The system of differential equations , , is said to be permanent if there exists a compact set such that all solutions starting in ultimately enter and remain in .
The authors study the following predator-pray model in a patchy environment
where denotes the species in patch ; , , , , are continuous -periodic functions; is the dispersal coefficient of the species from patch to patch , ; the predator functional response is bounded as , , and . Necessary and sufficient conditions for the permanence of the above system are presented. Sufficient conditions for the permanence of the single-species system in the absence of a predator () is also obtained. The approach is based on the well-known properties of the periodic logistic model .
The biological interpretion of the main results is given.