# Spatial patterns emerging from a stochastic process near criticality

## TL;DR

This paper introduces a minimal stochastic model for community dynamics near criticality, analytically deriving spatial distribution properties that align with empirical biodiversity patterns in tropical rain forests.

## Contribution

It provides an analytical framework for understanding spatial patterns in communities near critical points, incorporating fluctuations and violating detailed balance.

## Key findings

- Analytical probability density functions for local populations derived.
- Model predictions match observed biodiversity patterns.
- Simulations confirm analytical results in relevant regimes.

## Abstract

There is mounting empirical evidence that many communities of living organisms display key features which closely resemble those of physical systems at criticality. We here introduce a minimal model framework for the dynamics of a community of individuals which undergoes local birth-death, immigration and local jumps on a regular lattice. We study its properties when the system is close to its critical point. Even if this model violates detailed balance, within a physically relevant regime dominated by fluctuations, it is possible to calculate analytically the probability density function of the number of individuals living in a given volume, which captures the close-to-critical behavior of the community across spatial scales. We find that the resulting distribution satisfies an equation where spatial effects are encoded in appropriate functions of space, which we calculate explicitly. The validity of the analytical formul{\ae} is confirmed by simulations in the expected regimes. We finally discuss how this model in the critical-like regime is in agreement with several biodiversity patterns observed in tropical rain forests.