Dynamic Behaviors Of Four Kinds Of Competition And Predator-prey Systems With Feedback Controls

This paper consists of four parts:Firstly, we consider a non-autonomous allelopathic phytoplankton model with feedback controls. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, a set of sufficient conditions which guarantee the permanence and global attractivity of the system are obtained. Results of numerical simulations are provided.Secondly, a two species Lotka-Volterra competitive system with feedback controls and infinite delays is studied. The traditional Lotka-Volterra competitive system has three cases:(i) globally stable, (ii) extinct and (iii) bistable. By constructing a suitable Lyapunov functional, we show that if the Lotka-Volterra competitive system is bistable (in the absence of feedback controls), then by choosing the suitable values of feedback control variables, one of the species is driven to extinction while the other one becomes globally stable. Our results complement those of Li, Han and Chen[14], Examples together with their numerical simulations are presented to verify the feasibility of our results.Thirdly, a two species Lotka-Volterra competitive system with single feedback control variable and infinite delays is studied, such a strategy has influence on both species. Sufficient conditions which ensure the extinction and global stability of the system are obtained, respectively. Our extinction result shows that for the traditional Lotka-Volterra competitive system, in any one of the cases:globally stable, extinct and bistable, by choosing the suitable feedback control variable, one of the species will be driven to extinction, while the other one will stabilize at a positive equilibrium. Specially, by choosing the suitable feedback control variable, the extinct species in original system could become permanent, while the permanent species in original system will be driven to extinction. This is the first time that such a phenomenon been discussed. Our stability result shows that under certain conditions, the extinct species can become globally stable and the stable species still keep the property of stability. Our findings may be used to protect the endangered species. Examples together with their numerical simulations show the feasibility of our results.Finally, we study a Leslie-Gower predator-prey system with feedback control on predator species. The global stability of the system is investigated by constructing a suitable Lyapunov function. The result indicates that feedback control variable only changes the position of the interior equilibrium--the predator’s density decreases while the prey’s increases and has no influence on the stability property of the system. An example together with its numerical simulations shows the feasibility of our main result.