Synchronization in spatially extend

Synchronization in spatially extended ecological systems has been a topic of continuous interest [1–14]. In a variety of ecosystems, cyclic patterns across space that persist in time are ubiquitous, in which synchronous dynamics are believed to play an important role [2,3,11,14]. For example, in a network of predator–prey systems, chaotic phase synchronization was uncovered, providing an explanation for a class of ecological cycles, e.g. the hare-lynx cycle [15–19], in which the populations in different spatial regions oscillate synchronously and periodically in phase but their peak abundances are different and vary erratically with time [2,3]. More recently, synchronous dynamics were exploited to explain the correlations across space of cyclic dynamics in ecology, especially in terms of yield from pistachio trees [11,14]. Based on a large data set from over 6500 trees in a pistachio orchard in California, the authors established a surprising link between the spatially networked system of pistachio trees and the Ising model in statistical physics, with the common trait that local, neighbor-to-neighbor interactions (root grafting for the former and spin interactions for the latter) can generate correlation and synchronization over large distances