  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.
