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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 $$\aligned \dot x_i(t)=x_i(t)[b_i(t)-\sum_{j=1}^n a_{ij}(t)x_j(t)-d_i(t)u_i(t)],\\ \dot u_i(t)=r_i(t)-e_i(t)u_i(t)+f_i(t)x_i(t),\quad i=1,2,\dots,n, \endaligned \tag1$$ 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 $g_M=\sup\{g(t)\vert c\leq t<+\infty\}, g_L=\inf\{g(t)\vert c\leq t<+\infty\}.$ 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 {\it J. Zhao, J. Jiang} and {\it 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)
Full Text: DOI
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