# Stable cycling in quasi-linkage equilibrium: fluctuating dynamics under gene conversion and selection

## TL;DR

This paper investigates complex genetic dynamics, including stable cycling, under gene conversion and selection, revealing conditions where allele frequencies oscillate rather than stabilize, with implications for understanding genetic variation.

## Contribution

It introduces a model inspired by molecular biology that demonstrates stable cycling and complex dynamics under quasi-linkage equilibrium, with analytical stability analysis.

## Key findings

- Stable cycling occurs near heteroclinic cycles.
- Discrete-time models show always stable cycles.
- Continuous-time models show always unstable cycles.

## Abstract

Genetic systems with multiple loci can have complex dynamics. For example, mean fitness need not always increase and stable cycling is possible. Here, we study the dynamics of a genetic system inspired by the molecular biology of recognition-dependent double strand breaks and repair as it happens in recombination hotspots. The model shows slow-fast dynamics in which the system converges to the quasi-linkage equilibrium (QLE) manifold. On this manifold, sustained cycling is possible as the dynamics approach a heteroclinic cycle, in which allele frequencies alternate between near extinction and near fixation. We find a closed-form approximation for the QLE manifold and use it to simplify the model. For the simplified model, we can analytically calculate the stability of the heteroclinic cycle. In the discrete-time model the cycle is always stable; in a continuous-time approximation, the cycle is always unstable. This demonstrates that complex dynamics are possible under quasi-linkage equilibrium.