# American Institute of Mathematical Sciences

June  2020, 40(6): 3813-3836. doi: 10.3934/dcds.2020160

## Turing type instability in a diffusion model with mass transport on the boundary

 1 Department of Applied Mathemarics and Informatics, Ryukoku University, Seta Otsu 520-2194, Japan 2 Department of Mathematical and Life Science, Hiroshima University, 1-3-1 Kagamiyama Higashi-Hiroshima, 739-8526, Japan

* Corresponding author: Kunimochi Sakamoto

Received  January 2019 Revised  January 2020 Published  March 2020

Fund Project: The first author was partially supported by JSPS KAKENHI Grant JP18H01139 and the second author was partially supported by JSPS KAKENHI Grant JP19K03564

Some reaction-diffusion models describing the cell polarity are proposed, where the system has two independent variables standing for the concentration of proteins in the membrane and the cytosol respectively. In this article we deal with such a polarity model consisting of one equation on a unit sphere and the other one in the ball inside the sphere. The two equations are coupled through a nonlinear boundary condition and the total mass is conserved. We investigate the linearized stability of a constant steady state and provide conditions under which a Turing type instability takes place, namely, the constant state is stable against spatially uniform perturbations on the sphere for all choices of diffusion rates, while unstable against nonuniform perturbations on the sphere as the diffusion coefficient of the equation on the sphere becomes small relative to the one in the ball.

Citation: Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
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