Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds

• Published in: Bulletin of mathematical biology Vol. 81; no. 3; pp. 759 - 799
• Main Authors: Krause, Andrew L., Ellis, Meredith A., Van Gorder, Robert A.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2019; Springer Nature B.V
• Subjects: Animals; Anisotropy; Biological evolution; Cell Biology; Computer Simulation; Curvature; Diffusion; Domains; Finite Element Analysis; Kinetics; Life Sciences; Manifolds; Mathematical and Computational Biology; Mathematical Concepts; Mathematics; Mathematics and Statistics; Models, Biological; Nonlinear Dynamics; Pattern Recognition, Automated - statistics & numerical data; Reaction kinetics; Species diffusion; Systems Biology; Anisotropic growth; Pattern selection; Growing surfaces; Pattern robustness; Turing patterns
• ISSN: 0092-8240; 1522-9602
• DOI: 10.1007/s11538-018-0535-y

We study two-species reaction–diffusion systems on growing manifolds, including situations where the growth is anisotropic yet dilational in nature. In contrast to the literature on linear instabilities in such systems, we study how growth and anisotropy impact the qualitative properties of nonlinear patterned states which have formed before growth is initiated. We produce numerical solutions to numerous reaction–diffusion systems with varying reaction kinetics, manner of growth (both isotropic and anisotropic), and timescales of growth on both planar elliptical and curved ellipsoidal domains. We find that in some parameter regimes, some of these factors have a negligible effect on the long-time patterned state. On the other hand, we find that some of these factors play a role in determining the patterns formed on surfaces and that anisotropic growth can produce qualitatively different patterns to those formed under isotropic growth.