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The permanence and global attractivity of Lotka–Volterra competition system with feedback controls. (English) Zbl 1103.34038

The author studies the following Lotka-Volterra type competitive system with feedback control

x ˙ i (t)=x i (t)[b i (t)- j=1 n a ij (t)x j (t)-d i (t)u i (t)],u ˙ i (t)=r i (t)-e i (t)u i (t)+f i (t)x i (t),i=1,2,,n,(1)

where x i (t) represents the density of the ith species at time t, respectively, i=1,2,,n, and u i (t) is the control variable, i=1,2,,n. a ij (t),b i (t),d i (t), r i (t),e i (t),f i (t), i,j=1,2,,n, are continuous functions defined on [c,+). Given a function g(t) defined on [c,+), let g M =sup{g(t)|ct<+},g L =inf{g(t)|ct<+}· It is assumed in (1) that a ij (t)0,a ijM <+,a ijL >0,b i (t)>0,b iM <+,b iL >0,d i (t)0,d iM <+,d iL 0,r i (t)0,r iM <+,r iL 0,e i (t)>0,e iM <+,e iL >0,f i (t)>0,f iM <+ and f iL >0.

Some average conditions for the permanence of system (1) and sufficient conditions for the global attractivity of positive solutions of system (1) are derived, respectively. The results developed by J. Zhao, J. Jiang and A. C. Lazer [Nonlinear Anal., Real World Appl. 5, 265–276 (2004; Zbl 1085.34040)] are generalized.

34D05Asymptotic stability of ODE
34C25Periodic solutions of ODE
92D25Population dynamics (general)
34D20Stability of ODE
34D40Ultimate boundedness (MSC2000)