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Permanence of a discrete \(n\)-species food-chain system with time delays. (English) Zbl 1109.92048

From the paper: We consider the permanence of the following nonautonomous discrete \(n\)-species food-chain system with time delays of the form: \[ x_1(k+1)= x_1(k)\exp\left\{r_1(k)-a_{11}(k)x_1(k-\tau_{11})-\frac{a_{12}(k)x_2(k)}{1+m_1 x_1(k)}\right\}, \]
\[ x_j(k+1)=x_j(k)\exp\left\{-r_j(k)+\frac{a_{j,j-1}(k)x_{j-1} (k-\tau_{j,j-1})}{1+m_{j-1}x_{j-1}(k-\tau_{j,j-1})}-a_{jj}(k)x_j(k-\tau_{jj})\right.+ \]
\[ \left.-\frac{a_{j,j+1}(k)x_{j+1}(k)}{1+m_{j+1}x_{j+1}(k)}\right\},\quad \text{with}\;1<j<n, \]
\[ x_n(k+1) =x_n(k)\exp\left\{-r_n(k)+\frac{a_{n,n-1}(k)x_{n-1}(k-\tau_{n,n-1})}{1+m_{n-1} x_{n-1}(k-\tau_{n,n-1})}-a_{nn}(k)x_n(k-\tau_{nn})\right\}, \] where \(x_i(k)\) \((i=1,\dots,n)\) is the density of the \(i\) th species \(X_i\). By applying the comparison theorem for difference equations, sufficient conditions are obtained for the permanence of the system.

MSC:

92D40 Ecology
39A11 Stability of difference equations (MSC2000)
37N25 Dynamical systems in biology
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