  We investigate a suitable application of Model Order Reduction (MOR)
techniques for the numerical approximation of Turing patterns, that are
stationary solutions of reaction-diffusion PDE (RD-PDE) systems. We show that
solutions of surrogate models built by classical Proper Orthogonal
Decomposition (POD) exhibit an unstable error behaviour over the dimension of
the reduced space. To overcome this drawback, first of all, we propose a
POD-DEIM technique with a correction term that includes missing information in
the reduced models. To improve the computational efficiency, we propose an
adaptive version of this algorithm in time that accounts for the peculiar
dynamics of the RD-PDE in presence of Turing instability. We show the
effectiveness of the proposed methods in terms of accuracy and computational
cost for a selection of RD systems, i.e. FitzHugh-Nagumo, Schnackenberg and the
morphochemical DIB models, with increasing degree of nonlinearity and more
structured patterns.
