# Bacterial range expansions on a growing front: Roughness, Fixation, and Directed Percolation

## TL;DR

This paper models how bacterial populations expanding on a surface exhibit rough growth fronts and fixation, using coupled stochastic equations and renormalization group analysis to understand their critical behavior.

## Contribution

It introduces a novel coupled stochastic PDE framework linking surface roughness with directed percolation in bacterial range expansions.

## Key findings

- Roughness influences the critical behavior of bacterial fixation.
- Scaling laws at the DP transition are derived through renormalization group analysis.
- The model predicts specific growth patterns and genealogical structures.

## Abstract

Directed Percolation (DP) is a classic model for nonequilibrium phase transitions into a single absorbing state (fixation). It has been extensively studied by analytical and numerical techniques in diverse contexts. Recently, DP has appeared as a generic model for the evolutionary/ecological dynamics of competing bacterial populations. Range expansion -- the stochastic reproduction of bacteria competing for space to be occupied by their progeny -- leads to a fluctuating and rough growth front, which is known from experiment and simulation to affect the underlying critical behavior of the DP transition. In this work, we employ symmetry arguments to construct a pair of non-linear stochastic partial differential equations describing the co-evolution of surface roughness with the composition field of DP. Macroscopic manifestations (phenomenology) of these equations on growth patterns and genealogical tracks of range expansion are discussed; followed by a renormalization group analysis of possible scaling behaviors at the DP transition.