A possible mechanism for the attainment of out-of-phase periodic dynamics in two chaotic subpopulations coupled at low dispersal rate

Much research in metapopulation dynamics has concentrated on identifying factors that affect coherence in spatially structured systems. In this regard, conditions for the attainment of out-of-phase dynamics have received considerable attention, due to the stabilizing effect of asynchrony on global dynamics. At low to moderate rates of dispersal, two coupled subpopulations with intrinsically chaotic dynamics tend to go out-of-phase with one another and also become periodic in their dynamics. The onset of out-of-phase dynamics and periodicity typically coincide. Here, we propose a possible mechanism for the onset of out-of-phase dynamics, and also the stabilization of chaos to periodicity, in two coupled subpopulations with intrinsically chaotic dynamics. We suggest that the onset of out-of-phase dynamics is due to the propensity of chaotic subpopulations governed by a steep, single-humped one-dimensional population growth model to repeatedly reach low subpopulation sizes that are close to a value Nt=A (A≠equilibrium population size, K) for which Nt+1=K. Subpopulations with very similar low sizes, but on opposite sides of A, will become out-of-phase in the next generation, as they will end up at sizes on opposite sides of K, resulting in positive growth for one subpopulation and negative growth for the other. The key to the stabilization of out-of-phase periodic dynamics in this mechanism is the net effect of dispersal placing upper and lower bounds to subpopulation size in the two coupled subpopulations, once they have become out-of-phase. We tested various components of this proposed mechanism by simulations using the Ricker model, and the results of the simulations are consistent with predictions from the hypothesized mechanism. Similar results were also obtained using the logistic and Hassell models, and with the Ricker model incorporating the possibility of extinction, suggesting that the proposed mechanism could be key to the attainment and maintenance of out-of-phase periodicity in two-patch metapopulations where each patch has local dynamics governed by a single-humped population growth model.