Fronts and patterns with a dynamic parameter ramp

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.