  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 in 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, that is 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.
