Modelling collective invasion with reaction–diffusion equations: When does domain curvature matter?

Real-world cellular invasion processes often take place in curved geometries. Such problems are frequently simplified in models to neglect the curved geometry in favour of computational simplicity, yet doing so risks inaccuracies in any model-based predictions. To quantify the conditions under which neglecting a curved geometry is justifiable, we explore the dynamics of a system of reaction-diffusion equations (RDEs) on a two-dimensional annular geometry analytically. Defining <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϵ</mml:mi></mml:math> as the ratio of the annulus thickness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>δ</mml:mi></mml:math> and radius <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msub> </mml:math> we derive, through an asymptotic expansion, the conditions under which it is appropriate to ignore the domain curvature for a general system of reaction-diffusion equations. To highlight the consequences of these results, we simulate solutions to the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) model, a paradigm nonlinear RDE typically used to model spatial invasion, on an annular geometry. Thus, we quantify the size of the deviation from an analogous simulation on the rectangle, and how this deviation changes across the width of the annulus. We further characterise the nature of the solutions through numerical simulations for different values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>δ</mml:mi></mml:math> . Our results provide insight into when it is appropriate to neglect the domain curvature in studying travelling wave behaviour in RDEs.