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<p>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 <i>strongly coupled</i> infinite differential systems. By introducing suitable changes of basis as well as the Gershgorin disks theorem for infinite matrices, we first show that all <i>N</i>-th Gershgorin disks lie completely on the left half-plane for sufficiently large <i>N</i>. For the remaining finitely many disks, a computer-assisted proof shows that if the intensity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by a detailed study of this eigenvalue as a function of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>, we obtain a sharp threshold <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\delta ^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>δ</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> which is the bifurcation point for Turing instability.</p>

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Turing Instability for Nonlocal Heterogeneous Reaction-Diffusion Systems: A Computer-Assisted Proof Approach

  • 期刊论文

  • 发表日期：

    2026年2月10日

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

摘要

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 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 \(\delta \) δ of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by a detailed study of this eigenvalue as a function of \(\delta \) δ , we obtain a sharp threshold \(\delta ^*\) δ ∗ which is the bifurcation point for Turing instability.