Fronts and patterns with a dynamic parameter ramp

TL;DR

This paper investigates how slowly changing parameters influence invasion fronts and pattern formation in reaction-diffusion systems, providing analytical predictions and rigorous results for front dynamics and wavenumber selection.

Contribution

It introduces a unified approach combining linearized analysis, comparison principles, and modulation techniques to analyze dynamic parameter effects on invasion fronts and pattern formation.

Findings

Derived asymptotic predictions for front position and decay.

Established rigorous spreading results for unbounded parameter variation.

Predicted wavenumber selection and delayed invasion phenomena.

Abstract

We examine the effect of a slowly-varying time-dependent parameter on invasion fronts for which an unstable homogeneous equilibrium is invaded by either another homogeneous state or a spatially periodic state. We first explain and motivate our approach by studying asymptotically constant invasion fronts in a scalar FKPP equation with time-dependent parameter which controls the stability of the trivial state. Following recent works in the area, we use a linearized analysis to derive formal predictions for front position and leading-edge spatial decay. We then use a comparison principle approach to establish a rigorous spreading result in the case of an unbounded temporal parameter. We then consider patterned-invasion in the complex Ginzburg-Landau equation with dynamic bifurcation parameter, a prototype for slow passage through a spatio-temporal Hopf instability. Linearized analysis once again gives front position and decay asymptotics, but also the selected spatial wavenumber at the leading edge. We then use a Burger's modulation analysis to predict the slowly-varying wavenumber in the wake of the front. Finally, in both equations, we used the recently developed concept of a space-time memory curve to characterize delayed invasion in the case where the parameter is initially stable before a subsequent slow passage through instability and invasion. We also provide preliminary results studying invasion in other prototypical pattern formation models modified with a dynamic parameter, as well as numerical results for delayed transition between pushed and pulled fronts in Nagumo's equation with dynamic parameter.