The Turing pattern is a key concept in the modern study of reaction-diffusion systems, with Turing patterns proposed a possible explanation for the spatial structure observed in myriad physical, chemical, and biological systems. Real-world systems are not always so clean as idealized Turing systems, and in this talk we will take up the case of more messy reaction-diffusion systems involving explicit space or time dependence in diffusion or reaction terms. Turing systems of this nature arise in several applications, such as when a Turing system is studied on a growing substrate, is subjected to a temperature gradient, or is immersed within a fluid flow. The analysis of these messy Turing systems is not as straightforward as in the idealized case, with the explicit space or time dependence greatly complicating or even preventing most standard routes of analysis. Motivated by patterning in phenomena involving explicit space or time dependence, and by the interesting mathematical challenges inherent in the study of such systems, in this talk we consider the following questions:

• Is it possible to obtain generalizations of the Turing instability conditions for non-autonomous, spatially heterogeneous reaction-diffusion systems?
• What can one say about predicting nascent patterns in these systems?
• What role does explicit space or time dependence play in selecting fully developed patterns in these systems?
• Is it possible to exploit this space or time dependence in order to manipulate the form of emergent patterns?

We will also highlight some of the applications opened up by the analysis of heterogeneous Turing systems.