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 (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 2-D 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 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.