Unstable patterns in reaction–diffusion model of early carcinogenesis

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 carcinogenesis proposed by Marciniak-Czochra and Kimmel [Comput. Math. Methods Med. 7 (2006) 189–213], [Math. Models Methods Appl. Sci. 17 (suppl.) (2007) 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, i.e., regular stationary solutions, are unstable in the Lyapunov sense. Furthermore, we show existence of discontinuous stationary solutions, which are also unstable.