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Turing patterns play a fundamental role in morphogenesis and population dynamics, encoding key information about the underlying biological mechanisms. Yet, traditional inverse problems have largely relied on non-biological data such as…

Analysis of PDEs · Mathematics 2025-09-10 Yuhan Li , Hongyu Liu , Catharine W. K. Lo

Including pairwise interactions between the predictors of a regression model can produce better predicting models. However, to fit such interaction models on typical data sets in biology and other fields can often require solving enormous…

Methodology · Statistics 2023-02-14 Guo Yu , Jacob Bien , Ryan Tibshirani

The model of the physical system with discrete interactions is based on the postulates that (i) parameters of the physical system are defined in process of its interaction; (ii) the process of interaction is discrete. Consequently ordering…

Quantum Physics · Physics 2007-05-23 M Yudin

This paper investigates the conditions for the stability and emergence of patterns in a new three-component reaction-diffusion system. The system describes the coexistence and interaction of water reservoirs, vegetation, and bushfire…

Analysis of PDEs · Mathematics 2026-04-14 Serena Dipierro , Enrico Valdinoci

We introduce the idea of weakly coherent collisional models, where the elements of an environment interacting with a system of interest are prepared in states that are approximately thermal, but have an amount of coherence proportional to a…

Reaction-diffusion processes on networked systems have received mounting attention in the past two decades, and the corresponding theory of network dynamics has been continuously enriched with the advancement of network science. Recently,…

Pattern Formation and Solitons · Physics 2025-04-30 Junyuan Shi , Linhe Zhu

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 theory of patterns formation for a reaction-diffusion system defined on a multiplex is developed by means of a perturbative approach. The intra-layer diffusion constants act as small parameter in the expansion and the unperturbed state…

Statistical Mechanics · Physics 2015-06-22 Malbor Asllani , Daniel M. Busiello , Timoteo Carletti , Duccio Fanelli , Gwendoline Planchon

Many cellular patterns exhibit a reaction-diffusion component, suggesting that Turing instability may contribute to pattern formation. However, biological gene-regulatory pathways are more complex than simple Turing activator-inhibitor…

Molecular Networks · Quantitative Biology 2024-12-05 Hazlam S. Ahmad Shaberi , Aibek Kappassov , Antonio Matas-Gil , Robert G. Endres

The approach to a substantiation of thermodynamics is offered. A conservative system of interacting elements, which is not in equilibrium, is used as a model. This system is then split into small subsystems that are accepted as being in…

Statistical Mechanics · Physics 2007-05-23 V. M. Somsikov

The Turing instability is a paradigmatic route to patterns formation in reaction-diffusion systems. Following a diffusion-driven instability, homogeneous fixed points can become unstable when subject to external perturbation. As a…

Pattern Formation and Solitons · Physics 2015-09-02 Joseph D. Challenger , Raffaella Burioni , Duccio Fanelli

Spontaneous pattern formation in homogeneous systems is ubiquitous in nature. Although Turing demonstrated that spatial patterns can emerge in reaction-diffusion (RD) systems when the homogeneous state becomes linearly unstable, it remains…

Biological Physics · Physics 2024-08-20 Shuonan Wu , Bing Yu , Yuhai Tu , Lei Zhang

Turing patterns are a cornerstone of biological self-organization, yet their emergence typically requires finely tuned parameters occupying narrow regions of high-dimensional space. This poses a fundamental challenge: how can evolving…

Cell Behavior · Quantitative Biology 2026-05-18 Seyoon Kim , Antonio Matas-Gil , Robert G. Endres

Nature is a blossoming of regular structures, signature of self-organization of the underlying microscopic interacting agents. Turing theory of pattern formation is one of the most studied mechanisms to address such phenomena and has been…

Pattern Formation and Solitons · Physics 2025-10-22 Riccardo Muolo , Lorenzo Giambagli , Hiroya Nakao , Duccio Fanelli , Timoteo Carletti

A system responding to a stochastic driving signal can be interpreted as computing, by means of its dynamics, an implicit model of the environmental variables. The system's state retains information about past environmental fluctuations,…

Statistical Mechanics · Physics 2012-10-09 Susanne Still , David A. Sivak , Anthony J. Bell , Gavin E. Crooks

In an equilibrium thermal environment, random elastic collisions between background particles and a tracer establish the picture of Brownian motion fulfilling the celebrated Einstein relation between diffusivity and mobility. In nature,…

These lecture notes are designed to provide a brief introduction into the phenomenology of active matter and to present some of the analytical tools used to rationalize the emergent behavior of active systems. Such systems are made of…

Soft Condensed Matter · Physics 2018-07-02 Étienne Fodor , M. Cristina Marchetti

In his seminal paper on morphogenesis (1952), Alan Turing demonstrated that different spatio-temporal patterns can arise due to instability of the homogeneous state in reaction-diffusion systems, but at least two species are necessary to…

Formal Languages and Automata Theory · Computer Science 2014-06-03 Yuri Shalygo

A common approach to the provably stable design of reactive behavior, exemplified by operational space control, is to reduce the problem to the design of virtual classical mechanical systems (energy shaping). This framework is widely used,…

Robotics · Computer Science 2021-06-28 Nathan D. Ratliff , Karl Van Wyk , Mandy Xie , Anqi Li , Muhammad Asif Rana

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