Abstract
Turing patterns in reaction-diffusion systems on growing or otherwise evolving space domains are useful models for applications in developmental biology, ecology and chemistry. Common growth mechanisms include uniform growth (comprising global expansion over the whole domain) and apical growth (growth concentrated near a domain boundary). While both evolution mechanisms have their applications, to study Turing patterns under more complex yet realistic domain evolution, we combine aspects of each, considering regions of uniform evolution separated by nodes at which apical evolution occurs. The result is a hybrid of both evolution mechanisms. We study the Turing instability and resulting pattern formation under this hybrid uniform-apical domain evolution along one space dimension, modelling how local domain growth or contraction modifies the development and persistence of Turing patterns. We highlight key differences in the evolution of Turing patterns near an apical node (which births or destroys peaks) or within subregions of uniform evolution (causing either splitting or coalescence of peaks), while also showing that certain simultaneous combinations of growth and contraction allow for the transport of Turing patterns across the domain. Possibilities for the extension of our methods and results to evolving space domains of higher dimension are briefly discussed.