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A theory of pattern formation for reactiondiffusion systems on temporal networks
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences ( IF 2.9 ) Pub Date : 2021-03-24 , DOI: 10.1098/rspa.2020.0753
Robert A. Van Gorder 1
Affiliation  

Networks have become ubiquitous in the modern scientific literature, with recent work directed at understanding ‘temporal networks’—those networks having structure or topology which evolves over time. One area of active interest is pattern formation from reaction–diffusion systems, which themselves evolve over temporal networks. We derive analytical conditions for the onset of diffusive spatial and spatio-temporal pattern formation on undirected temporal networks through the Turing and Benjamin–Feir mechanisms, with the resulting pattern selection process depending strongly on the evolution of both global diffusion rates and the local structure of the underlying network. Both instability criteria are then extended to the case where the reaction–diffusion system is non-autonomous, which allows us to study pattern formation from time-varying base states. The theory we present is illustrated through a variety of numerical simulations which highlight the role of the time evolution of network topology, diffusion mechanisms and non-autonomous reaction kinetics on pattern formation or suppression. A fundamental finding is that Turing and Benjamin–Feir instabilities are generically transient rather than eternal, with dynamics on temporal networks able to transition between distinct patterns or spatio-temporal states. One may exploit this feature to generate new patterns, or even suppress undesirable patterns, over a given time interval.



中文翻译:

时间网络上反应扩散系统的模式形成理论

在现代科学文献中,网络已经变得无处不在,最近的工作是着眼于理解“时间网络”,即那些具有随着时间而发展的结构或拓扑的网络。感兴趣的领域之一是反应扩散系统的模式形成,它们本身是通过时间网络演化的。我们通过图灵和本杰明·费尔机制,推导了在无向时间网络上发生扩散性时空格局形成的分析条件,由此产生的模式选择过程强烈依赖于总体扩散率和局部结构的演化。基础网络。然后将这两个不稳定性标准扩展到反应扩散系统是非自治的情况下,这使我们能够研究时基状态的模式形成。我们通过多种数值模拟对我们提出的理论进行了说明,这些数值模拟突显了网络拓扑结构的时间演化,扩散机制和非自主反应动力学对模式形成或抑制的作用。一个基本的发现是,图灵和本杰明·费尔不稳定性通常是短暂的,而不是永恒的,时态网络的动力学能够在不同的模式或时空状态之间转换。在给定的时间间隔内,可以利用此功能生成新的图案,甚至抑制不想要的图案。扩散机制和模式形成或抑制的非自主反应动力学。一个基本的发现是,图灵和本杰明·费尔不稳定性通常是短暂的,而不是永恒的,时态网络的动力学能够在不同的模式或时空状态之间转换。在给定的时间间隔内,可以利用此功能生成新的图案,甚至抑制不想要的图案。扩散机制和模式形成或抑制的非自主反应动力学。一个基本的发现是,图灵和本杰明·费尔不稳定性通常是短暂的,而不是永恒的,时态网络的动力学能够在不同的模式或时空状态之间转换。在给定的时间间隔内,可以利用此功能生成新的图案,甚至抑制不想要的图案。

更新日期:2021-03-24
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