Patchiness and Demographic Noise in Three Ecological Examples

TL;DR

This paper introduces a simple method to incorporate demographic stochasticity into classic ecological models, demonstrating that stochastic effects can lead to patchy distributions and phase transitions similar to directed percolation.

Contribution

It presents an accessible way to add demographic noise to deterministic ecological models and analyzes the resulting spatial patterns and phase transitions.

Findings

Models exhibit patchy configurations due to demographic noise.

Systems undergo phase transitions in the directed percolation universality class.

The integration scheme effectively handles stochastic equations with demographic noise.

Abstract

Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only -at most- local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals. Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the "square-root" intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.