Mean field analysis of Williams-Bjerknes type growth

TL;DR

This paper uses mean field analysis to study stochastic competition models with an advantaged phase, providing analytical insights that align well with lattice simulations, and connects to classical growth models like Eden.

Contribution

It introduces a mean field approach to analyze Williams-Bjerknes type growth models with competitive advantage, extending understanding of tumor growth and related processes.

Findings

Analytical equilibrium populations match simulation results

Regression probabilities for extinction are accurately approximated

Models connect to Eden variants in certain limits

Abstract

We investigate a class of stochastic growth models involving competition between two phases in which one of the phases has a competitive advantage. The equilibrium populations of the competing phases are calculated using a mean field analysis. Regression probabilities for the extinction of the advantaged phase are calculated in a leading order approximation. The results of the calculations are in good agreement with simulations carried out on a square lattice with periodic boundaries. The class of models are variants of the Williams- Bjerknes model for the growth of tumours in the basal layer of an epithelium. In the limit in which only one of the phases is unstable the class of models reduces to the well known variants of the Eden model.