Effects of motion in structured populations

TL;DR

This paper investigates how individual motion in structured populations influences natural selection, revealing that motion generally suppresses selection in certain updating rules and can either amplify or suppress it depending on graph structure and parameters.

Contribution

It introduces a formal analysis of motion in evolutionary graphs, showing its suppressive effect under specific updating rules and its complex role on weighted and dynamic graphs.

Findings

Motion suppresses natural selection on cycles with death-birth updating.

No effect of motion on fixation probability under pure birth-death updating.

Motion can either amplify or suppress selection on weighted graphs depending on parameters.

Abstract

In evolutionary processes, population structure has a substantial effect on natural selection. Here, we analyze how motion of individuals affects constant selection in structured populations. Motion is relevant because it leads to changes in the distribution of types as mutations march toward fixation or extinction. We describe motion as the swapping of individuals on graphs, and more generally as the shuffling of individuals between reproductive updates. Beginning with a one-dimensional graph, the cycle, we prove that motion suppresses natural selection for death-birth updating or for any process that combines birth-death and death-birth updating. If the rule is purely birth-death updating, no change in fixation probability appears in the presence of motion. We further investigate how motion affects evolution on the square lattice and weighted graphs. In the case of weighted graphs we find that motion can be either an amplifier or a suppressor of natural selection. In some cases, whether it is one or the other can be a function of the relative reproductive rate, indicating that motion is a subtle and complex attribute of evolving populations. As a first step towards understanding less restricted types of motion in evolutionary graph theory, we consider a similar rule on dynamic graphs induced by a spatial flow and find qualitatively similar results indicating that continuous motion also suppresses natural selection.