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Systems of dynamical interactions between competing species can be used to model many complex systems, and can be mathematically described by {\em random} networks. Understanding how patterns of activity arise in such systems is important…

Adaptation and Self-Organizing Systems · Physics 2016-01-21 Nick McCullen , Thomas Wagenknecht

It is well known that simple reaction-diffusion systems can display very rich pattern formation behavior. Here we have studied two examples of such systems in three dimensions. First we investigate the morphology and stability of a generic…

Statistical Mechanics · Physics 2009-11-07 Teemu Leppanen , Mikko Karttunen , Kimmo Kaski , Rafael A. Barrio , Limei Zhang

Long after Turing's seminal Reaction-Diffusion (RD) model, the elegance of his fundamental equations alleviated much of the skepticism surrounding pattern formation. Though Turing model is a simplification and an idealization, it is one of…

Machine Learning · Computer Science 2020-12-09 Litu Rout

Many real systems are strongly characterized by collective cooperative phenomena whose existence and properties still need a satisfactory explanation. Coherently with their collective nature, they call for new and more accurate descriptions…

Physics and Society · Physics 2020-07-30 Giulio Burgio , Joan T. Matamalas , Sergio Gómez , Alex Arenas

A class of systems is considered, where immobile species associated to distinct patches, the nodes of a network, interact both locally and at a long-range, as specified by an (interaction) adjacency matrix. Non local interactions are…

Pattern Formation and Solitons · Physics 2020-06-30 Giulia Cencetti , Federico Battiston , Timoteo Carletti , Duccio Fanelli

Self-organization in natural and engineered systems causes the emergence of ordered spatio-temporal motifs. In presence of diffusive species, Turing theory has been widely used to understand the formation of such patterns on continuous…

Statistical Mechanics · Physics 2025-10-22 Marie Dorchain , Riccardo Muolo , Timoteo Carletti

The problem of pattern formation in a generic two species reaction--diffusion model is studied, under the hypothesis that only one species can diffuse. For such a system, the classical Turing instability cannot take place. At variance, by…

Statistical Mechanics · Physics 2013-09-16 Laura Cantini , Claudia Cianci , Duccio Fanelli , Emma Massi , Luigi Barletti

The Turing patterning mechanism is believed to underly the formation of repetitive structures in development, such as zebrafish stripes and mammalian digits, but it has proved difficult to isolate the specific biochemical species…

Molecular Networks · Quantitative Biology 2018-03-22 Stephen Smith , Neil Dalchau

The problem of Turing pattern formation has attracted much attention in nonlinear science as well as physics, chemistry and biology. So far all Turing patterns have been observed in stationary and oscillatory media only. In this letter we…

Pattern Formation and Solitons · Physics 2007-05-23 Jinghua Xiao , Junzhong Yang , Gang Hu

We hereby develop the theory of Turing instability for reaction-diffusion systems defined on m-directed hypergraphs, the latter being generalization of hypergraphs where nodes forming hyperedges can be shared into two disjoint sets, the…

Pattern Formation and Solitons · Physics 2025-10-22 Marie Dorchain , Wilfried Segnou , Riccardo Muolo , Timoteo Carletti

The interplay between causal mechanisms and emerging collective behaviors is a central aspect of understanding, controlling, and predicting complex networked systems. In our work, we investigate the relationship between higher-order…

General conditions are established under which reaction-cross-diffusion systems can undergo spatiotemporal pattern-forming instabilities. Recent work has focused on designing systems theoretically and experimentally to exhibit patterns with…

Pattern Formation and Solitons · Physics 2025-08-26 Edgardo Villar-Sepúlveda , Alan R. Champneys , Andrew L. Krause

The spontaneous emergence of ordered structures, known as Turing patterns, in complex networks is a phenomenon that holds potential applications across diverse scientific fields, including biology, chemistry, and physics. Here, we present a…

Physics and Society · Physics 2023-09-26 Jiaying Zhou , Yong Ye , Alex Arenas , Sergio Gómez , Yi Zhao

We study the effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems. For this purpose, the Gierer-Meinhardt model of pattern formation is considered. The cases we study are: (i)randomness in the underlying…

Statistical Mechanics · Physics 2009-10-30 Indrani Bose , Indranath Chaudhuri

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental…

Pattern Formation and Solitons · Physics 2022-07-11 Robert A. Van Gorder , Václav Klika , Andrew L. Krause

Representing social systems as networks, starting from the interactions between individuals, sheds light on the mechanisms governing their dynamics. However, networks encode only pairwise interactions, while most social interactions occur…

Physics and Society · Physics 2024-08-29 Iacopo Iacopini , Márton Karsai , Alain Barrat

The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose…

There is a growing interest, inspired by advances in technology, in the low temperature physics of thin films. These quasi-2D systems show a wide range of ordering effects including formation of striped states, reorientation transitions,…

Statistical Mechanics · Physics 2009-11-13 Alessandro Giuliani , Joel L. Lebowitz , Elliott H. Lieb

Cooperation enables teams to solve complex problems that one individual alone cannot address. In science, collaborative teams have become the predominant way through which progress is achieved. These scientific collaborations arise though…

Physics and Society · Physics 2026-03-04 Emma Zajdela , Nicholas W. Landry

The process of pattern formation for a multi-species model anchored on a time varying network is studied. A non homogeneous perturbation superposed to an homogeneous stable fixed point can amplify, as follows a novel mechanism of…

Statistical Mechanics · Physics 2017-10-11 Julien Petit , Ben Lauwens , Duccio Fanelli , Timoteo Carletti