Fisher-KPP waves and the minimal speed on hexagonal lattice

TL;DR

The study analyzes Fisher–KPP propagation on a hexagonal lattice, formulating a lattice differential equation with a hexagonal diffusion operator. It establishes existence and uniqueness of traveling waves for speeds at or above the minimal value $c^*(\alpha)$, and shows $c^*(\alpha)$ is π/3-periodic with monotone segments in angle. The authors derive the dependence of the minimal speed on direction via an implicit-function approach and corroborate the theory with numerical simulations, which indicate the spreading speed matches the minimal wave speed in all directions. They also adapt the framework to a square lattice, revealing analogous angular periodicity and monotonicity, and discuss extensions to bistable dynamics and nonlocal dispersal. The results illuminate how lattice geometry governs propagation and open paths for further rigorous analysis and applications in ecological spreading on complex networks.

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.

Figures (5)

• Figure 1: A hexagonal lattice described by the coordinates $x$-$y$.
• Figure 2: The minimal wave speed $c^*$ with respect to $\alpha$ in the polar coordinate system. The coordinates on the radius represent the minimal wave speed with range $[4.5,4.65]$.
• Figure 3: The expression $\Phi_n(\alpha)$ with respect to $\alpha$ for different $n$.
• Figure 4: A square lattice with a diagonal line described by the coordinates $i$-$j$.
• Figure 5: Speeds $\bar{c}_1$, $\bar{c}_2$, $\bar{c}_3$ and $c^*$ with respect to $\alpha$ in the polar coordinate system. The coordinates on the radius represent the minimal wave speed or spreading speed with range $[30,48]$.

Theorems & Definitions (23)

• Theorem 2.1: Existence of traveling waves
• Theorem 2.2: Uniqueness of traveling waves
• Theorem 2.3: Angle-dependent speed
• Proposition 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof
• Lemma 4.1