Logarithmic Lipschitz norms and diffusion-induced instability

This paper proves that contractive ordinary differential equation systems remain contractive when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems. An important biochemical system is shown to satisfy the required conditions.

[1]  G. Söderlind The logarithmic norm. History and modern theory , 2006 .

[2]  B N Kholodenko,et al.  Spatial gradients of cellular phospho‐proteins , 1999, FEBS letters.

[3]  B. Kholodenko Cell-signalling dynamics in time and space , 2006, Nature Reviews Molecular Cell Biology.

[4]  D.L. Elliott,et al.  Feedback systems: Input-output properties , 1976, Proceedings of the IEEE.

[5]  Eduardo Sontag,et al.  Modular cell biology: retroactivity and insulation , 2008, Molecular systems biology.

[6]  Hans G. Othmer,et al.  Synchronized and Differentiated Modes of Cellular Dynamics , 1980 .

[7]  L. E. Scriven,et al.  Interactions of reaction and diffusion in open systems , 1969 .

[8]  Pablo A Iglesias,et al.  Cells navigate with a local-excitation, global-inhibition-biased excitable network , 2010, Proceedings of the National Academy of Sciences.

[9]  Mihailo R. Jovanovic,et al.  A Passivity-Based Approach to Stability of Spatially Distributed Systems With a Cyclic Interconnection Structure , 2008, IEEE Transactions on Automatic Control.

[10]  Dulos,et al.  Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. , 1990, Physical review letters.

[11]  Chris Cosner,et al.  Book Review: Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems , 1996 .

[12]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[13]  I. Joó,et al.  Note on my paper “a simple proof for von Neumann's minimax theorem” , 1984 .

[14]  Guy Dewel,et al.  Turing Bifurcations and Pattern Selection , 1995 .

[15]  G. W. Cross Three types of matrix stability , 1978 .

[16]  Peter N. Brown,et al.  Decay to Uniform States in Ecological Interactions , 1980 .

[17]  P. Haccou Mathematical Models of Biology , 2022 .

[18]  Karsten Weis,et al.  Visualization of a Ran-GTP Gradient in Interphase and Mitotic Xenopus Egg Extracts , 2002, Science.

[19]  Murat Arcak,et al.  Certifying spatially uniform behavior in reaction-diffusion PDE and compartmental ODE systems , 2011, Autom..

[20]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[21]  D. Hoff,et al.  LARGE TIME BEHAVIOR OF SOLUTIONS OF SYSTEMS OF NONLINEAR REACTION-DIFFUSION EQUATIONS* , 1978 .

[22]  S. Basu,et al.  A synthetic multicellular system for programmed pattern formation , 2005, Nature.

[23]  H. Meinhardt,et al.  A theory of biological pattern formation , 1972, Kybernetik.

[24]  A. Gierer Generation of biological patterns and form: some physical, mathematical, and logical aspects. , 1981, Progress in biophysics and molecular biology.

[25]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[26]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[27]  Jean-Jacques E. Slotine,et al.  Contraction Analysis of Nonlinear Distributed Systems , 2004 .

[28]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[29]  J. L. Jackson,et al.  Dissipative structure: an explanation and an ecological example. , 1972, Journal of theoretical biology.

[30]  J. Morgan,et al.  Stability and Lyapunov functions for reaction-diffusion systems , 1997 .

[31]  Miles Miller,et al.  Modular Design of Artificial Tissue Homeostasis: Robust Control through Synthetic Cellular Heterogeneity , 2012, PLoS Comput. Biol..

[32]  F. Browder Nonlinear functional analysis , 1970 .

[33]  P. Maini,et al.  Turing instabilities in general systems , 2000, Journal of mathematical biology.

[34]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[35]  採編典藏組 Society for Industrial and Applied Mathematics(SIAM) , 2008 .

[36]  Liming Wang,et al.  A passivity-based stability criterion for reaction diffusion systems with interconnected structure , 2011 .

[37]  Gustaf Stiderlind Bounds on nonlinear operators in finite-dimensional banach spaces , 1986 .

[38]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[39]  Mario di Bernardo,et al.  Global Entrainment of Transcriptional Systems to Periodic Inputs , 2009, PLoS Comput. Biol..