Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations
Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations
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DOI：
10.1007/s00285-014-0779-6
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发表时间：
2014-03
影响因子：
1.9
通讯作者：
A. Madzvamuse;H. S. Ndakwo;R. Barreira
中科院分区：
文献类型：
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作者：
A. Madzvamuse;H. S. Ndakwo;R. Barreira
通过对具有活化剂耗尽反应动力学的两组分反应扩散系统引入线性交叉扩散（Gierer和Meinhardt，Kybernetik 12：30-39，1972; Prigogine和Lefever，J Chem Phys 48：1695-1700，1968; Schnakenberg，J Theor Biol 81：389-400，1979），我们导出了交叉扩散驱动的不稳定性条件，并表明它们是不存在交叉扩散的经典扩散驱动的不稳定性条件的推广。我们最具启发性的结果是，与没有交叉扩散的经典反应扩散系统相比，不再需要强制其中一个物种比另一个物种扩散快得多。此外，不再需要活化剂-抑制剂机制作为图案形成的前提，活化剂-活化剂、通道-通道或反应动力学以及短程抑制和长程活化都有可能引起交叉扩散驱动的不稳定性。为了支持我们的理论研究结果，我们计算交叉扩散引起的参数空间，并证明使用标准的反应扩散理论所获得的相似性和差异。对平面正方形区域进行了有限元数值模拟，以支持理论预测。对于所提出的数值模拟，我们选择的参数值和经典图灵扩散驱动的不稳定空间之外，这些被选择属于交叉扩散驱动的不稳定参数空间。我们的数值实验验证了我们的理论预测，即由交叉扩散引起的反应扩散系统的和组件的参数空间大得多，并不同于那些没有交叉扩散。此外，没有交叉扩散的参数空间是交叉扩散诱导参数空间的子空间。我们的结果使实验者能够拥有更广泛的参数空间，从中选择反应动力学参数值，这些参数值将在存在交叉扩散的情况下产生空间图案。
By introducing linear cross-diffusion for a two-component reaction-diffusion system withactivator-depletedreaction kinetics (Gierer and Meinhardt, Kybernetik 12:30–39, 1972; Prigogine and Lefever, J Chem Phys 48:1695–1700, 1968; Schnakenberg, J Theor Biol 81:389–400, 1979), we derivecross-diffusion-driveninstability conditions and show that they are a generalisation of the classical diffusion-driven instability conditions in the absence of cross-diffusion. Our most revealing result is that, in contrast to the classical reaction-diffusion systems without cross-diffusion,it is no longer necessary to enforce that one of the species diffuse much faster than the other. Furthermore,it is no longer necessary to have an activator–inhibitor mechanism as premises for pattern formation, activator–activator,inhibitor–inhibitorreaction kinetics as well asshort-range inhibitionandlong-range activationall have the potential of giving rise to cross-diffusion-driven instability. To support our theoretical findings, we compute cross-diffusion induced parameter spaces and demonstrate similarities and differences to those obtained using standard reaction-diffusion theory. Finite element numerical simulations on planary square domains are presented to back-up theoretical predictions. For the numerical simulations presented, we choose parameter values from and outside the classical Turing diffusively-driven instability space; outside, these are chosen to belong to cross-diffusively-driven instability parameter spaces. Our numerical experiments validate our theoretical predictions that parameter spaces induced by cross-diffusion in both theandcomponents of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spaces of the cross-diffusion induced parameter spaces. Our results allow experimentalists to have a wider range of parameter spaces from which to select reaction kinetic parameter values that will give rise to spatial patterning in the presence of cross-diffusion.