The Moran process on 2-chromatic graphs

TL;DR

This paper investigates how resource heterogeneity, modeled by graph coloring, influences evolutionary dynamics in populations, proving equivalence among certain graph structures and exploring effects of dynamic resource fluctuations.

Contribution

It establishes that all properly two-colored, undirected, regular graphs are evolutionarily equivalent and analyzes how resource heterogeneity impacts evolutionary outcomes.

Findings

Properly two-colored, regular graphs are evolutionarily equivalent.

Background heterogeneity affects fixation probabilities.

Dynamic resource fluctuations can diminish heterogeneity effects.

Abstract

Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where "properly colored" means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring.