Nonclassical Turing instabilities induced by superdiffusive transport in FitzHugh-Nagumo dynamics

TL;DR

This paper explores how superdiffusive transport modeled by fractional Laplacians affects pattern formation in FitzHugh-Nagumo systems, revealing new instability mechanisms and nonlinear behaviors.

Contribution

It introduces a fractional Laplacian-based model for reaction-diffusion systems, analyzing how superdiffusion alters classical Turing instability conditions and pattern scales.

Findings

Superdiffusion modifies the band of unstable modes.

Spatial scale of patterns depends on diffusion orders and domain size.

Superdiffusion can induce instabilities even when activator diffuses faster.

Abstract

We investigate diffusion-driven instabilities in a FitzHugh-Nagumo reaction-diffusion system with superdiffusive transport, modeled by fractional Laplacian operators with different diffusion orders for the activator and the inhibitor. A linear stability analysis yields explicit expressions for the instability threshold and the critical wavenumber and shows that superdiffusion modifies the band of unstable modes and the characteristic spatial scale of emerging patterns. We show that the threshold depends only on the ratio of the fractional exponents and on the kinetic parameters, while the spatial scale is controlled by the diffusion orders and the domain size. When the diffusion orders differ, spatial instabilities may occur even in regimes where the activator diffuses faster than the inhibitor, due to the combined effect of diffusion rates, anomalous scaling and system size. This leads to instability mechanisms that depart from the classical activator-inhibitor framework. A weakly nonlinear analysis near threshold provides the amplitude equation governing nonlinear saturation and reveals that superdiffusion promotes subcritical behavior. We also analyze the interaction between stationary and oscillatory instabilities near Turing-Hopf codimension-two points. All analytical results are supported by numerical simulations.