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.


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


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