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<p>The diffusive transmembrane exchange of chemical species constitutes a fundamental interaction mechanism in chemical and biological systems, thereby motivating extensive investigations into activator–inhibitor coupling mediated by membrane transport processes. In this paper, we examine a general coupled reaction–diffusion (RD) system wherein membrane coupling is achieved through diffusive transport between reactants contained within two identical compartments. A salient characteristic of this system is the coexistence of two distinct diffusion modalities: inter-compartmental diffusion, governing species exchange between reactors, and intra-compartmental diffusion, regulating spatial redistribution within individual reactors. Under specific conditions pertaining to intra-compartmental diffusion coefficients, the system admits synchronized layered steady-state solutions as well as synchronized periodic solutions arising from Hopf bifurcations that exhibit “breathing” oscillatory dynamics. The principal focus of our investigation concerns the stability analysis of these solutions, with particular emphasis on Turing-type instabilities driven by inter-compartmental diffusions. Our analytical framework integrates the singular limit eigenvalue problem methodology, Floquet theory, and regular perturbation techniques. To demonstrate the applicability of our theoretical results, we provide an illustrative example based on the coupled RD FitzHugh–Nagumo model.</p>

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Turing instability of synchronized layered steady states and “breathing” periodic solutions in membrane-coupled reaction–diffusion systems

  • 期刊论文

  • 发表日期：

    2026年5月11日

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  • Qidong Wu,
  • Fengqi Yi

摘要

The diffusive transmembrane exchange of chemical species constitutes a fundamental interaction mechanism in chemical and biological systems, thereby motivating extensive investigations into activator–inhibitor coupling mediated by membrane transport processes. In this paper, we examine a general coupled reaction–diffusion (RD) system wherein membrane coupling is achieved through diffusive transport between reactants contained within two identical compartments. A salient characteristic of this system is the coexistence of two distinct diffusion modalities: inter-compartmental diffusion, governing species exchange between reactors, and intra-compartmental diffusion, regulating spatial redistribution within individual reactors. Under specific conditions pertaining to intra-compartmental diffusion coefficients, the system admits synchronized layered steady-state solutions as well as synchronized periodic solutions arising from Hopf bifurcations that exhibit “breathing” oscillatory dynamics. The principal focus of our investigation concerns the stability analysis of these solutions, with particular emphasis on Turing-type instabilities driven by inter-compartmental diffusions. Our analytical framework integrates the singular limit eigenvalue problem methodology, Floquet theory, and regular perturbation techniques. To demonstrate the applicability of our theoretical results, we provide an illustrative example based on the coupled RD FitzHugh–Nagumo model.