Entropy evaluation sheds light on ecosystem complexity

Preserving biodiversity and ecosystem stability is a challenge that can be pursued through modern statistical mechanics modeling. Here we introduce a variational maximum entropy-based algorithm to evaluate the entropy of a minimal ecosystem on a lattice in which two species struggle for survival. The method quantitatively reproduces the scale-free law of the prey shoals size, where the simpler mean-field approach fails: the direct near-neighbor correlations are found to be the fundamental ingredient describing the system self-organized behavior. Furthermore, entropy allows the measurement of structural ordering we found to be a key ingredient in characterizing two different coexistence behaviors, one where predators form localized patches in a sea of preys and another where species display more complex patterns. The general nature of the introduced method paves the way for its application in many other systems of interest.