Madzvamuse Anotida, Chung Andy H W, Venkataraman Chandrasekhar
School of Mathematical and Physical Sciences, Department of Mathematics , University of Sussex , Brighton BN19QH, UK.
Proc Math Phys Eng Sci. 2015 Mar 8;471(2175):20140546. doi: 10.1098/rspa.2014.0546.
In this article, we formulate new models for coupled systems of bulk-surface reaction-diffusion equations on stationary volumes. The bulk reaction-diffusion equations are coupled to the surface reaction-diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction-diffusion system can induce patterning on the surface independent of whether the surface reaction-diffusion system produces or not, patterning. On the other hand, the surface reaction-diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction-diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.
在本文中，我们为固定体积上的体 - 表面反应 - 扩散方程耦合系统建立了新模型。体反应 - 扩散方程通过线性罗宾型边界条件与表面反应 - 扩散方程耦合。然后，我们陈述并证明了耦合系统扩散驱动不稳定性的必要条件。由于体动力学和表面动力学之间耦合的性质，我们能够将体动力学和表面动力学的稳定性分析解耦。在适当选择模型参数值的情况下，体反应 - 扩散系统可以在表面上诱导图案形成，而与表面反应 - 扩散系统是否产生图案无关。另一方面，在没有体反应 - 扩散系统产生的图案的情况下，表面反应 - 扩散系统不能在体的所有地方生成图案。对于这种情况，图案只能在靠近表面膜的区域诱导产生。我们进行了各种数值实验来支持我们的理论发现。我们最具启发性的数值结果是，罗宾型边界条件似乎引入了一个耦合体动力学和表面动力学的边界层。
