Abstract
We present conditions that guarantee spatial uniformity of the solutions of reaction-diffusion partial differential equations. These equations are of central importance to several diverse application fields concerned with pattern formation. The conditions make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators on the given spatial domain. We present analogous conditions that apply to the solutions of diffusively-coupled networks of ordinary differential equations. We derive numerical tests making use of linear matrix inequalities that are useful in certifying these conditions. We discuss examples relevant to enzymatic cell signaling and biological oscillators. From a systems biology perspective, the paper’s main contributions are unified verifiable relaxed conditions that guarantee spatial uniformity of biological processes.
The authors Zahra Aminzare and Yusef Shafi contributed equally.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aminzare Z, Sontag ED (2012) Logarithmic Lipschitz norms and diffusion-induced instability. arXiv:12080326v2
Arcak M, Sontag E (2006) Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica 42(9):1531–1537
Arcak M (2011) Certifying spatially uniform behavior in reaction-diffusion pde and compartmental ode systems. Automatica 47(6):1219–1229
Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory, vol 15. Society for Industrial Mathematics
Cantrell RS, Cosner C (2003) Spatial ecology via reaction-diffusion equations. Wiley series in mathematical and computational biology
Elowitz M, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403(6767):335–338
Ge X, Arcak M, Salama K (2010) Nonlinear analysis of ring oscillator and cross-coupled oscillator circuits. Dyn Continuous Discrete Impulsive Syst Ser B: Appl Algorithms 17(6): 959–977
Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1): 30–39
Gierer A (1981) Generation of biological patterns and form: some physical, mathematical, and logical aspects. Prog Biophys Mol Biol 37(1):1–47
Godsil C, Royle G, Godsil C (2001) Algebraic graph theory, vol 8. Springer, New York
Hale J (1997) Diffusive coupling, dissipation, and synchronization. J Dyn Differ Equ 9(1):1–52
Hartman P (1961) On stability in the large for systems of ordinary differential equations. Can J Math 13:480–492
Henrot A (2006) Extremum problems for eigenvalues of elliptic operators. Birkhauser
Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge
Lewis DC (1949) Metric properties of differential equations. Am J Math 71:294–312
Lohmiller W, Slotine JJE (1998) On contraction analysis for non-linear systems. Automatica 34:683–696
Lozinskii SM (1959) Error estimate for numerical integration of ordinary differential equations. I Izv Vtssh Uchebn Zaved Mat 5:222–222
Pavlov A, Pogromvsky A, van de Wouv N, Nijmeijer H (2004) Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst Control Lett 52:257–261
Russo G, di Bernardo EDSM (2010) Global entrainment of transcriptional systems to periodic inputs. PLoS Comput Biol 6(4)
Scardovi L, Arcak M, Sontag E (2010) Synchronization of interconnected systems with applications to biochemical networks: an input-output approach. IEEE Trans Autom Control 55(6):1367–1379
Schöll E (2001) Nonlinear spatio-temporal dynamics and Chaos in Semiconductors. Cambridge University Press, Cambridge
Smith H (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. American Mathematical Society
Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond Ser B, Biol Sci 237(641):37–72
Yang XS (2003) Turing pattern formation of catalytic reaction-diffusion systems in engineering applications. Model Simul Mater Sci Eng 11(3):321
Acknowledgments
Work supported in part by grants NIH 1R01GM086881 and 1R01GM100473, AFOSR FA9550-11-1-0247 and FA9550-11-1-0244, and NSF ECCS-1101876.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Aminzare, Z., Shafi, Y., Arcak, M., Sontag, E.D. (2014). Guaranteeing Spatial Uniformity in Reaction-Diffusion Systems Using Weighted \(L^2\) Norm Contractions. In: Kulkarni, V., Stan, GB., Raman, K. (eds) A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9041-3_3
Download citation
DOI: https://doi.org/10.1007/978-94-017-9041-3_3
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-017-9040-6
Online ISBN: 978-94-017-9041-3
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)