# American Institute of Mathematical Sciences

May  2014, 19(3): 697-714. doi: 10.3934/dcdsb.2014.19.697

## Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems

 1 Department of Mathematics, Tamkang University, No. 151, Yingzhuan Rd., Tamsui Dist., New Taipei City 25137, Taiwan 2 Division of System Engineering for Mathematics, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran 050-8585, Japan 3 School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan

Received  September 2013 Revised  December 2013 Published  February 2014

In this study, we consider the traveling spots that were observed in the photosensitive Belousov-Zhabotinsky reaction experiment conducted by Mihailuk et al. in 2001. First, we introduce the interface equation by the singular limit analysis of a FitzHugh--Nagumo-type reaction-diffusion system. Then, we obtain the profile of the support of the solution. Next, we prove the uniqueness of the traveling spot by studying ordinary differential equations that describe its front and back. In addition, we provide an upper bound for the width of the spot. Furthermore, we compare the singular limit problem with the wave front interaction model proposed by Zykov and Showalter in 2005 and obtain traveling fingers.
Citation: Yan-Yu Chen, Yoshihito Kohsaka, Hirokazu Ninomiya. Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 697-714. doi: 10.3934/dcdsb.2014.19.697
##### References:
 [1] E. Ammelt, Y. A. Astrov and H.-G. Purwins, Hexagon structures in a two-dimensional dc-driven gas discharge system, Physical Review E, 58 (1988), 7109-7117. doi: 10.1103/PhysRevE.58.7109. [2] C. D. Bain, G. D. Burnett-Hall and R. R. Montgomerie, Rapid motion of liquid drops, Nature, 372 (1994), 414-415. doi: 10.1038/372414a0. [3] Y.-Y. Chen, J.-S. Guo and H. Ninomiya, Existence and uniqueness of rigidly rotating spiral waves by a wave front interaction model, Physica D: Nonlinear Phenomena, 241 (2012), 1758-1766. doi: 10.1016/j.physd.2012.08.004. [4] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Diff. Geom., 33 (1991), 635-681. [5] J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagating wave segments in wave front interaction model, Physica D: Nonlinear Phenomena, 239 (2010), 230-239. doi: 10.1016/j.physd.2009.11.001. [6] J.-S. Guo, H. Ninomiya and C.-C. Wu, Existence of a rotating wave pattern in a disk for a wave front interaction model, Comm. Pure Applied Anal., 12 (2013), 1049-1063. doi: 10.3934/cpaa.2013.12.1049. [7] P. Hartman, Ordinary Differential Equations, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222. [8] A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: The effects of front bifurcations, Nonlinearity, 7 (1994), 805-835. doi: 10.1088/0951-7715/7/3/006. [9] K. Krischer and A. Mikhailov, Bifurcation to traveling spots in reaction-diffusion systems, Physical Review Letters, 73 (1994), 3165-3168. doi: 10.1103/PhysRevLett.73.3165. [10] W. F. Loomis, The Development of Dioctyostelium Discoideum, Academic Press, New York, 1982. [11] E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Experimental and theoretical studies of feedback stabilization of propagating wave segments, Faraday Discuss, 120 (2001), 383-394. doi: 10.1039/b103431f. [12] E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Feedback stabilization of unstable propagating waves, Phys. Review E., 65 (2002), 065602. doi: 10.1103/PhysRevE.65.065602. [13] M. Or-Guil, M. Bode, C. P. Schenk and H.-G. Purwins, Spot bifurcations in three-component reaction-diffusion systems: The onset of propagation, Phys. Review E., 57 (1998), 6432-6437. doi: 10.1103/PhysRevE.57.6432. [14] T. Ohta, M. Mimura and R. Kobayashi, Higher-dimensional localized patterns in excitable media, Physica D, 34 (1989), 115-144. [15] L. M. Pismen, Nonlocal boundary dynamics of traveling spots in a reaction-diffusion system, Physical Review Letters, 86 (2001), 548-551. doi: 10.1103/PhysRevLett.86.548. [16] J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation, Biophysical Journal, 13 (1973), 1313-1337. [17] C. P. Schenk, M. Or-Guil, M. Bode and H.-G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Physical Review Letters, 78 (1997), 3781-3784. doi: 10.1103/PhysRevLett.78.3781. [18] H. Willebrand, T. Hünteler, F.-J. Niedernostheide, R. Dohmen and H.-G. Purwins, Periodic and turbulent behavior of solitary structures in distributed active media, Phys. Rev. A, 45 (1992), 8766-8775. doi: 10.1103/PhysRevA.45.8766. [19] V. S. Zykov and K. Showalter, Wave front interaction model of stabilized propagating wave segments, Phys. Review Letters, 94 (2005), 068302. doi: 10.1103/PhysRevLett.94.068302.

show all references

##### References:
 [1] E. Ammelt, Y. A. Astrov and H.-G. Purwins, Hexagon structures in a two-dimensional dc-driven gas discharge system, Physical Review E, 58 (1988), 7109-7117. doi: 10.1103/PhysRevE.58.7109. [2] C. D. Bain, G. D. Burnett-Hall and R. R. Montgomerie, Rapid motion of liquid drops, Nature, 372 (1994), 414-415. doi: 10.1038/372414a0. [3] Y.-Y. Chen, J.-S. Guo and H. Ninomiya, Existence and uniqueness of rigidly rotating spiral waves by a wave front interaction model, Physica D: Nonlinear Phenomena, 241 (2012), 1758-1766. doi: 10.1016/j.physd.2012.08.004. [4] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Diff. Geom., 33 (1991), 635-681. [5] J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagating wave segments in wave front interaction model, Physica D: Nonlinear Phenomena, 239 (2010), 230-239. doi: 10.1016/j.physd.2009.11.001. [6] J.-S. Guo, H. Ninomiya and C.-C. Wu, Existence of a rotating wave pattern in a disk for a wave front interaction model, Comm. Pure Applied Anal., 12 (2013), 1049-1063. doi: 10.3934/cpaa.2013.12.1049. [7] P. Hartman, Ordinary Differential Equations, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898719222. [8] A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: The effects of front bifurcations, Nonlinearity, 7 (1994), 805-835. doi: 10.1088/0951-7715/7/3/006. [9] K. Krischer and A. Mikhailov, Bifurcation to traveling spots in reaction-diffusion systems, Physical Review Letters, 73 (1994), 3165-3168. doi: 10.1103/PhysRevLett.73.3165. [10] W. F. Loomis, The Development of Dioctyostelium Discoideum, Academic Press, New York, 1982. [11] E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Experimental and theoretical studies of feedback stabilization of propagating wave segments, Faraday Discuss, 120 (2001), 383-394. doi: 10.1039/b103431f. [12] E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Feedback stabilization of unstable propagating waves, Phys. Review E., 65 (2002), 065602. doi: 10.1103/PhysRevE.65.065602. [13] M. Or-Guil, M. Bode, C. P. Schenk and H.-G. Purwins, Spot bifurcations in three-component reaction-diffusion systems: The onset of propagation, Phys. Review E., 57 (1998), 6432-6437. doi: 10.1103/PhysRevE.57.6432. [14] T. Ohta, M. Mimura and R. Kobayashi, Higher-dimensional localized patterns in excitable media, Physica D, 34 (1989), 115-144. [15] L. M. Pismen, Nonlocal boundary dynamics of traveling spots in a reaction-diffusion system, Physical Review Letters, 86 (2001), 548-551. doi: 10.1103/PhysRevLett.86.548. [16] J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation, Biophysical Journal, 13 (1973), 1313-1337. [17] C. P. Schenk, M. Or-Guil, M. Bode and H.-G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Physical Review Letters, 78 (1997), 3781-3784. doi: 10.1103/PhysRevLett.78.3781. [18] H. Willebrand, T. Hünteler, F.-J. Niedernostheide, R. Dohmen and H.-G. Purwins, Periodic and turbulent behavior of solitary structures in distributed active media, Phys. Rev. A, 45 (1992), 8766-8775. doi: 10.1103/PhysRevA.45.8766. [19] V. S. Zykov and K. Showalter, Wave front interaction model of stabilized propagating wave segments, Phys. Review Letters, 94 (2005), 068302. doi: 10.1103/PhysRevLett.94.068302.
 [1] John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851 [2] Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 [3] Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639 [4] Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150 [5] Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic and Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052 [6] Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216 [7] Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203 [8] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [9] Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 [10] Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101 [11] Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251 [12] Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441 [13] Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205 [14] Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118 [15] Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations and Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028 [16] Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172 [17] Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072 [18] Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations and Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027 [19] Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 781-795. doi: 10.3934/dcdss.2020044 [20] Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397

2021 Impact Factor: 1.497