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

The emergence of localized 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 localized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo form="prefix" stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo form="postfix" stretchy="false">)</mml:mo> </mml:mrow> </mml:math> -dimensional radial patterns in general two-component reaction–diffusion systems near a Turing instability, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> is taken to be a continuous parameter. We determine the explicit dependence of each pattern’s radial profile on the dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>n</mml:mi> </mml:math> through the introduction of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo form="prefix" stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo form="postfix" stretchy="false">)</mml:mo> </mml:mrow> </mml:math> -dimensional Bessel functions, revealing a deep connection between the formation of localized radial patterns in different spatial dimensions.