A discrete predator-prey system with age-structure for predator and natural barriers for prey

We analyze a two species discrete predator-prey model in which the prey dis perses between two patches of a heterogeneous environment with barriers and the mature predator disperses between the patches with no barrier. By usin g the discrete dynamical system generated by a monotone, concave maps for s ubcommunity of prey, we obtain the subcommunity of prey exists an equilibri um which attracts all positive solutions, and using the stability trichotom y results on the monotone and continuous operator, we obtain some sufficien t conditions for the permanence of species. These results are applied to th e models with rational growth functions and exponential growth functions. W e also present numerical examples to illustrate the dynamic complexity of s ystems.