Effect of a membrane on diffusion-driven Turing instability

TL;DR

This paper extends classical Turing instability analysis to systems with permeable membranes, deriving new constraints and demonstrating pattern formation through theoretical and numerical methods.

Contribution

It introduces a novel extension of Turing theory to membrane-separated domains, including a diagonalization approach for membrane operators and analysis of pattern formation.

Findings

Membrane permeability affects Turing instability conditions.

Discontinuous patterns emerge with varying membrane and diffusion parameters.

Theoretical results are validated by numerical simulations.

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.