Linear spreading speeds from nonlinear resonant interaction

TL;DR

This paper introduces a new mechanism for invasion speed determination in spatially extended systems, based on nonlinear resonant interactions at the leading edge, which can be predicted from linear dispersion relations and confirmed through analysis and simulations.

Contribution

It identifies a novel resonant interaction mechanism for invasion speeds, providing a speed criterion based on linear dispersion relations and validating it across various models.

Findings

Resonant invasion speeds can be predicted from linear dispersion relations.

Front speeds slower than the resonant speed are unstable.

Numerical validation in diverse models, including neural fields.

Abstract

We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on the complex linear dispersion relation at the unstable equilibrium, but rely on the presence of a nonlinear term that facilitates the resonant coupling. We prove that these resonant speeds give the correct invasion speed in a simple example, we show that fronts with speeds slower than the resonant speed are unstable, and corroborate our speed criterion numerically in a variety of model equations, including a nonlocal scalar neural field model.