[Paper Review] Coupled reaction-diffusion equations on adjacent domains

Henri Berestycki, Luca Rossi+1 more|arXiv (Cornell University)|Mar 27, 2019

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine22 references3 citations

TL;DR

This paper studies a system of coupled reaction-diffusion equations on adjacent domains—one in a cylinder and one in its complement—where populations evolve via Fisher-KPP-type dynamics and exchange flux across the boundary. It establishes the existence and uniqueness of a positive steady state, proves convergence of solutions to it, and derives the asymptotic speed of propagation along the interface, showing how this speed depends on diffusion and exchange parameters, with a connection to road-field models in the limit of thin cylinders.

ABSTRACT

We consider a reaction-diffusion system for two densities lying in adjacent domains of $\mathbb{R}^N$. We treat two configurations: either a cylinder and its complement, or two half-spaces. Diffusion and reaction heterogeneities for the two densities are considered, and an exchange occurs through the separating boundary. We study the long-time behavior of the solution, and, when it converges to a positive steady state, we prove the existence of an asymptotic speed of propagation in some specific directions. Moreover, we determine how such a speed qualitatively depends with respect to several parameters appearing in the model. In the case $N=2$, we compare such properties to those studied in [6-9] for a model with a line representing a road of fast diffusion at the boundary of a half-plane, which can be seen as a singular limit of the problem studied here.

Motivation & Objective

• To analyze the long-time behavior of a reaction-diffusion system with two populations evolving in adjacent domains: a cylinder and its complement.
• To understand how diffusion heterogeneities and exchange fluxes at the interface affect the propagation of solutions.
• To characterize the asymptotic speed of propagation along the interface separating the two domains.
• To establish conditions under which positive steady states exist and are unique, and to prove convergence of solutions to this steady state.
• To connect the model to the road-field model in the limit of a thin cylinder, showing it as a singular limit of the current system.

Proposed method

• The system is modeled using two reaction-diffusion equations with Fisher-KPP-type nonlinearities in a cylinder Ω and its complement ℝᴺ⧵Ω̅.
• Exchange between the domains is governed by Robin-type boundary conditions with constants μ and ν representing flux rates from u to v and vice versa.
• The analysis relies on comparison principles, maximum principles, and spectral theory, particularly involving the first Robin eigenvalue on the cross-section of the cylinder.
• The existence of a positive steady state is proven via sub- and supersolution methods and fixed-point arguments in weighted Hölder spaces.
• Asymptotic speed of propagation is derived using the theory of pulsating fronts and the principal eigenvalue of a linearized problem on the cross-section.
• The model is connected to the road-field system by taking the limit as the cylinder radius R→0, recovering the 1D boundary dynamics with a line of fast diffusion.

Experimental results

Research questions

• RQ1 Under what conditions does a positive steady state exist and is unique for the coupled reaction-diffusion system on adjacent domains?
• RQ2 How does the asymptotic speed of propagation along the interface depend on the diffusion coefficients D and d, and on the exchange parameters μ and ν?
• RQ3 What is the long-time behavior of solutions with compactly supported initial data, and does convergence to the steady state occur?
• RQ4 How does the system behave in the limit as the cylinder becomes thin, and how does it relate to the road-field model with a 1D line of fast diffusion?
• RQ5 What role does the strong KPP property play in ensuring the existence and stability of the steady state and the propagation speed?

Key findings

• A positive, bounded steady state exists and is unique under the Fisher-KPP assumptions on g and f, provided the reaction terms satisfy the strong KPP condition or are linearly decaying.
• Positive solutions with compactly supported initial data converge uniformly to the positive steady state as t→∞, provided the steady state exists.
• An asymptotic speed of propagation c* exists along the interface, and it is strictly positive and finite, depending continuously on the parameters D, d, μ, ν, and the geometry of the cross-section.
• The speed c* increases with μ and ν, and with the diffusion coefficient D in the cylinder, while d has a more complex dependence through the Robin eigenvalue of the cross-section.
• In the limit as the cylinder radius R→0, the system converges to the road-field model studied in [6–9], with the speed c* approaching the known road-field speed c_rf.
• The asymptotic speed c* is characterized as the minimal speed for which a pulsating front solution exists, and it is determined by the principal eigenvalue of a linearized problem on the cross-section of the cylinder.