Significance of non-normality-induced patterns: Transient growth versus asymptotic stability

Reaction-diffusion models following the original idea of Turing are widely applied to study the propensity of a system to develop a pattern. To this end, an asymptotic analysis is typically performed via the so-called dispersion relation that relates the spectral properties of a spatial operator (diffusion) to the temporal behaviour of the whole initial-boundary value reaction-diffusion problem. Here, we amend this approach by studying the transient growth due to non-normality that can also lead to a pattern development in non-linear systems. We conclude by identification of the significance of this transient growth and by assessing the plausibility of the standard spectral approach. Particularly, the non-normality-induced patterns are possible but require fine parameter tuning.