Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system
The main result is the following Theorem 1: If the inequalities \[ r_1\sum^m_{j=1} b_{2j}-r_2 \sum^m_{j=1} b_{1j}>0,\;r_1\sum^m_{j =1} a_{2j}+r_2 \sum^m_{j=1} a_{1j}>0 \] hold, then the Lotka-Volterra discrete predator-prey system \[ \begin{gathered} x(n+1)= x(n)\exp \left[r_1-\sum^m_{j=1} a_{1j}x (n-k_{ij}) -\sum^m_{j=1} b_{1j}y (n-k_{2j})\right],\\ y(n+1)= y(n)\exp \left[r_2 +\sum^m_{j=1} a_{2j}x(n-l_{1j}) -\sum^m_{j=1} b_{2j} y(n-l_{2j}) \right],\end{gathered} \] is permanent for nonnegative integers \(k_{ij}\) and \(l_{ij}\) \((i=1, 2,j=1,\dots,m)\). Here \(r_1\) and \(r_2\) are constants and \(a_{ij}\) and \(b_{ij}\) are nonnegative constants. The existence of an equilibrium and the impersistence of the system are considered too.
Lotka-Volterra discrete predator-prey system
scientific article; zbMATH DE number 1788036