FKPP dynamics mediated by a parent field with a delay

TL;DR

This paper studies a biologically motivated FKPP model with delay, showing how delay affects wave speed, which is always slower than classical FKPP waves, with maximum speed at zero delay.

Contribution

It introduces a natural science-based FKPP model with delay, analyzing how delay impacts wave speed and revealing it is always slower than classical FKPP fronts.

Findings

Wave speed decreases with increasing delay.

Maximum wave speed is half of classical FKPP speed.

Wave speed is highest when delay is zero and rates are matched.

Abstract

We examine a modification of the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) process in which the diffusing substance requires a parent density field for reproduction. A biological example would be the density of diffusing spores (propagules) and the density of a stationary fungus (parent). The parent produces propagules at a certain rate, and the propagules turn into the parent substance at another rate. We model this evolution by the FKPP process with delay, which reflects a finite time typically required for a new parent to mature before it begins to produce propagules. While the FKPP process with other types of delays have been considered in the past as a pure mathematical construct, in our work a delay in the FKPP model arises in a natural science setting. The speed of the resulting density fronts is shown to decrease with increasing delay time, and has a non-trivial dependence on the rate of conversion of propagules into the parent substance. Remarkably, the fronts in this model are always slower than Fisher waves of the classical FKPP model. The largest speed is half of the classical value, and it is achieved at zero delay and when the two rates are matched.