Turing pattern or system heterogeneity? A numerical continuation approach to assessing the role of turing...

Turing patterns in reaction-diffusion (RD) systems have traditionally been studied only in systems that do not explicitly depend on independent variables such as space. In practice, many systems in which Turing patterning is important are not homogeneous and do not possess ideal boundary conditions. In heterogeneous systems with stable steady states, the steady states themselves are also necessarily heterogeneous, which is problematic for analytical approaches. Whilst there has been a large body of work extending Turing analysis to certain heterogeneous systems, it remains difficult-especially on small domains-to determine whether a stable patterned state arises purely from system heterogeneity or whether a Turing instability plays a role. This also complicates numerical investigations into critical domain lengths for such instabilities. In this work, we propose a framework that uses numerical continuation to map heterogeneous RD systems onto a nearby homogeneous system. This framework may be used to analyse the role of Turing instabilities in generating patterns in heterogeneous RD systems. We study the Schnakenberg and Gierer-Meinhardt models with spatially heterogeneous production as test problems. Our investigation reveals the following features. For sufficiently large system heterogeneity (i.e., large-amplitude spatial variations in morphogen production), it is possible for Turing-patterned and base states to become coincident and therefore indistinguishable. This only occurs when a Turing instability is present in a nearby homogeneous reaction-diffusion system. In fact, an instability must occur in a mode that is at least resonant with, or of higher frequency than, the spatial frequency of the system heterogeneity, implying that a resonance effect governs the breakdown of the base state definition. Otherwise, a base state-by our definition-can always be found. Furthermore, we provide numerical evidence that, in the case of large domains, the homotopy-based base state definition we propose agrees with that found in the literature. We then use this base state definition to numerically investigate critical domain lengths in systems with spatial heterogeneity, which give rise to regions that locally support Turing patterning.