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Effect of a membrane on diffusion-driven Turing instability

Abstract : Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To this aim, we study a reaction–diffusion system with zero-flux boundary conditions on the external boundary and Kedem-Katchalsky membrane conditions on the inner membrane. We use the same approach as in the classical Turing analysis but applied to membrane operators. The introduction of a diagonalization theory for compact and self-adjoint membrane operators is needed. Here, Turing instability is proven with the addition of new constraints, due to the presence of membrane permeability coefficients. We perform an explicit one-dimensional analysis of the eigenvalue problem, combined with numerical simulations, to validate the theoretical results. Finally, we observe the formation of discontinuous patterns in a system which combines diffusion and dissipative membrane conditions, varying both diffusion and membrane permeability coefficients. The case of a fast reaction-diffusion system is also considered.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03231369
Contributor : Giorgia Ciavolella Connect in order to contact the contributor
Submitted on : Thursday, October 14, 2021 - 3:30:36 PM
Last modification on : Saturday, October 16, 2021 - 3:55:13 AM

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  • HAL Id : hal-03231369, version 3

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Giorgia Ciavolella. Effect of a membrane on diffusion-driven Turing instability. 2021. ⟨hal-03231369v3⟩

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