Competition and cooperation in one-dimensional stepping stone models

TL;DR

This paper investigates how spatial structure, density, and migration influence the persistence of mutualism in one-dimensional models, revealing phase transitions and conditions under which mutualism is maintained or lost.

Contribution

It introduces a detailed analysis of mutualism dynamics in one-dimensional stepping stone models, highlighting the roles of density and migration in phase transitions.

Findings

Mutualism persists only at high density and migration rates.

Loss of mutualism occurs via a directed percolation transition.

The phase diagram is significantly affected by an exceptional DP2 transition.

Abstract

Cooperative mutualism is a major force driving evolution and sustaining ecosystems. Although the importance of spatial degrees of freedom and number fluctuations is well-known, their effects on mutualism are not fully understood. With range expansions of microbes in mind, we show that, even when mutualism confers a distinct selective advantage, it persists only in populations with high density and frequent migrations. When these parameters are reduced, mutualism is generically lost via a directed percolation process, with a phase diagram strongly influenced by an exceptional DP2 transition.