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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 \begin{aligned} \dot x_i(t)=x_i(t)[b_i(t)-\sum_{j=1}^n a_{ij}(t)x_j(t)-d_i(t)u_i(t)],\\ \dot u_i(t)=r_i(t)-e_i(t)u_i(t)+f_i(t)x_i(t),\quad i=1,2,\dots,n, \end{aligned} \tag{1} where $$x_i(t)$$ represents the density of the $$i$$th species at time $$t$$, respectively, $$i=1,2,\dots,n$$, and $$u_i(t)$$ is the control variable, $$i=1,2,\dots,n$$. $$a_{ij}(t), b_i(t), d_i(t),$$ $$r_i(t), e_i(t), f_i(t)$$, $$i,j=1,2,\dots, n$$, are continuous functions defined on $$[c,+\infty)$$. Given a function $$g(t)$$ defined on $$[c,+\infty)$$, let $$g_M=\sup\{g(t)| c\leq t<+\infty\}, g_L=\inf\{g(t)| c\leq t<+\infty\}.$$ It is assumed in (1) that $$a_{ij}(t)\geq 0, a_{ijM}<+\infty, a_{ijL}>0, b_i(t)>0, b_{iM}<+\infty, b_{iL}>0, d_i(t)\geq 0, d_{iM}<+\infty, d_{iL}\geq 0, r_i(t)\geq 0, r_{iM}<+\infty, r_{iL}\geq 0, e_i(t)>0, e_{iM}<+\infty, e_{iL}>0, f_i(t)>0, f_{iM}<+\infty$$ 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.

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 92D25 Population dynamics (general) 34D20 Stability of solutions to ordinary differential equations 34D40 Ultimate boundedness (MSC2000)
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