competitive system with the effect of toxic substances and feedback controls.Our results show that the feedback controls and toxic substances have no effect on the permanence and extinction of species.Moreover,by constructing a suitable discrete Lyapunov function,we derive sufficient conditions which ensure the global stability of the positive solutions.