Tohoku Mathematical Journal

Steady states of FitzHugh-Nagumo system with non-diffusive activator and diffusive inhibitor

Ying Li, Anna Marciniak-Czochra, Izumi Takagi, and Boying Wu

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In this paper, we consider a diffusion equation coupled to an ordinary differential equation with FitzHugh-Nagumo type nonlinearity. We construct continuous spatially heterogeneous steady states near, as well as far from, constant steady states and show that they are all unstable. In addition, we construct various types of steady states with jump discontinuities and prove that they are stable in a weak sense defined by Weinberger.The results are quite different from those for classical reaction-diffusion systems where all species diffuse.

Article information

Tohoku Math. J. (2), Volume 71, Number 2 (2019), 243-279.

First available in Project Euclid: 21 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B36: Pattern formation
Secondary: 35K57: Reaction-diffusion equations 35B35: Stability

FitzHugh-Nagumo model reaction-diffusion-ODE system pattern formation bifurcation analysis steady states global behaviour of solution branches instability


Li, Ying; Marciniak-Czochra, Anna; Takagi, Izumi; Wu, Boying. Steady states of FitzHugh-Nagumo system with non-diffusive activator and diffusive inhibitor. Tohoku Math. J. (2) 71 (2019), no. 2, 243--279. doi:10.2748/tmj/1561082598.

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  • M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Anal. 8 (1971), 321–340.
  • C. N. Chen, S. I. Ei, Y. P. Lin and S. Y. Kung, Standing waves joining with Turing patterns in FitzHugh-Nagumo type systems, Comm. Partial Differential Equations 36 (2011), 998–1015.
  • C. N. Chen, C. C. Chen and C. C. Huang, Traveling waves for the FitzHugh-Nagumo system on an infinite channel, J. Differential Equations 261 (2016), 3010–3041.
  • L. H. Chuan, T. Tsujikawa and A. Yagi, Stationary solutions to forest kinematic model, Glasgow Mathematical Journal 51(1) (2009-01), 1–17.
  • E. N. Dancer and S. Yan, Solutions with interior and boundary peaks for the Neumann problem of an elliptic system of FitzHugh-Nagumo type, Indiana Univ. Math. J. 55 (2006), 217–258.
  • R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics. 17 (1955), 257–278.
  • R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical 1 (1961), 445–466.
  • R. Gordon and L. Beloussov, From observations to paradigms; the importance of theories and models: An interview with Hans Meinhardt, International J. Developmental Biology 50 (2006), 103–111.
  • S. Härting, A. Marciniak-Czochra, Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Meth. Appl. Sci. 37 (2013), 1377–1391.
  • S. Härting, A. Marciniak-Czochra and I. Takagi, Stable patterns with jump discontinuity in systems with Turing instability and hysteresis, Discrete Contin. Dyn. Syst. 37 (2017), 757–800.
  • S. Hock, Y. Ng, J. Hasenauer, D. Wittmann, D. Lutter, D. Traembach, W. Wurst, N. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Syst. Biol. 7 (2013), 1–14.
  • V. Klika, R. Baker, D. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation, Bull. Math. Biol. 74 (2012), 935–957.
  • S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science 329 (2010), 1616–1620.
  • T. Kostova, R. Ravindran and M. Schonbek, Fitzhugh-Nagumo revisited: Types of bifurcations, periodical forcing and stability regions by a Lyapunov functional, Internati. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), 913–925.
  • Y. Li and A. Marciniak-Czochra, I. Takagi and B. Y. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J. 47 (2017), 217–247.
  • P. K. Maini, R. E. Baker and C. M. Chuong, The Turing Model Comes of Molecular Age, Science 314 (2006), 1397–1398.
  • A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of turing patterns in reaction-diffusion-ODE systems, J. Math. Biol. 74 (2017), 583–618.
  • A. Marciniak-Czochra, M. Nakayama and I. Takagi, Pattern formation in a diffusion-ode model with hysteresis, Differential Integral Equations 28 (2015), 655–694.
  • A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci. 17 (2007), 1693–1719.
  • M. Mimura, M. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal. 11 (1980), 613–631.
  • J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. Inst. Radio Engineers 50 (1962), 2061–2070.
  • Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal. 13 (1982), 555–593.
  • Y. Nishiura, Coexistence of infinitely many stable solutions to reaction-diffusion systems in the singlular limit, pp. 25–103, in Dynamics Reported: Expositions in Dynamical Systems, Vol. 3, C.R.K.T. Jones, U. Kirchgraber, H.O. Walther Eds., Springer, New York, 1994.
  • Y. Oshita, On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions, J. Differential Equations 188 (2003), 110–134.
  • K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H. M. Byrne, V. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: instability in a simple reaction-diffusion model for the migration/proliferation dichotomy, J. Biol. Dyn. 6 (2011), 54–71.
  • X. Ren and J. Wei, Nucleation in the FitzHugh-Nagumo system: Interface-spike solutions, J. Differential Equations 209 (2005), 266–301.
  • C. Rocsoreanu, A. Georgescu and N. Giurgiteanu, The FitzHugh-Nagumo model: Bifurcation and dynamics, Kluwer Academic Publishers, Boston, 2000.
  • J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second edition, Springer-Verlag, New York, 1994.
  • K. U. Torii, Two-dimensional spatial patterning in developmental systems, Trends in Cell Biology 22 (2012) 438–446.
  • A. M. Turing, The chemical basis of morphogenesis, Philo. Trans. Roy. Soc. London Ser. B 237 (1952), 37–72.
  • D. M. Umulis, M. Serpe, M. B. O'Connor and H. G. Othmer, Robust, bistable patterning of the dorsal surface of the Drosophila embryo, Proc.Nat. Acad. Sci. 103 (2006), 11613–11618.
  • H. F. Weinberger, A simple system with a continuum of stable inhomogeneous steady states, Nonlinear Partial Differential Equations in Applied Science; Proceedings of the U.S.-Japan Seminar (Tokyo, 1982), 345–359, North-Holland Math. Stad. 81, Lecture Notes Numer. Appl. Anal. 5, North-Holland, Amsterdam, 1983.