Symmetry in models of natural selection

TL;DR

This paper formalizes the concept of symmetry in mathematical models of natural selection, showing how symmetry can simplify analysis by reducing the state space of the underlying Markov chain.

Contribution

It introduces a formal definition of symmetry in natural selection models and characterizes how these symmetries can be used to reduce the complexity of the models.

Findings

Symmetries form a group acting on population states.

Symmetry-based reduction bounds the number of states in the Markov chain.

Formal framework applies to a broad class of natural selection models.

Abstract

Symmetry arguments are frequently used -- often implicitly -- in mathematical modeling of natural selection. Symmetry simplifies the analysis of models and reduces the number of distinct population states to be considered. Here, I introduce a formal definition of symmetry in mathematical models of natural selection. This definition applies to a broad class of models that satisfy a minimal set of assumptions, using a framework developed in previous works. In this framework, population structure is represented by a set of sites at which alleles can live, and transitions occur via replacement of some alleles by copies of others. A symmetry is defined as a permutation of sites that preserves probabilities of replacement and mutation. The symmetries of a given selection process form a group, which acts on population states in a way that preserves the Markov chain representing selection. Applying classical results on group actions, I formally characterize the use of symmetry to reduce the states of this Markov chain, and obtain bounds on the number of states in the reduced chain.