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Turing instability and pattern formation of a fractional Hopfield reaction–diffusion neural network with transmission delay

Jiazhe Lin, Jiapeng Li, Rui Xu
2022 Nonlinear Analysis: Modelling and Control  
In this paper, we mainly investigate the Turing pattern in a delayed reactiondiffusion neural network with Caputo-type fractional derivative.  ...  It is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation.  ...  We would like to thank the referees and the editor for their careful reading of the original manuscript and their valuable comments and suggestions that improved the presentation of this work.  ... 
doi:10.15388/namc.2022.27.27473 fatcat:37pmpgid5jg5vh4g6p7p2ilwn4

A graph-theoretic method for detecting potential Turing bifurcations

Maya Mincheva, Marc R. Roussel
2006 Journal of Chemical Physics  
The conditions for diffusion-driven ͑Turing͒ instabilities in systems with two reactive species are well known.  ...  General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed.  ...  Turing bifurcation. 1 Finding the Turing instabilities in biochemical models involves the analysis of a system of reaction-diffusion equations.  ... 
doi:10.1063/1.2397073 pmid:17144685 fatcat:xtfdruzairgp3i6wbvggvyzole


1999 Journal of the Society of Materials Science Japan  
The mesosco evolution of the immobile dislocation density is driven by the space-oriented bifurcation (Turing instabilit and results in the ladder-type self-organization.  ...  The local rule of CA taken in the prese paper is the weighted addition modulo two rule.  ...  The local rule of the CA is the weighted addition modulo two rule whose weights are chosen in referring to the reaction-diffusion equations.  ... 
doi:10.2472/jsms.48.12appendix_258 fatcat:hor7ucfq25gxnhilzpwpvkivpm

Multigraph Conditions for Multistability, Oscillations and Pattern Formation in Biochemical Reaction Networks

M. Mincheva, G. Craciun
2008 Proceedings of the IEEE  
We present as an example a model for cell cycle and apoptosis along with bifurcation diagrams and sample solutions that confirm the predictions obtained with the help of the multigraph network conditions  ...  We also derive corresponding graphtheoretic conditions for pattern formation for the respective reaction-diffusion models.  ...  In a recent article we reviewed graph-theoretic conditions for Turing instabilities for n-dimensional mass action reaction-diffusion systems [36] , first derived in [27] .  ... 
doi:10.1109/jproc.2008.925474 fatcat:n6wch3w2l5bddcb3z4kzbqfngm

Turing instability in reaction–diffusion models on complex networks

Yusuke Ide, Hirofumi Izuhara, Takuya Machida
2016 Physica A: Statistical Mechanics and its Applications  
In this paper, the Turing instability in reaction-diffusion models defined on complex networks is studied.  ...  From analysis of the Laplacian matrices of graphs generated by these models, we numerically reveal that stable and unstable regions of a homogeneous steady state on the parameter space of two diffusion  ...  In this paper, since we focus on the Turing instability of reaction-diffusion models on networks, the associated linear systems with network structures were considered.  ... 
doi:10.1016/j.physa.2016.03.055 fatcat:t5cssz2cjfcl7dpi4dvsflzr6e

Page 576 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
In the present work, which is part of a series, perturbation and numerical methods are used to analyse the bifurcation from a sim- ple eigenvalue in a one-dimensional, two-variable, cubic reaction- diffusion  ...  reaction-diffusion spectroscopy—a new method for the determination of chemical mechanisms, Presses Univ.  ... 

Key Features of Turing Systems are Determined Purely by Network Topology

Xavier Diego, Luciano Marcon, Patrick Müller, James Sharpe
2018 Physical Review X  
Essential properties of a Turing system, such as the conditions for the existence of patterns and the mechanisms of pattern selection are well understood in small networks.  ...  Here we provide a first general theory of Turing network topology, which proves why three key features of a Turing system are directly determined by the topology: the type of restrictions that apply to  ...  The reaction-diffusion graph of a reaction-diffusion system is a weighted digraph associated to the matrix F RD ðqÞ ¼ J R ðr 0 Þ − q 2 D, obtained from the linear approximation of the reaction-diffusion  ... 
doi:10.1103/physrevx.8.021071 fatcat:t7svrqwga5enxenyboj7c2xbxe

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems [article]

Andrew L. Krause, Eamonn A. Gaffney, Philip K. Maini, Václav Klika
2021 arXiv   pre-print
Here we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion  ...  In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion  ...  (b) Reaction-Diffusion Systems on Networks Turing's paper also analyzed the case of a ring of discrete cells coupled by diffusion, with reactions confined to the nodes.  ... 
arXiv:2106.08375v2 fatcat:os5dbssjafeydhpo4gkwclqogi

Hybrid reaction-diffusion and clock-and-wavefront model for the arrest of oscillations in the somitogenesis segmentation clock [article]

Jesús Pantoja-Hernández, Víctor F. Breña-Medina, Moisés Santillán
2021 arXiv   pre-print
Cotterell et al. called it a progressive oscillatory reaction-diffusion (PORD) model.  ...  In this work, we propose an equivalent Meinhardt-PORD model, and then amend it to couple it with a wavefront consisting of a receding morphogen gradient.  ...  JP-H thanks CONACYT-México for granting him a doctoral scholarship. VFBM thanks the financial support by Asociación Mexicana de Cultura AC.  ... 
arXiv:2009.02364v4 fatcat:uxuuyrn735gv5mbkpbpu7h2qoi

Pattern Formation of Competing Microorganisms in Sediments

Y. Schmitz, M. Baurmann, B. Engelen, U. Feudel
2007 Mathematical Modelling of Natural Phenomena  
Considering diffusion to be the main transport process, we obtain a reaction diffusion system (RDS) which we study in terms of spontaneous pattern formation.  ...  We present a three species model describing the degradation of substrate by two competing populations of microorganisms in a marine sediment.  ...  We thank Jan Holstein for beneficial and instructive discussions about the biogeochemical background of our model.  ... 
doi:10.1051/mmnp:2008023 fatcat:ukc5ogshejdsvlyambhbtpdsxi


Jesus Pantoja-Hernndez, Victor F. Brena-Medina, Moises Santillan
2020 bioRxiv   pre-print
Recently, Cotterell et al. (2015) reported a conceptually different mathematical model for somitogenesis. The authors called it a progressive oscillatory reaction-diffusion (PORD) model.  ...  In this work, we propose a modified version of the PORD model in order to overcome this and others inconveniences.  ...  In Appendix, A we analyze the stability of the PDE system when depending of β and d a parameters, and found that for a fixed value for d a = 5 × 10 −5 the system undergoes a bifurcation from a stable to  ... 
doi:10.1101/2020.09.04.283515 fatcat:lqhq5p5ksbhwjiftawaf5n7nne

Turing patterns in network-organized activator–inhibitor systems

Hiroya Nakao, Alexander S. Mikhailov
2010 Nature Physics  
Turing instability in activator-inhibitor systems provides a paradigm of nonequilibrium pattern formation; it has been extensively investigated for biological and chemical processes.  ...  Turing pattern formation should furthermore be possible in network-organized systems, such as cellular networks in morphogenesis and ecological metapopulations with dispersal connections between habitats  ...  Acknowledgements This work was supported by the Volkswagen Foundation, Germany, and by the MEXT, Japan (Global COE Program 'The Next Generation of Physics, Spun from Universality and Emergence' and Kakenhi  ... 
doi:10.1038/nphys1651 fatcat:demvqaapmnaxdgr5rnop7zlf4i

Page 4725 of Mathematical Reviews Vol. , Issue 98H [page]

1998 Mathematical Reviews  
R‘ (3671-3681); Milan Kuéera, Bifurcation of solutions to reaction-diffusion systems with condi- tions described by inequalities and inclusions (3683-3694); Julian Lopez-Gomez and Rosa Pardo [Rosa Pardo  ...  Mirzai, A type of fuzzy neural networks based on fuzzy arithmetic (21-27); Parviz Azimi, Isomorphic classification of X,, spaces (29-31); Zahra Afsharne- jad, Perturbation of Hamiltonian system (33-38)  ... 

Turing Patterns of Non-linear S-I Model on Random and Real-Structure Networks with Diarrhea Data

Prama Setia Putra, Hadi Susanto, Nuning Nuraini
2019 Scientific Reports  
Using a real structure network, the system also exhibits a comparable disease spread pattern of Diarrhea in Jakarta.  ...  To cover the lack of spatial sense in the method, one uses statistical modeling, reaction-diffusion in continuous medium, or multi-patch model to depict epidemic activities in several connected locations  ...  Following the report on pattern formation in chemical reactions 19 , Turing explained the emergence of pattern to be driven by the diffusion in the system that changed the stability of the equilibria  ... 
doi:10.1038/s41598-019-45069-3 pmid:31221999 pmcid:PMC6586790 fatcat:q3a7yznmtfdjvctha5yojrfuhy

Stability of nonlinear waves and patterns and related topics

Anna Ghazaryan, Stephane Lafortune, Vahagn Manukian
2018 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
One contribution of 14 to a theme issue 'Stability of nonlinear waves and patterns and related topics' .  ...  of kinetics of the reaction-diffusion system at the wave.  ...  Beyn & Otten [40] in this issue study stability properties of exponentially localized rotating waves that a class of reaction-diffusion systems support.  ... 
doi:10.1098/rsta.2018.0001 pmid:29507181 pmcid:PMC5869617 fatcat:jhumyml6yvbubdofn3dmnijkqu
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