The Moran process on 2-chromatic graphs

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.

Counterintuitive properties of fixation probability and fixation time in population structures with spatially periodic resource distribution

“…Ref. [24] showed that this does not happen for two-chromatic graphs (T = 2 in our case). This is confirmed from figure 5 a.…”

“…Similarly fitness values at a red node are a R and b R . This is a color-to-fitness map [24] and can be summarized into a simple matrix form,…”

“…Spatial variations in resources requires a unified model that incorporates spatial structure and environmental interactions in one framework. This has been the focus of recent works in literature [12,13,15,23,24,31,32,36] . Maciejewski et al, and Kaveh et al, suggested to represent a population structure with variable distribution of resource as a colored graph [24,31].…”

“…This has been the focus of recent works in literature [12,13,15,23,24,31,32,36] . Maciejewski et al, and Kaveh et al, suggested to represent a population structure with variable distribution of resource as a colored graph [24,31]. Color of each node represents the concentration level of resource at that node.…”

Martingales and the characteristic functions of absorption time on bipartite graphs

“…We can consider birthdeath processes with more than two competing species, each with different fitnesses [43,44]. We can consider heterogeneous graphs, where fitness is attributed to nodes on the graph, as well as the species of the individual occupying it [45]. We can consider evolutionary games on graphs, where individuals connected by graph edges compete in games for some pay-off [25,42,[46][47][48].…”