# Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches

## TL;DR

This paper compares stochastic and deterministic models of horizontal gene transfer to evaluate how well the simplified Hamilton-Jacobi approach captures the complex dynamics observed in individual-based simulations, especially regarding evolutionary rescue.

## Contribution

It provides a numerical comparison between stochastic jump processes, deterministic integrodifferential equations, and Hamilton-Jacobi models for horizontal gene transfer, highlighting their similarities and differences.

## Key findings

- Hamilton-Jacobi model reproduces key qualitative behaviors of stochastic models
- Deterministic models effectively approximate large population dynamics
- Insights into the phenomenon of evolutionary rescue in bacterial populations

## Abstract

Horizontal gene Transfer (HT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HT is believed to be the main instrument of developing the antibiotic resistance. In this work, we consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integrod-ifferential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal of this paper is to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular.