Turing-Hopf patterns on growing domains: The torus and the sphere (original) (raw)

Journal of theoretical biology, 2018

Abstract

This paper deals with the study of spatial and spatio-temporal patterns in the reaction-diffusion FitzHugh-Nagumo model on growing curved domains. This is carried out on two exemplar cases: a torus and a sphere. We compute bifurcation boundaries for the homogeneous steady state when the homogeneous system is monostable. We exhibit Turing and Turing-Hopf bifurcations, as well as additional patterning outside of these bifurcation regimes due to the multistability of patterned states. We consider static and growing domains, where the growth is slow, isotropic, and exponential in time, allowing for a simple analytical calculation of these bifurcations in terms of model parameters. Numerical simulations allow us to discuss the role played by the growth and the curvature of the domains on the pattern selection on the torus and the sphere. We demonstrate parameter regimes where the linear theory can successfully predict the kind of pattern (homogeneous and heterogeneous oscillations and st...

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