Transition states of two-cycle ecological oscillators: Blume-Capel representation

Many spatially-extended systems of ecological oscillators exhibit spatial synchrony with periodic oscillations in time. If the individual oscillators have two-cycle behavior, the transition to synchrony as a function of noise and coupling strength is in the Ising universality class. It was shown that a dynamical Ising model with memory does a good job in representing such ecological systems and predicting their future states[1]. In the Ising representation, the two phases of oscillations (high at odd times or high at even times) of an individual oscillator are represented by spin-up and spin-down. However, the behavior of an individual ecological oscillator suggests the existence of a transition state along with the two phases of oscillations. The oscillations at this transition state have amplitude very close to zero. To study such systems, we use Blume-Capel representation where the spin can take three values S={+1,-1,0} with S=0 as the transition state and S={-1,+1} as the two phases of oscillations. We model the spatially-extended ecological systems with coupled lattice maps in two-cycle regime and represent them with the Blume-Capel model. We also discuss maximum likelihood methods to infer the Blume-Capel representation.
1.V.Nareddy,et.al,J R Soc Interface2020