Ecological Dynamics I

Many ecological populations are known to display a cyclic behavior with period 2. Previous work has shown that when a group of metapopulations with such dynamics is allowed to interact via nearest neighbor dispersal in two dimensions, they undergo a phase transition from disorder (spatially asynchronous) to order (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 annealed 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 both Ising spins and ecological oscillators. Our results indicate that, in both cases, increasing the amount of long-range interaction favors the ordered regime, but for Ising spins the critical temperature saturates at a small value of p. We also show that, even for very small values of p, the phase transition falls under the mean field universality class. Finally, we argue that most transitions to synchrony in ecology are likely described by a mean field phase transition.