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Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach

Maxime Breden, Maxime Payan, Cordula Reisch, Bao Quoc Tang

TL;DR

This work develops a computer-assisted, rigorous spectral analysis for a nonlocal heterogeneous reaction-diffusion system that exhibits Turing instability driven by nonlocal terms. By formulating the linearization as an infinite matrix with compact resolvent and applying two successive basis changes, the authors obtain finite, verifiable Gershgorin disk estimates and isolate a single unstable eigenvalue for large nonlocal intensity $\delta$, while ensuring stability for small $\delta$. A Newton-Kantorovich framework provides tight enclosures for the leading eigenvalue and its $\delta$-derivative, proving the existence and uniqueness of a sharp threshold $\delta^*$ at which the trivial equilibrium loses stability. The analysis is grounded in a liver inflammation model, highlighting nonlocal immune-response effects; the approach readily extends to other nonlocal RD systems. Overall, the paper delivers a rigorous, computable pathway from infinite-dimensional spectral problems to concrete instability criteria and bifurcation points with practical relevance to pattern formation in heterogeneous media.

Abstract

This paper provides a computer-assisted proof for the Turing instability induced by heterogeneous nonlocality in reaction-diffusion systems. Due to the heterogeneity and nonlocality, the linear Fourier analysis gives rise to \textit{strongly coupled} infinite differential systems. By introducing suitable changes of basis as well as the Gershgorin disks theorem for infinite matrices, we first show that all $N$-th Gershgorin disks lie completely on the left half-plane for sufficiently large $N$. For the remaining finitely many disks, a computer-assisted proof shows that if the intensity $δ$ of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by detailed study of this eigenvalue as a function of $δ$, we obtain a sharp threshold $δ^*$ which is the bifurcation point for Turing instability.

Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach

TL;DR

This work develops a computer-assisted, rigorous spectral analysis for a nonlocal heterogeneous reaction-diffusion system that exhibits Turing instability driven by nonlocal terms. By formulating the linearization as an infinite matrix with compact resolvent and applying two successive basis changes, the authors obtain finite, verifiable Gershgorin disk estimates and isolate a single unstable eigenvalue for large nonlocal intensity , while ensuring stability for small . A Newton-Kantorovich framework provides tight enclosures for the leading eigenvalue and its -derivative, proving the existence and uniqueness of a sharp threshold at which the trivial equilibrium loses stability. The analysis is grounded in a liver inflammation model, highlighting nonlocal immune-response effects; the approach readily extends to other nonlocal RD systems. Overall, the paper delivers a rigorous, computable pathway from infinite-dimensional spectral problems to concrete instability criteria and bifurcation points with practical relevance to pattern formation in heterogeneous media.

Abstract

This paper provides a computer-assisted proof for the Turing instability induced by heterogeneous nonlocality in reaction-diffusion systems. Due to the heterogeneity and nonlocality, the linear Fourier analysis gives rise to \textit{strongly coupled} infinite differential systems. By introducing suitable changes of basis as well as the Gershgorin disks theorem for infinite matrices, we first show that all -th Gershgorin disks lie completely on the left half-plane for sufficiently large . For the remaining finitely many disks, a computer-assisted proof shows that if the intensity of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by detailed study of this eigenvalue as a function of , we obtain a sharp threshold which is the bifurcation point for Turing instability.

Paper Structure

This paper contains 22 sections, 21 theorems, 128 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main results and key ideas
  4. Localization of the spectrum.
  5. A Gershgorin theorem for infinite matrices having compact resolvent
  6. Infinite matrices and basis changes
  7. Gershgorin disks and estimation of radius bounds
  8. Proof of Theorem \ref{['thm:spectrum']}
  9. Existence of the threshold
  10. The map . First properties
  11. Change of basis and definitions: a new point of view
  12. Zero of
  13. Formalism and construction of
  14. Derivation of the bounds
  15. Proof of Proposition \ref{['prop:lambda1']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Consider system eq:system with $\Omega = (0,2)$, $\Omega_1 = (\pi/4, \pi/2)$, $\Omega_2=( \pi/5, \pi/2 + 1/4)$, $a=-3$, $b=2$, $c = 3$, $d=-3$ and $\vartheta =1$. There exists $\delta^*\in [2.428,2.46]$ such that the equilibrium state $(0,0)$ is linearly stable for all $\delta \in [0,\delta^*)$, and

Figures (4)

  • Figure 1: Numerical simulations of system \ref{['eq:system']}, for the domain and parameter values used in Theorem \ref{['thm:main']}. We observe convergence to the trivial equilibrium for $\delta=1$, but the apparition of nontrivial patterns for $\delta=4$, in accordance with Theorem \ref{['thm:main']}.
  • Figure 2: Illustration of some of the quantitative statements obtained within the proof of Theorem \ref{['thm:spectrum']} (see Section \ref{['sec:proofspectrum']}). We display here a threshold $\mu = \mu(\delta)$ satisfying point 1. of Theorem \ref{['thm:spectrum']}, together with a narrow enclosure of $d_0$ given by lower and upper-bounds $d_0^-$ and $d_0^+$, satisfying $d_0^- \leq d_0 \leq d_0^+$ . Dark green indicates values of $\delta$ for which point $2.$ of Theorem \ref{['thm:spectrum']} holds. Light green indicates values of $\delta$ for which point $3.$ of Theorem \ref{['thm:spectrum']} holds. Black indicates values of $\delta$ for which the sign of $d_0$ remains undetermined in Theorem \ref{['thm:spectrum']}.
  • Figure 3: Two examples showing the first $2N$ Gershgorin disks and the bound $\overline{m}_{N,p}$, illustrating the proof of Theorem \ref{['thm:spectrum']}. In each case, the second picture is a zoom in close to the origin. Note that we intentionally took $N$ ten times smaller than in the proof of Theorem \ref{['thm:spectrum']}, and $p = 1.7$, in order to get disks that are not too small and therefore easier to visualize. The estimates obtained in the proof of Theorem \ref{['thm:spectrum']} are actually much sharper.
  • Figure 4: Upper- and lower-bounds on the map $\delta \mapsto d_0(\delta)$ for $\delta \in\left[\delta_0,\delta_1\right]$, with $k=607,\dots,614$. These are the functions $\underline{d_0}$ and $\overline{d_0}$ from Proposition \ref{['prop:lambda1']}, which give sharper enclosures on $d_0$ than the ones obtained in Section \ref{['sec:Gersh']} using Gershgorin disks, and shown on Figure \ref{['fig:d0']}.

Theorems & Definitions (73)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 63 more