The role of spatial dimension in the emergence of localised radial patterns from a Turing instability

The emergence of localised radial patterns from a Turing instability has been well studied in two and three dimensional settings and predicted for higher spatial dimensions. We prove the existence of localised (n+1) -dimensional radial patterns in general two-component reaction-diffusion systems near a Turing instability, where n>0 is taken to be a continuous parameter. We determine explicit dependence of each pattern's radial profile on the dimension n through the introduction of (n+1) -dimensional Bessel functions, revealing a deep connection between the formation of localised radial patterns in different spatial dimensions.