Inhomogeneous Branching Random Walks: Incorporating Genealogy and Density Effects

TL;DR

This paper introduces inhomogeneous Branching Random Walk models that incorporate genealogy and density effects to better understand complex growth patterns like bacterial colonies, combining simulations with real data comparison.

Contribution

It presents a new framework for modeling growth processes with genealogy-dependent inhomogeneity, extending beyond traditional models that ignore genealogical effects.

Findings

Genealogy-dependent inhomogeneous BRWs can replicate complex growth patterns.

Simulation results align with real bacterial colony images.

The model reveals long-term behavior influenced by genealogy and density effects.

Abstract

In this paper, we introduce a novel framework using inhomogeneous Branching Random Walks (BRWs) to model growth processes, specifically introducing genealogy-dependence in branching rates and displacement distributions to model phenomena like bacterial colony growth. Current stochastic models often either assume independent and identical behavior of individual agents or incorporate only spatiotemporal inhomogeneity, ignoring the effect of genealogy-based inhomogeneity on the long-time behavior of these processes. Such long-time asymptotics are of independent mathematical interest and are crucial in understanding the effect of patterns. We propose several inhomogeneous BRW models in 2D space where displacement distributions and branching rates vary with time, space, and genealogy. A combined model then uses a weighted average of positions given by these separate models to study the shape of the growth patterns. Using computer simulations, we tune parameters from these models, which are based on genealogical and spatiotemporal factors, observe the resulting structures, and compare them with images of real bacterial colonies.