Inside dynamics of pulled and pushed fronts

TL;DR

This paper analyzes the internal structure of reaction-diffusion fronts, distinguishing between pulled and pushed types, and provides new insights into their convergence behavior and spreading properties across different reaction types.

Contribution

It offers a comprehensive analysis of the inside dynamics of pulled and pushed fronts, extending the definitions to general transition waves and providing detailed convergence and spreading speed results.

Findings

Localized components of pulled fronts decay to zero over time.

Components of pushed fronts converge to a positive proportion of the front.

Uniform convergence and precise spreading speed estimates are established.

Abstract

We investigate the inside structure of one-dimensional reaction-diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.