  Many natural or human-made systems encompassing local reactions and diffusion
processes exhibit spatially distributed patterns of some relevant dynamical
variable. These interactions, through self-organization and critical phenomena,
give rise to power-law distributions, where emergent patterns and structures
become visible across vastly different scales. Recent observations reveal
power-law distributions in the spatial organization of, e.g., tree clusters and
forest patch sizes. Crucially, these patterns do not follow a spatially
periodic order but rather a statistical one. Unlike the spatially periodic
patterns elucidated by the Turing mechanism, the statistical order of these
particular vegetation patterns suggests an incomplete understanding of the
underlying mechanisms. Here, we present a novel self-segregation mechanism,
driving the emergence of power-law scalings in pattern-forming systems. The
model incorporates an Allee-logistic reaction term, responsible for the local
growth, and a nonlinear diffusion process accounting for positive interactions
and limited resources. According to a self-organized criticality (SOC)
principle, after an initial decrease, the system mass reaches an analytically
predictable threshold, beyond which it self-segregates into distinct clusters,
due to local positive interactions that promote cooperation. Numerical
investigations show that the distribution of cluster sizes obeys a power-law
with an exponential cutoff.
