The duality of spatial death-birth and birth-death processes and limitations of the isothermal theorem

TL;DR

This paper examines the limitations of the isothermal theorem in evolutionary graph models, showing that duality between birth-death and death-birth processes affects fixation probabilities, especially on spatial lattices.

Contribution

It demonstrates that the isothermal theorem applies only under specific conditions for birth-death and death-birth processes and highlights differences in fixation probabilities on spatial lattices.

Findings

Isothermal theorem holds only when mutants differ by death rate in DB processes.

Differences between BD and DB processes are significant on 1D and 2D lattices.

Generating function approach can analyze fixation probabilities on arbitrary graphs.

Abstract

Evolutionary models on graphs, as an extension of the Moran process, have two major implementations: birth-death (BD) models (or the invasion process) and death- birth (DB) models (or voter models). The isothermal theorem states that the fixation probability of mutants in a large group of graph structures (known as isothermal graphs, which include regular graphs) coincides with that for the mixed population. This result has been proven by Lieberman et al (Nature 433: 312-316, 2005) in the case of BD processes, where mutants differ from the wild types by their birth rate (and not by their death rate). In this paper we discuss to what extent the isothermal theorem can be formulated for DB processes, proving that it only holds for mutants that differ from the wild type by their death rate (and not by their birth rate). For more general BD and DB processes with arbitrary birth and death rates of mutants, we show that the fixation probabilities of mutants are different from those obtained in the mass-action populations. We focus on spatial lattices and show that the difference between BD and DB processes on 1D and 2D lattices are non-small even for large population sizes. We support these results with a generating function approach that can be generalized to arbitrary graph structures. Finally, we discuss several biological applications of the results.