Turing Bifurcations on Elliptical Domains: A Deviation from Rectangular and Circular Limits

Abstract Turing patterns have been extensively studied on simple geometries such as lines, squares, rectangles, and circles. Consequently, many biological and physical applications of Turing’s theory approximate their domains to have simple geometries. In particular, thin domains are often approximated as one-dimensional lines or rectangles, whereas the actual geometry may be curved and closer to a stretched ellipse. Thus, we investigate Turing patterns on ellipses and show that they exhibit two distinct limiting behaviours: (i) they tend to those on the circular domain as the ellipse’s aspect ratio approaches unity; (ii) they do not converge to the behaviour of a one-dimensional line as the ellipse becomes thin. This contrasts with rectangular domains, where the bifurcation structure smoothly tends to that of a one-dimensional line as the rectangle’s height is reduced. Using a combination of analytical methods involving Mathieu equations and numerical bifurcation tracking, we demonstrate that the bifurcation modes in an elliptical domain are intrinsically coupled in both radial and angular directions, preventing simple interpolation between circular and linear limits. The results provide insights into the role of domain geometry in governing Turing instabilities and pattern selection, highlighting the distinctive behaviour of ellipses compared to other commonly studied geometries.