Article
Mathematics, Applied
Alexander Kolinichenko, Lev Ryashko
Summary: This paper investigates a spatially extended stochastic reaction-diffusion model, demonstrating a theoretical approach and comparing statistically obtained data on stochastic sensitivity functions for stable nonhomogeneous stationary patterns. It discusses variations in pattern sensitivity to noise and the phenomenon of stochastic preference in different patterns in the Brusselator.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Edgardo Villar-Sepulveda, Alan Champneys
Summary: This paper provides explicit calculations to justify the existence and criticality of Turing bifurcations in a general class of activator-inhibitor reaction-diffusion equations on a one-dimensional infinite domain. The calculations consider two distinct scalings of parameters and reveal that the Turing bifurcation can be either subcritical or supercritical, leading to the existence of a codimension-two degenerate bifurcation. The sign of a fifth-order normal form coefficient is computed and shown to be correct for the birth of homoclinic snaking.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Engineering, Mechanical
Wei Gou, Zhen Jin
Summary: This study investigates the epidemiological patterns in spatial networks through the development of an epidemic reaction-diffusion model based on spatially embedded networks. By systematically studying factors such as network size, connectivity, and degree heterogeneity, the study provides new insights into the formation of patterns in spatial networks. It is found that degree heterogeneity does not trigger fundamental changes in pattern types, and randomly connected links in spatial networks act as a mechanism to induce irregular stationary patterns, narrowing prevalence differences and preventing Turing instability.
NONLINEAR DYNAMICS
(2021)
Article
Multidisciplinary Sciences
Robert A. Van Gorder
Summary: Turing instability and Turing patterns, first proposed by Turing in 1952, are key tools for studying diffusion-driven pattern formation. Spatial heterogeneity in reaction-diffusion systems is identified as one route to obtaining irregular patterns, with increasing interest in understanding irregular patterns. The study investigates pattern formation from systems involving spatial heterogeneity through analytical and numerical techniques, extending classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern.
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2021)
Article
Mathematics, Interdisciplinary Applications
Riccardo Muolo, Luca Gallo, Vito Latora, Mattia Frasca, Timoteo Carletti
Summary: Turing theory is widely used to explain the spatio-temporal structures observed in Nature. This paper proposes a method to include group interactions in reaction-diffusion systems and examines their effects on Turing pattern formation. Results demonstrate the mechanisms of pattern formation in systems with many-body interactions and provide a basis for further extensions of the Turing framework.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Applied
Alan R. Champneys, Fahad Al Saadi, Victor F. Brena-Medina, Veronica A. Grieneisen, Athanasius F. M. Maree, Nicolas Verschueren, Bert Wuyts
Summary: This paper presents a synthesis of recent research on the formation of localized patterns, isolated spots, or sharp fronts in models of natural processes governed by reaction-diffusion equations. It contrasts with the well-known Turing mechanism of periodic pattern formation and provides a general picture in one spatial dimension for models on long domains exhibiting sub-critical Turing instabilities.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Chemistry, Multidisciplinary
Zerong Xing, Genpei Zhang, Jianye Gao, Jiao Ye, Zhuquan Zhou, Biying Liu, Xiaotong Yan, Xueqing Chen, Minghui Guo, Kai Yue, Xuanze Li, Qian Wang, Jing Liu
Summary: This study reveals the mechanism of Turing instability using liquid metal-solid metal reaction-diffusion systems. By designing metal pairs with appropriate reaction kinetics and diffusion coefficients, labyrinths, stripes, and spots-like Turing structures can be generated.
ADVANCED MATERIALS
(2023)
Article
Physics, Multidisciplinary
Joshua S. Ritchie, Andrew L. Krause, Robert A. Van Gorder
Summary: This study explores diffusive instabilities and resulting pattern formation in hyperbolic reaction-diffusion equations, finding that additional temporal terms affect the formation and existence regions of Turing patterns, and are necessary for the emergence of spatiotemporal patterns under wave instability. The parameters leading to wave instabilities are mutually exclusive to those leading to stationary Turing patterns.
Article
Multidisciplinary Sciences
Hiroyuki Shima, Yoshitaka Umeno, Takashi Sumigawa
Summary: Spontaneous pattern formation by dislocations in metal fatigue under cyclic straining shows a crossover from a 2D spot-scattered pattern to a 1D ladder-shaped pattern. However, the physical mechanism behind this crossover remains unclear. This study proposes a bifurcation diagram that explains the crossover using a weakly nonlinear stability analysis. It is found that inequalities with respect to nonlinearity parameters can describe the selection rule between the two dislocation patterns, spot and ladder-shaped.
Article
Engineering, Mechanical
Rui Yang
Summary: This paper investigates the Turing instability conditions driven by cross-diffusion in the Schnakenberg system and reveals that long-range inhibition and short-range activation are no longer necessary for Turing instability with the introduction of cross-diffusion. The amplitude equations at the critical value of Turing bifurcation are derived using the multiple scales method, which helps to determine the parameter space where certain patterns emerge. Numerical simulations in the Turing instability region and Turing-Hopf region demonstrate the variety of patterns that the system can exhibit, and different initial conditions are employed to enhance understanding of the complex patterns.
NONLINEAR DYNAMICS
(2022)
Article
Biology
Edgardo Villar-Sepulveda, Alan R. R. Champneys
Summary: This study provides necessary and sufficient conditions for a diffusion-driven instability of a stable equilibrium in a reaction-diffusion system with n components and diagonal diffusion matrix. These instabilities can be either Turing or wave instabilities. The known conditions for a diffusion rate that causes a Turing bifurcation of a stable homogeneous state in the absence of diffusion are reproducible. The method of proof used here gives a constructive approach for selecting diffusion constants, based on studying dispersion relations in different limits.
JOURNAL OF MATHEMATICAL BIOLOGY
(2023)
Article
Computer Science, Artificial Intelligence
Yiqing Jia, Qili Zhao, Hongqiang Yin, Shan Guo, Mingzhu Sun, Zhuo Yang, Xin Zhao
Summary: This study provided a reinterpretation of the relationship between dendritic spine pattern abnormalities and multiple nervous system diseases, and used a reaction-diffusion model to simulate the formation process of dendritic spines, investigating the factors affecting spine patterns. The research found that the consumption rate of substrates by the cytoskeleton, as well as the amount of an exogenous activator and inhibitor, are key factors in regulating spine shape and density. The study also analyzed the inner mechanism of how these factors regulate the dendritic spine pattern through Turing instability analysis, providing insights for potential treatment research for diseases related to dendritic spine pattern abnormalities.
FRONTIERS IN NEUROROBOTICS
(2021)
Article
Engineering, Mechanical
Yuxi Li, Zhouchao Wei
Summary: A stochastic reaction-diffusion epidemic model is used to analyze and control the infectious disease COVID-19, discussing the stationary distribution and Turing instability. The derivation of amplitude equations, determination of Turing patterns, study of optimal quarantine control, and comparison between models are also conducted. Optimal control theory is applied to obtain the existence and uniqueness of the optimal control and solution, which are then verified through numerical simulation.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Mingrui Song, Shupeng Gao, Chen Liu, Yue Bai, Lei Zhang, Beilong Xie, Lili Chang
Summary: Predator-prey models have attracted attention in various disciplines and the theory of pattern formation in monolayer networks has been extensively studied. This study extends the theory to multiplex networks, which are common in diverse areas. Furthermore, the model is enhanced with cross-diffusion to account for specific movement tendencies of each species. The linear analysis reveals the theoretical Turing instability region and demonstrates that either the multiplex network topology or cross-diffusion can destabilize the homogeneous fixed point, resulting in various Turing patterns. The experimental simulation results confirm the validity of the theoretical analysis.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Computer Science, Information Systems
Le He, Linhe Zhu, Zhengdi Zhang
Summary: The paper studied the spreading of rumors and crowd diffusion behavior on network structure through a Suspicious-Infected (SI) model with Allee effect, and conducted extensive numerical evaluations under different networks. The results showed that the diffusion coefficient can significantly change patterns, and discussed the necessary conditions for Turing pattern to appear in space.
INFORMATION SCIENCES
(2021)