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.

中文翻译：

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通过时间分数阶导数形成模式。

众所周知，时间一阶导数反应扩散系统会产生各种引人入胜的图灵模式。但是，已经发现，许多物理，化学和生物系统都可以通过时间分数-导数反应-扩散方程来很好地描述。自然会产生这样的系统是否以及如何形成空间模式的问题。为了清楚地解决这个问题，我们考虑具有Holling II功能响应的经典捕食者-扩散系统的扩散模型，其中根据猎物和捕食者的行为的记忆特性引入时间分数导数。在本文中，我们证明了该分数阶导数系统可以稳定地形成空间模式，即使其一阶导数对应物不能表现出任何稳定模式。