Unstable patterns in reaction-diffusion model of early carcinogenesis

TL;DR

This paper investigates pattern formation in a reaction-diffusion model of early cancer development, revealing that all regular and discontinuous stationary patterns are unstable, despite exhibiting diffusion-driven instability.

Contribution

It proves the instability of all Turing-type and discontinuous stationary solutions in a reaction-diffusion model of early carcinogenesis.

Findings

All Turing-type patterns are Lyapunov unstable.

Existence of unstable discontinuous stationary solutions.

Patterns emerge but are inherently unstable.

Abstract

Motivated by numerical simulations showing the emergence of either periodic or irregular patterns, we explore a mechanism of pattern formation arising in the processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. We focus on a basic model of early cancerogenesis proposed by Marciniak-Czochra and Kimmel [Comput. Math. Methods Med. {\bf 7} (2006), 189--213], [Math. Models Methods Appl. Sci. {\bf 17} (2007), suppl., 1693--1719], but the theory we develop applies to a wider class of pattern formation models with an autocatalytic non-diffusing component. The model exhibits diffusion-driven instability (Turing-type instability). However, we prove that all Turing-type patterns, {\it i.e.,} regular stationary solutions, are unstable in the Lyapunov sense. Furthermore, we show existence of discontinuous stationary solutions, which are also unstable.