Yadav A, Horsthemke Werner
Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Dec;74(6 Pt 2):066118. doi: 10.1103/PhysRevE.74.066118. Epub 2006 Dec 28.
We derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by Vlad and Ross [Phys. Rev. E 66, 061908 (2002)]. Reaction and diffusion processes are separable in a typical reaction-diffusion system, and their combined influence on the evolution of the density of a species is a simple sum. Our derivation shows that this is no longer true for subdiffusive entities undergoing reactions. The strong memory effects in the transport process, i.e., the non-Markovian nature of subdiffusion, results in a nontrivial combination of reactions and spatial dispersal, which we discuss in detail. We carry out a linear stability analysis of the derived reaction-subdiffusion system to understand the effects of memory on pattern formation. We find that the Turing instability persists in the subdiffusive system. However, the memory modifies the Turing threshold and the characteristics of the band of unstable modes close to this threshold.
基于弗拉德和罗斯提出的非线性连续时间随机游走形式（《物理评论E》66卷，061908页，2002年），我们推导了反应和亚扩散实体的一般动力学方程。在典型的反应扩散系统中，反应和扩散过程是可分离的，它们对物种密度演化的综合影响是简单的相加。我们的推导表明，对于发生反应的亚扩散实体，情况并非如此。传输过程中的强记忆效应，即亚扩散的非马尔可夫性质，导致了反应和空间扩散的非平凡组合，我们将对此进行详细讨论。我们对推导得到的反应 - 亚扩散系统进行了线性稳定性分析，以了解记忆对图案形成的影响。我们发现图灵不稳定性在亚扩散系统中持续存在。然而，记忆改变了图灵阈值以及接近该阈值的不稳定模式带的特征。
