Yin Hongwei, Wen Xiaoqing
School of Science, Nanchang University, Nanchang, 330031, P.R. China.
Numerical Simulation and High-Performance Computing Laboratory, Nanchang University, Nanchang, 330031, P.R. China.
Sci Rep. 2018 Mar 22;8(1):5070. doi: 10.1038/s41598-018-23470-8.
It is well known that temporal first-derivative reaction-diffusion systems can produce various fascinating Turing patterns. However, it has been found that many physical, chemical and biological systems are well described by temporal fractional-derivative reaction-diffusion equations. Naturally arises an issue whether and how spatial patterns form for such a kind of systems. To address this issue clearly, we consider a classical prey-predator diffusive model with the Holling II functional response, where temporal fractional derivatives are introduced according to the memory character of prey's and predator's behaviors. In this paper, we show that this fractional-derivative system can form steadily spatial patterns even though its first-derivative counterpart can't exhibit any steady pattern. This result implies that the temporal fractional derivatives can induce spatial patterns, which enriches the current mechanisms of pattern formation.
众所周知，时间一阶导数反应扩散系统能够产生各种迷人的图灵模式。然而，人们发现许多物理、化学和生物系统都可以用时间分数阶导数反应扩散方程来很好地描述。自然而然地就出现了这样一个问题：对于这类系统，空间模式是否会形成以及如何形成。为了清楚地解决这个问题，我们考虑一个具有Holling II功能反应的经典猎物 - 捕食者扩散模型，其中根据猎物和捕食者行为的记忆特性引入了时间分数阶导数。在本文中，我们表明这个分数阶导数系统能够形成稳定的空间模式，尽管其一阶导数对应的系统不会表现出任何稳定模式。这一结果意味着时间分数阶导数能够诱导空间模式，从而丰富了当前的模式形成机制。
