Representing spatial ecological oscillators by dynamical Ising model with memory

Spatial synchronization in many biological systems are known to develop from short-range interactions of local oscillators. Locally-coupled ecological oscillators with noise and two-cycle behavior undergo a phase transition from incoherence to synchrony. These phase transitions exist in the Ising universality class, ensuring that the stationary properties of the ecological systems can be replicated by the simple Ising model. The universal properties shared by all the models in the universality class match that of the Ising model. Here we are interested in studying the dynamical properties shared between the ecological oscillators and the Ising model as synchronization is a dynamic phenomenon. We show that we need to go beyond the simple Ising model with nearest neighbor coupling and add a memory term to explain the tendency of local oscillators to maintain their phase of oscillations. We infer the Ising parameters using maximum likelihood methods by representing the ecological oscillators with the dynamical Ising model with memory. This correspondence to the dynamical Ising model is useful as it reveals that the spatial properties arise independent of local dynamics and the Ising parameters play a clear role in both understanding and predicting the dynamics of the ecological system. We study the location of phase transition in Ising parameter space and the ability of the dynamical Ising model to predict the future dynamics. We find that the simple dynamical Ising model is reasonable good at representing the ecological oscillators. This agreement between the dynamics of spatially-coupled ecological oscillators and the dynamical Ising model suggests the potential for simplification of many complex biological systems.