Locked fronts in a discrete time discrete space population model

TL;DR

This paper investigates velocity locking in a discrete population model on a lattice, constructing specific locked fronts, analyzing their stability, and deriving parameter space boundaries for locking regions.

Contribution

It constructs explicit locked fronts for a piecewise linear reproduction function and derives the boundaries of locking regions in parameter space.

Findings

Locked fronts are linear combinations of exponentially decaying solutions.

Explicit expressions for locking region boundaries are derived.

Strict spectral stability of the fronts is established.

Abstract

A model of population growth and dispersal is considered where the spatial habitat is a lattice and reproduction occurs generationally. The resulting discrete dynamical systems exhibits velocity locking, where rational speed invasion fronts are observed to persist as parameters are varied. In this article, we construct locked fronts for a particular piecewise linear reproduction function. These fronts are shown to be linear combinations of exponentially decaying solutions to the linear system near the unstable state. Based upon these front solutions, we then derive expressions for the boundary of locking regions in parameter space. We obtain leading order expansions for the locking regions in the limit as the migration parameter tends to zero. Strict spectral stability in exponentially weighted spaces is also established.