Behavior of Ising spins and ecological oscillators on dynamically rewired small-world networks

Many ecological populations are known to display a cyclic behavior with period 2. Previous work has shown that when a metapopulation (a group of coupled populations) with such dynamics is allowed to interact via nearest-neighbor dispersal in two dimensions, it undergoes a phase transition from disordered (spatially asynchronous) to ordered (spatially synchronous) that falls under the two-dimensional Ising universality class. While nearest-neighbor dispersal may satisfactorily describe how most individuals migrate between habitats, we should expect a small fraction of individuals to venture on a journey to further locations. We model this behavior by considering ecological oscillators on dynamically rewired small-world networks, in which at each time step a fraction p of the nearest-neighbor interactions is replaced by a new interaction with a random node on the network. We measure how this connectivity change affects the critical point for synchronizing ecological oscillators. Our results indicate that increasing the amount of long-range interaction (increasing p) favors the ordered regime, but the presence of memory in ecological oscillators leads to quantitative differences in how much long-range dispersal is needed to order the network, relative to an analogous network of Ising spins. We also show that, even for very small values of p, the phase transition falls into the mean-field universality class, and we argue that ecosystems where dispersal can occasionally happen across the system's length scale will display a phase transition in the mean-field universality class.