Fisher-KPP waves and the minimal speed on hexagonal lattice

2025/11/28 by Jian Fang, Yifei Li, Fang, Jian +5
Computer Science · Mathematics · Medicine · #34A33 #35C07 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation

paper · pdf · doi:10.48550/arxiv.2511.22932

openalex publication_date 2025/11/28 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

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 ℝ² and ℤ². 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.

Citations

• Spreading properties for SIR models on homogeneous trees
• On a population model in discrete periodic habitat. I. spreading speed and optimal dispersal strategy
• Multidimensional Stability of Waves Travelling Through Rectangular Lattices in Rational Directions
• Traveling wave front for a two-component lattice dynamical system arising in competition models
• Bistable Traveling Waves for Monotone Semiflows with Applications
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• A Discrete Convolution Model¶for Phase Transitions
• Traveling Waves in Lattice Dynamical Systems
• Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

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• Transition Fronts of Fisher-KPP Equations in Locally Spatially Inhomogeneous Patchy Environments I: Existence and Non-existence
• Explicit pulsating fronts and minimal speeds in periodic Fisher-KPP equations
• Traveling fronts for Fisher-KPP lattice equations in almost periodic media
• Planar pulsating traveling wave solutions of non-cooperative Fisher--KPP systems in space-time periodic media