  Long-range synchrony from short-range interactions is a familiar pattern in
biological and physical systems, many of which share a common set of
``universal'' properties at the point of synchronization. Common biological
systems of coupled oscillators have been shown to be members of the Ising
universality class, meaning that the very simple Ising model replicates certain
spatial statistics of these systems at stationarity. This observation is useful
because it reveals which aspects of spatial pattern arise independently of the
details governing local dynamics, resulting in both deeper understanding of and
a simpler baseline model for biological synchrony. However, in many situations
a system's dynamics are of greater interest than their static spatial
properties. Here, we ask whether a dynamical Ising model can replicate
universal and non-universal features of ecological systems, using noisy coupled
metapopulation models with two-cycle dynamics as a case study. The standard
Ising model makes unrealistic dynamical predictions, but the Ising model with
memory corrects this by using an additional parameter to reflect the tendency
for local dynamics to maintain their phase of oscillation. By fitting the two
parameters of the Ising model with memory to simulated ecological dynamics, we
assess the correspondence between the Ising and ecological models in several of
their features (location of the critical boundary in parameter space between
synchronous and asynchronous dynamics, probability of local phase changes, and
ability to predict future dynamics). We find that the Ising model with memory
is reasonably good at representing these properties of ecological
metapopulations. The correspondence between these models creates the potential
for the simple and well-known Ising class of models to become a valuable tool
for understanding complex biological systems.
