Turing patterns in simplicial complexes

Shupeng Gao, Lili Chang, Matjaž Perc, and Zhen Wang
Phys. Rev. E 107, 014216 – Published 27 January 2023

Abstract

The spontaneous emergence of patterns in nature, such as stripes and spots, can be mathematically explained by reaction-diffusion systems. These patterns are often referred as Turing patterns to honor the seminal work of Alan Turing in the early 1950s. With the coming of age of network science, and with its related departure from diffusive nearest-neighbor interactions to long-range links between nodes, additional layers of complexity behind pattern formation have been discovered, including irregular spatiotemporal patterns. Here we investigate the formation of Turing patterns in simplicial complexes, where links no longer connect just pairs of nodes but can connect three or more nodes. Such higher-order interactions are emerging as a new frontier in network science, in particular describing group interaction in various sociological and biological systems, so understanding pattern formation under these conditions is of the utmost importance. We show that a canonical reaction-diffusion system defined over a simplicial complex yields Turing patterns that fundamentally differ from patterns observed in traditional networks. For example, we observe a stable distribution of Turing patterns where the fraction of nodes with reactant concentrations above the equilibrium point is exponentially related to the average degree of 2-simplexes, and we uncover parameter regions where Turing patterns will emerge only under higher-order interactions, but not under pairwise interactions.

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  • Received 18 June 2022
  • Revised 3 November 2022
  • Accepted 6 December 2022

DOI:https://doi.org/10.1103/PhysRevE.107.014216

©2023 American Physical Society

Physics Subject Headings (PhySH)

Interdisciplinary PhysicsNetworksNonlinear Dynamics

Authors & Affiliations

Shupeng Gao1,2, Lili Chang3,4,*, Matjaž Perc5,6,7,8,9, and Zhen Wang1,2,†

  • 1School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China
  • 2School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, China
  • 3Complex Systems Research Center, Shanxi University, Taiyuan 030006, China
  • 4Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis for Disease Control and Prevention, Taiyuan 030006, China
  • 5Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
  • 6Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan
  • 7Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia
  • 8Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria
  • 9Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea

  • *Corresponding author: changll@sxu.edu.cn
  • Corresponding author: w-zhen@nwpu.edu.cn

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Vol. 107, Iss. 1 — January 2023

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