Abstract
In this paper, we study a predator-prey system with random predator dispersal over two habitat patches. We show that in most cases the dispersal delay does not affect the stability and instability of the coexistence equilibrium. However, if the mean time that the predator spent in one patch is much shorter than the timescale of reproduction of the prey and is larger than the double mean time of capture of prey, the dispersal delay can induce stability switches such that an otherwise unstable coexistence equilibrium can be stabilized over a finite number of stability intervals.
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