The joint impacts of dispersal delay and dispersal patterns on the stability of predator-prey metacommunities

A predator-prey metapopulation model over arbitrary number of patches is considered in this paper. The model assumes that only prey move (with a dispersal delay) between all connected patches. Two cases of dispersal patterns are considered. For the case the dispersal of prey is due to random effect only (independent of predator density), we show that either the dispersal delay is harmless in the sense that it does not affect the stability of the metacommunity, or the dispersal delay can induce stability switches with finite number of stability intervals. For the case the dispersal of prey is due to predator-avoidance (dependent on predator density), we show that the interplay of density-dependent dispersal and dispersal delay may also induce finite number of stability switches. This indicates that the combination of the density-dependent dispersal and dispersal delay can exhibit both stabilizing and destabilizing effects on the stability of the coexistence equilibrium. Our results show that the delay and the patterns of prey dispersal jointly affect the stability of predator-prey metacommunities and can induce multiple stability switches.