Selection mechanisms in front invasion

TL;DR

The paper develops a dynamical-systems framework for front invasion into unstable states, unifying linear and nonlinear mechanisms of selection for both monotone and pattern-forming fronts. It introduces and exploits the concepts of linear and nonlinear marginal stability, pinched double roots, and exponential weights to predict linear spreading speeds and nonlinear front selection, including pushed and pulled fronts and their wakes. It provides rigorous results for pushed-front selection and outlines rigorous, matched-asymptotics proofs for pulled-front selection, while also addressing pattern-forming fronts, modulated fronts, and wakes with diffusive pattern selection; practical numerical continuation and robustness analyses are presented to implement these theories. The work has broad implications for predicting invasion speeds and pattern outcomes across reaction-diffusion systems, buffered by practical computational methods and openness to future exploration of resonances, staged invasions, and higher-dimensional settings.

Abstract

We review progress on questions related to front propagation into unstable states and point out open problems in the area. We strive to highlight different theoretical perspectives and challenges while also addressing more practical questions with examples and guides to computational methods. Throughout we take a dynamical systems point of view that focuses on the ability of invasion processes to act as a selection mechanism in complex systems.

Figures (21)

• Figure 1.1: Solution to \ref{['e:fkpp']} with small Gaussian initial condition near $x=10$, snap shots of profiles (left) and logarithms of profiles (center) at times $t=0,5,9,15,30,45,60$. Also shown in the center are parabolas corresponding to Gaussians solving the linear equation (black dashed). The space-time plot (right) shows spreading with speed 2 (black line) and even better agreement when including logarithmic corrections $x=2t-\frac{2}{2}\log t$ (red).
• Figure 1.2: Phase portrait sketches in the $u-u_x$-plane of \ref{['e:kpptw']}, $f(u)=u(1-u)$ (left), with $c$ decreasing from top to bottom. Also shown are spectra of the linearized operator in $L^p_{\mathrm{exp},c/2}$ (right); see \ref{['e:Lpweighted']} for function spaces. The marginally stable front is in fact the steepest front as illustrated in the bottom panel with plot sketches of $u_*(x;c)$.
• Figure 1.3: Phase portrait sketches in the $u-u_x$-plane of \ref{['e:kpptw']}, $f(u)=u(1-u)(u+a)$ for $a<\frac{1}{2}$ (left), with $c$ decreasing from top to bottom. Also shown are spectra of the linearized operator in $L^p_{\mathrm{exp},c/2}$ (right); see \ref{['e:Lpweighted']} for function spaces. The marginally stable front is in fact the steepest front as illustrated in the bottom panel with plot sketches of $u_*(x;c)$.
• Figure 1.4: Sub- and super-solutions that trap solutions between speeds $c_*\pm \varepsilon$ in the pulled (left, $c_*=c_\mathrm{lin}$) and in the pushed case (right, $c_*=c_\mathrm{push}$).
• Figure 1.5: Positive and negative fronts in the Nagumo equation \ref{['e:nagumo']} originating from a sign-changing initial condition at the center of the domain: balanced nonlinearity $a=1$ (left two panels) and imbalanced nonlinearity $a=.2$ (right two panels). Space-time plots and snapshot at $t=20$ in both case. Note the slightly different speeds of propagation in the imbalanced case due to a pushed front propagating to the left in the right two panels (illustrated by the red dashed line). Invasion fronts leave behind a stationary (left) or traveling (right) kink.

Theorems & Definitions (21)

• Definition 2.1: Pointwise growth modes
• Definition 2.2: Spreading speeds
• Definition 2.3: Pinched double roots
• Lemma 2.4: Pinched double roots and pointwise growth modes HolzerScheelPointwiseGrowth
• Definition 3.1
• Theorem 3.2: Marginal stability conjecture as1avery2
• Definition 5.1: Marginal spectral stability --- pushed fronts
• Theorem 5.2
• Corollary 5.3: Selection of rigid pushed fronts
• Definition 5.4: Marginal spectral stability --- pulled fronts