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Unstable patterns in reaction-diffusion model of early carcinogenesis

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.