  The modelling of linear and nonlinear reaction-subdiffusion processes is more
subtle than normal diffusion and causes different phenomena. The resulting
equations feature a spatial Laplacian with a temporal memory term through a
time fractional derivative. It is known that the precise form depends on the
interaction of dispersal and reaction, and leads to qualitative differences. We
refine these results by defining generalised spectra through dispersion
relations, which allows us to examine the onset of instability and in
particular inspect Turing type instabilities. These results are numerically
illustrated. Moreover, we prove expansions that imply for one class of
subdiffusion reaction equations algebraic decay for stable spectrum, whereas
for another class this is exponential.
