Spreading Properties of a City-Road Reaction-Diffusion Model on One-Dimensional Lattice

TL;DR

Spreading properties of a city-road reaction-diffusion model on a one-dimensional lattice develops an infinite PDE-ODE system on the lattice $\mathbb{Z}$ with diffusion on edges and logistic growth at vertices, connected by Robin-type exchanges. The authors prove global well-posedness, identify the unique positive stationary state $(\beta/\alpha,1)$ as the global attractor, and derive a linear spreading speed $c_*$ that also governs nonlinear spreading from compactly supported data. They analyze the large-diffusion limit, obtaining an asymptotic two-component system with speed $c_*^{\infty}$ and conjecturing $c_*\to c_*^{\infty}$ as $d\to\infty$, thereby linking discrete lattice dynamics to a continuum-graph limit. Together, these results provide a rigorous framework for understanding invasion fronts on infinite networks and connect discrete Fisher-KPP-type propagation with diffusion on metric graphs.

Abstract

We propose and study a new model to describe biological invasions constrained on infinite homogeneous one dimensional metric graphs. Our model consists of an infinite PDE-ODE system where, at each vertex of the one-dimensional lattice $\mathbb{Z}$, we have a logistic equation, and connections between vertices are given by diffusion equations on the edges supplemented with Robin like boundary conditions at the vertices. We establish the main properties of the system and study the long time behavior of the solutions, especially by characterizing an asymptotic spreading speed for the system. In the fast diffusion regime, we derive a novel asymptotic model which exhibits similar propagation properties as the classical discrete Fisher-KPP on the one-dimensional lattice $\mathbb{Z}$.

Figures (7)

• Figure 1: Schematic spatial configuration of the system \ref{['eq1']}-\ref{['bd']} where the unknowns $\rho_j$ are indexed on the lattice $\mathbb{Z}$ while each $v_j$ is locally defined on $(0,\ell)$.
• Figure 2: Numerically computed solution of system \ref{['eq1']}--\ref{['bd']} at time $t=50$ for $f(u)=u(1-u)$ and $(\alpha,\beta,d)=(1,1,1)$ starting from an initial condition where $h_j\equiv 0$ for all $j\in\mathbb{Z}$ and $\Lambda_j=1$ for $j\leq0$ and $\Lambda_j=0$ for $j\geq1$. The red dots represent $\rho_j$ located at position $j$ while each blue curve represents $v_j$ located on the interval $[j,j+1]$.
• Figure 3: Typical representation of the map $\Psi$ on $(\lambda_0,+\infty)$ with a unique global minimum at $\lambda=\lambda_*$. Here, parameters values are set to $(\alpha,\beta,d,f'(0))=(1,1,1,1)$.
• Figure 4: Plots of the linear spreading speed \ref{['eq1']} as parameters are varied.
• Figure 5: Left: Amplitude of the spreading speed $c_*$ as a function of $(\alpha,\beta)$ in the square $[0,20]\times[0,20]$. Several isolines (red curves) are also reported. Right: Linear spreading speed $c_*$ as a function of $\alpha=\beta$. Other values of the parameters are set to $(d,f'(0))=(1,1)$.

Theorems & Definitions (19)

• Theorem 1
• Proposition 2.1
• Proposition 2.2: Comparison principle
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5.1
• Theorem 5
• Proposition 6.1
• Proposition 6.2