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Turing patterns in reaction-diffusion (RD) systems have classically beenstudied only in RD systems which do not explicitly depend on independentvariables such as space. In practise, many systems for which Turing patterningis important are not homogeneous with ideal boundary conditions. Inheterogeneous systems with stable steady states, the steady states are alsonecessarily heterogeneous which is problematic for applying the classicalanalysis. Whilst there has been some work done to extend Turing analysis tosome heterogeneous systems, for many systems it is still difficult to determineif a stable patterned state is driven purely by system heterogeneity or if aTuring instability is playing a role. In this work, we try to define aframework which uses numerical continuation to map heterogeneous RD systemsonto a sensible nearby homogeneous system. This framework may be used fordiscussing the role of Turing instabilities in establishing patterns inheterogeneous RD systems. We study the Schnakenberg and Gierer-Meinhardt modelswith spatially heterogeneous production as test problems. It is shown that forsufficiently large system heterogeneity (large amplitude spatial variations inmorphogen production) it is possible that Turing-patterned and base statesbecome coincident and therefore impossible to distinguish. Other exoticbehaviour is also shown to be possible. We also study a novel scenario in whichmorphogen is produced locally at levels that could support Turing patterningbut on intervals/patches which are on the scale of classical critical domainlengths. Without classical domain boundaries, Turing patterns are allowed tobleed through; an effect noted by other authors. In this case, this phenomenaeffectively changes the critical domain length. Indeed, we even note that thisphenomena may also effectively couple local patches together and driveinstability in this way.

Turing pattern or system heterogeneity? A numerical continuation  approach to assessing the role of Turing instabilities in heterogeneous  reaction-diffusion systems