Abstract
In this paper, a May cooperative system with feedback controls is proposed and studied. The dynamic behaviors of the system are discussed by using the Lyapunov function method. If $b_{i}\neq0$ , $i=1,2$ , we show that feedback control variables have no influence on the global stability of the unique positive equilibrium of the system, which means that feedback control variables only change the position of the positive equilibrium and retain its global stability property. If $b_{i}= 0$ , $i=1,2$ , we can make the system which has a unique globally stable equilibrium or has unboundedly large solutions become globally stable. Some examples are given to illustrate the feasibility of the main results.
Highlights
May [ ] suggested the following set of equations to describe a pair of mutualists: dN dt = rN– N K + αN ( . ) – N K + βN, where N, N are the densities of the species, respectively. r, Ki, α, β, i =, are positive constants
U (t) = –e u (t) + d x (t), u (t) = –e u (t) + d x (t), where bi, aij, ci, ei, di, i, j =, are positive constants. xi(t), i =, denote the densities of the populations xi(t). ui(t), i =, denote feedback control variables. They showed that if the Lotka-Volterra competitive system is globally stable, the feedback control variables had no influence on the global stability of the system ( . )
If the Lotka-Volterra competitive system is showing extinction, they can make the extinct species become globally stable or still keep the property of extinction. They showed that the feedback control variables play an important role on the dynamic behavior of the system ( . )
Summary
Under the assumption that ri(t), ai(t), bi(t), ci(t), i = , , are all continuous T -periodic functions, and they obtained sufficient conditions which guarantee the existence of a unique globally asymptotically stable strictly positive periodic solution. In [ ], Chen et al first proposed the nonautonomous n-species cooperation system with continuous delays and feedback controls as follows: dxi(t) dt ri(t)xi(t) ). With regard to an autonomous system, since , Gopalsamy and Weng [ ] introduced feedback control variables into a two species competition system as follows: dx (t) dt
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