Dualities in population genetics: a fresh look with new dualities

TL;DR

This paper introduces a novel framework for understanding dualities in population genetics models, revealing new dualities between key diffusion processes and identifying self-duality in the Moran model.

Contribution

It applies a general duality method to population models, uncovering new dualities between Wright-Fisher and Moran processes using SU(1,1) operators.

Findings

New dualities between Wright-Fisher diffusion and Moran model.

Identification of self-duality in the Moran model.

Unified representation of classical and new dualities.

Abstract

We apply our general method of duality, introduced in [Giardina', Kurchan, Redig, J. Math. Phys. 48, 033301 (2007)], to models of population dynamics. The classical dualities between forward and ancestral processes can be viewed as a change of representation in the classical creation and annihilation operators, both for diffusions dual to coalescents of Kingman's type, as well as for models with finite population size. Next, using SU(1,1) raising and lowering operators, we find new dualities between the Wright-Fisher diffusion with $d$ types and the Moran model, both in presence and absence of mutations. These new dualities relates two forward evolutions. From our general scheme we also identify self-duality of the Moran model.