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Permanence and global stability in a discrete \(n\)-species competition system with feedback controls. (English) Zbl 1154.34352

Summary: We investigate the following discrete \(n\)-species competition system with feedback controls: \[ \begin{aligned} x_i(k+1)= & x_i(k)\text{exp} \left\{b_i(k)-\sum^n_{j=1} a_{ij}(k)x_j(k)-\sum^n_{j=1,j\neq i} c_{ij}(k)x_i(k)x_j(k)-d_i(k)u_i(k)\right\}, \\ \Delta u_i(k)=r_i & (k)-e_i(k)u_i(k)+f_i(k)x_i(k),\quad i=1,2,\cdots, n\end{aligned} \] which describes the effect of toxic substances and age structures simultaneously. Some sufficient conditions are established on the permanence and the global stability of the system. These results are also applied to some special cases.

MSC:

34D23 Global stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
93B52 Feedback control
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