Fisher-KPP waves and the minimal speed on hexagonal lattice

TL;DR

This paper investigates wave propagation in a hexagonal lattice using a lattice differential equation with Fisher-KPP dynamics, establishing existence, uniqueness, and properties of traveling waves, and confirming results through numerical simulations.

Contribution

It introduces a novel hexagonal lattice differential equation model and rigorously analyzes the existence, uniqueness, and properties of traveling waves in this setting.

Findings

Existence and uniqueness of traveling waves are proven.

The minimal wave speed depends periodically on an angle.

Numerical results confirm theoretical predictions and show the wave speed as the spreading speed.

Abstract

The hexagonal structure is ubiquitous in nature. The propagation phenomena occurring in a media with a hexagonal structure remain to be explored. One way of exploring this question is to formulate lattice dynamical systems and analyze the propagation dynamics. In this paper, we propose a lattice differential equation model featuring a discrete diffusion operator with the hexagonal structure, and a monostable nonlinear term known as the Fisher-KPP mechanism in modeling population growth. A rigorous analysis is conducted on the traveling waves, thoroughly establishing the existence and uniqueness (up to translation) of the traveling waves. Moreover, the periodicity and monotonicity of the minimal wave speed concerning an angle are demonstrated, which is different from the existing results of the minimal wave speed in $\mathbb{R}^2$ and $\mathbb{Z}^2$. Numerical results validate theoretical analysis and further suggest that the minimal wave speed is also the spreading speed of solutions with compactly supported initial values.