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January  2012, 11(1): 307-338. doi: 10.3934/cpaa.2012.11.307

Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system

1. 

Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0813, Japan

2. 

Faculty of Photonics Science, Chitose Institute of Science and Technology, Chitose 066-8655, Japan

3. 

Northeast Institute of Geography and Agroecology, Chinese Academy of Sciences, Harbin 150081, China

Received  February 2010 Revised  January 2011 Published  September 2011

Spatially localized patterns form a representative class of patterns in dissipative systems. We study how the dynamics of traveling spots in two-dimensional space change when heterogeneities are introduced in the media. The simplest but fundamental one is a line heterogeneity of jump type. When spots encounter the jump, they display various outputs including penetration, rebound, and trapping depending on the incident angle and its height. The system loses translational symmetry by the heterogeneity, but at the same time, it causes the emergence of various types of heterogeneity-induced-ordered-patterns (HIOPs) replacing the homogeneous constant state. We study these issues by using a three-component reaction-diffusion system with one activator and two inhibitors. The above outputs can be obtained through the interaction between the HIOPs and the traveling spots. The global bifurcation and eigenvalue behavior of HISPs are the key to understand the underlying mechanisms for the transitions among those dynamics. A reduction to a finite dimensional system is presented here to extract the model-independent nature of the dynamics. Selected numerical techniques for the bifurcation analysis are also provided.
Citation: Yasumasa Nishiura, Takashi Teramoto, Xiaohui Yuan. Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system. Communications on Pure and Applied Analysis, 2012, 11 (1) : 307-338. doi: 10.3934/cpaa.2012.11.307
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M. Bär, M. Eiswirth, H.-H. Rotermund and G. Ertl, Solitary-wave phenomena in an excitable surface reaction, Phys. Rev. Lett., 69 (1992), 945-948. doi: 10.1103/PhysRevLett.69.945.

[2]

M. Bär, E. Meron and C. Utzny, Pattern formation on anisotropic and heterogeneous catalytic surfaces, Chaos, 12 (2002), 204-214. doi: 10.1063/1.1450565.

[3]

M. Bode, Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions, Physica D, 106 (1997), 270-286. doi: 10.1016/S0167-2789(97)00050-X.

[4]

M. Bode, A. W. Liehr, C. P. Schenk and H.-G.Purwins, Interaction of dissipative solitons: particle-like behaviour of localized structures in a threecomponent reaction-diffusion system, Physica D, 161 (2002), 45-66. doi: 10.1016/S0167-2789(01)00360-8.

[5]

P. C. Bressloff, S. E. Folias, A. Prat and Y.-X. Li, Oscillatory waves in inhomogeneous neural media, Phys. Rev. Lett., 91 (2003), 178101. doi: 10.1103/PhysRevLett.91.178101.

[6]

E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Yu, B. Sandstede and X. Wang, AUTO 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), Technical report, California Institute of Technology, 2001.

[7]

A. Doelman, P. van Heijster and T. Kaper, Pulse dynamics in a three-component system: existence analysys, J. Dyn. Diff. Equat., 21 (2009), 73-115. doi: 10.1007/s10884-008-9125-2.

[8]

S.-I. Ei, M. Mumura and Nagayama, Interacting spots in reaction diffusion systems, Discrete Contin. Dyn. Syst., 14 (2006), 31-62.

[9]

G. B. Ermentrout and J. Rinzel, Reflected waves in an inhomogeneous excitable medium, SIAM J. Appl. Math., 56 (1996), 1107-1128. doi: 10.1137/S0036139994276793.

[10]

M. Gutman, I. Aviram and A. Rabinovitch, Pseudoreflection from interface between two oscillatory media: Extended driver, Phys. Rev. E, 69 (2004), 016211. doi: 10.1103/PhysRevE.69.016211.

[11]

A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Controlling domain patterns far from equilibrium, Phy. Rev. Lett., 76 (1996), 427-430. doi: 10.1103/PhysRevLett.76.427.

[12]

A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Order parameter equations for front bifurcations: planar and circular fronts, Phy. Rev. E, 55 (1997), 4450-4457. doi: 10.1103/PhysRevE.55.4450.

[13]

Y. Hayase and T Ohta, Self-replicating pulses and sierpinski gaskets in excitable media, Phy. Rev. E, 62 (2000), 5998-6003. doi: 10.1103/PhysRevE.62.5998.

[14]

P. van Heijster and B. Sandstede, Planar radial spots in a three-component FitzHugh-Nagumo system, J. Nonlinear Sci., 21 (2011), in press.

[15]

H. Ikeda and M. Mimura, Wave-blocking phenomena in bistable reaction-diffusion systems, SIAM J. Appl. Math., 49 (1989) 515-538 doi: 10.1103/PhysRevE.62.5998.

[16]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, 1998.

[17]

R. B. Lehoucq, D. C. Sorensen and C. Yang, "ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods," SIAM, 1998.

[18]

T. J. Lewis and J. P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316. doi: 10.1137/S0036139998349298.

[19]

H. Meinhardt, "The Algorithmmic Beauty of Sea Shells," PSpringer-Verlag, Heidelberg, 1995.

[20]

J. Miyazaki and S. Kinoshita, Stopping and initiation of a chemical pulse at the interface of excitable media with different diffusivity, Phys. Rev. E, 76 (2007), 066201. doi: 10.1103/PhysRevE.76.066201.

[21]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Dynamic transitions through scattors in dissipative systems, Chaos, 13 (2003), 962-972. doi: 10.1063/1.1592131.

[22]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering and separators in dissipative systems, Phys. Rev. E, 67 (2003), 056210. doi: 10.1103/PhysRevE.67.056210.

[23]

Y. Nishiura, T. Teramoto, X. Yuan and K.-I. Ueda, Dynamics of traveling pulses in heterogenous media, Chaos, 17 (2007), 037104. doi: 10.1063/1.2778553.

[24]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15 (2005), 047509. doi: 10.1063/1.2087127.

[25]

J. P. Pauwelussen, Nerve impulse propagation in a branching nerve system: a simple model, Physica D, 4 (1981), 67-88. doi: 10.1016/0167-2789(81)90005-1.

[26]

A. M. Pertsov, E. A. Ermakova and E. E. Shnol, On the diffraction of autowaves, Physica D, 44 (1990), 178-190. doi: 10.1016/0167-2789(90)90054-S.

[27]

A. Prat and Y.-X. Li and P. Bressloff, Inhomogeneity-induced bifurcation of stationary and oscillatory pulses, Physica D, 202 (2005), 177-199. doi: 10.1016/j.physd.2005.02.005.

[28]

H.-G. Purwins, H. U. Bödeker and Sh. Amiranashvili, Dissipative solitons, Advances in Physics, 59 (2010), 485-701. doi: 10.1080/00018732.2010.498228.

[29]

A. G. Salinger, N. M. Bou-Rabee, E. A. Burroughs, R. B. Lehoucq, R. P. Pawlowski, L. A. Romero and E. D. Wilkes, "LOCA 1.1: Library of Continuation Algorithms, Theory and Implementation Manual," Sandia National Laboratories, Albuquerque, USA, 2002. doi: 10.2172/800778.

[30]

C. P. Schenk, M. Or-Guil, M. Bode and H.-G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Phys. Rev. Lett., 78 (1997), 3781-3784. doi: 10.1103/PhysRevLett.78.3781.

[31]

P. Schütz, M. Bode and H. G. Purwins, Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities, Physica D, 82 (1995), 270-286.

[32]

R. Seydel, "Practical Bifurcation and Stability Analysis," Springer-Verlag, 1994.

[33]

A. J. Steele, M. Tinsley and K. Showalter, Collective behavior of stabilized reaction-diffusion waves, Chaos, 18 (2008), 026108. doi: 10.1063/1.2900386.

[34]

T. Teramoto, K. Suzuki and Y. Nishiura, Rotational motion of traveling spots in dissipative systems, Phys. Rev. E, 80 (2009), 046208. doi: 10.1103/PhysRevE.80.046208.

[35]

T. Teramoto, X. Yuan, M. Bär and Y. Nishiura, Onset of inidirectional pulse propagation in an excitable medium with asymmetric heterogeneity, Phys. Rev. E, 79 (2009), 046205. doi: 10.1103/PhysRevE.79.046205.

[36]

T. Teramoto, Traveling spots through a line of heterogeneity, unpublished.

[37]

T. Teramoto, K.-I. Ueda and Y. Nishiura, Phase-dependent output of scattering process for traveling breathers, Phys. Rev. E, 69 (2004), 056224. doi: 10.1103/PhysRevE.69.056224.

[38]

M. R. Tinsley, A. J. Steele and K. Showalter, Collective behavior of particle-like chemical waves, Eur. Phys. J. Special Topics, 165 (2008), 161-167. doi: 10.1140/epjst/e2008-00859-7.

[39]

R. S. Tuminaro, M. Heroux, S. A. Hutchinson and J. N. Shadid, "Official Aztec User's Guide: Version 2.1," Technical Report, SAND99-8801J, December, 1999.

[40]

V. K. Vanag and I. R. Epstein, Localized patterns in reaction-diffusion systems, Chaos, 17 (2007), 037110. doi: 10.1063/1.2752494.

[41]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

[42]

X. Yuan, T. Teramoto and Y. Nishiura, Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system, Phys. Rev. E, 75 (2007), 036220. doi: 10.1103/PhysRevE.75.036220.

[43]

A. M. Zhabotinsky, M. D. Eager and I. R. Epstein, Refraction and reflection of chemical waves, Phys. Rev. Lett., 71 (1993), 1526-1529. doi: 10.1103/PhysRevLett.71.1526.

show all references

References:
[1]

M. Bär, M. Eiswirth, H.-H. Rotermund and G. Ertl, Solitary-wave phenomena in an excitable surface reaction, Phys. Rev. Lett., 69 (1992), 945-948. doi: 10.1103/PhysRevLett.69.945.

[2]

M. Bär, E. Meron and C. Utzny, Pattern formation on anisotropic and heterogeneous catalytic surfaces, Chaos, 12 (2002), 204-214. doi: 10.1063/1.1450565.

[3]

M. Bode, Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions, Physica D, 106 (1997), 270-286. doi: 10.1016/S0167-2789(97)00050-X.

[4]

M. Bode, A. W. Liehr, C. P. Schenk and H.-G.Purwins, Interaction of dissipative solitons: particle-like behaviour of localized structures in a threecomponent reaction-diffusion system, Physica D, 161 (2002), 45-66. doi: 10.1016/S0167-2789(01)00360-8.

[5]

P. C. Bressloff, S. E. Folias, A. Prat and Y.-X. Li, Oscillatory waves in inhomogeneous neural media, Phys. Rev. Lett., 91 (2003), 178101. doi: 10.1103/PhysRevLett.91.178101.

[6]

E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Yu, B. Sandstede and X. Wang, AUTO 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), Technical report, California Institute of Technology, 2001.

[7]

A. Doelman, P. van Heijster and T. Kaper, Pulse dynamics in a three-component system: existence analysys, J. Dyn. Diff. Equat., 21 (2009), 73-115. doi: 10.1007/s10884-008-9125-2.

[8]

S.-I. Ei, M. Mumura and Nagayama, Interacting spots in reaction diffusion systems, Discrete Contin. Dyn. Syst., 14 (2006), 31-62.

[9]

G. B. Ermentrout and J. Rinzel, Reflected waves in an inhomogeneous excitable medium, SIAM J. Appl. Math., 56 (1996), 1107-1128. doi: 10.1137/S0036139994276793.

[10]

M. Gutman, I. Aviram and A. Rabinovitch, Pseudoreflection from interface between two oscillatory media: Extended driver, Phys. Rev. E, 69 (2004), 016211. doi: 10.1103/PhysRevE.69.016211.

[11]

A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Controlling domain patterns far from equilibrium, Phy. Rev. Lett., 76 (1996), 427-430. doi: 10.1103/PhysRevLett.76.427.

[12]

A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Order parameter equations for front bifurcations: planar and circular fronts, Phy. Rev. E, 55 (1997), 4450-4457. doi: 10.1103/PhysRevE.55.4450.

[13]

Y. Hayase and T Ohta, Self-replicating pulses and sierpinski gaskets in excitable media, Phy. Rev. E, 62 (2000), 5998-6003. doi: 10.1103/PhysRevE.62.5998.

[14]

P. van Heijster and B. Sandstede, Planar radial spots in a three-component FitzHugh-Nagumo system, J. Nonlinear Sci., 21 (2011), in press.

[15]

H. Ikeda and M. Mimura, Wave-blocking phenomena in bistable reaction-diffusion systems, SIAM J. Appl. Math., 49 (1989) 515-538 doi: 10.1103/PhysRevE.62.5998.

[16]

J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, 1998.

[17]

R. B. Lehoucq, D. C. Sorensen and C. Yang, "ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods," SIAM, 1998.

[18]

T. J. Lewis and J. P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316. doi: 10.1137/S0036139998349298.

[19]

H. Meinhardt, "The Algorithmmic Beauty of Sea Shells," PSpringer-Verlag, Heidelberg, 1995.

[20]

J. Miyazaki and S. Kinoshita, Stopping and initiation of a chemical pulse at the interface of excitable media with different diffusivity, Phys. Rev. E, 76 (2007), 066201. doi: 10.1103/PhysRevE.76.066201.

[21]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Dynamic transitions through scattors in dissipative systems, Chaos, 13 (2003), 962-972. doi: 10.1063/1.1592131.

[22]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering and separators in dissipative systems, Phys. Rev. E, 67 (2003), 056210. doi: 10.1103/PhysRevE.67.056210.

[23]

Y. Nishiura, T. Teramoto, X. Yuan and K.-I. Ueda, Dynamics of traveling pulses in heterogenous media, Chaos, 17 (2007), 037104. doi: 10.1063/1.2778553.

[24]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15 (2005), 047509. doi: 10.1063/1.2087127.

[25]

J. P. Pauwelussen, Nerve impulse propagation in a branching nerve system: a simple model, Physica D, 4 (1981), 67-88. doi: 10.1016/0167-2789(81)90005-1.

[26]

A. M. Pertsov, E. A. Ermakova and E. E. Shnol, On the diffraction of autowaves, Physica D, 44 (1990), 178-190. doi: 10.1016/0167-2789(90)90054-S.

[27]

A. Prat and Y.-X. Li and P. Bressloff, Inhomogeneity-induced bifurcation of stationary and oscillatory pulses, Physica D, 202 (2005), 177-199. doi: 10.1016/j.physd.2005.02.005.

[28]

H.-G. Purwins, H. U. Bödeker and Sh. Amiranashvili, Dissipative solitons, Advances in Physics, 59 (2010), 485-701. doi: 10.1080/00018732.2010.498228.

[29]

A. G. Salinger, N. M. Bou-Rabee, E. A. Burroughs, R. B. Lehoucq, R. P. Pawlowski, L. A. Romero and E. D. Wilkes, "LOCA 1.1: Library of Continuation Algorithms, Theory and Implementation Manual," Sandia National Laboratories, Albuquerque, USA, 2002. doi: 10.2172/800778.

[30]

C. P. Schenk, M. Or-Guil, M. Bode and H.-G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Phys. Rev. Lett., 78 (1997), 3781-3784. doi: 10.1103/PhysRevLett.78.3781.

[31]

P. Schütz, M. Bode and H. G. Purwins, Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities, Physica D, 82 (1995), 270-286.

[32]

R. Seydel, "Practical Bifurcation and Stability Analysis," Springer-Verlag, 1994.

[33]

A. J. Steele, M. Tinsley and K. Showalter, Collective behavior of stabilized reaction-diffusion waves, Chaos, 18 (2008), 026108. doi: 10.1063/1.2900386.

[34]

T. Teramoto, K. Suzuki and Y. Nishiura, Rotational motion of traveling spots in dissipative systems, Phys. Rev. E, 80 (2009), 046208. doi: 10.1103/PhysRevE.80.046208.

[35]

T. Teramoto, X. Yuan, M. Bär and Y. Nishiura, Onset of inidirectional pulse propagation in an excitable medium with asymmetric heterogeneity, Phys. Rev. E, 79 (2009), 046205. doi: 10.1103/PhysRevE.79.046205.

[36]

T. Teramoto, Traveling spots through a line of heterogeneity, unpublished.

[37]

T. Teramoto, K.-I. Ueda and Y. Nishiura, Phase-dependent output of scattering process for traveling breathers, Phys. Rev. E, 69 (2004), 056224. doi: 10.1103/PhysRevE.69.056224.

[38]

M. R. Tinsley, A. J. Steele and K. Showalter, Collective behavior of particle-like chemical waves, Eur. Phys. J. Special Topics, 165 (2008), 161-167. doi: 10.1140/epjst/e2008-00859-7.

[39]

R. S. Tuminaro, M. Heroux, S. A. Hutchinson and J. N. Shadid, "Official Aztec User's Guide: Version 2.1," Technical Report, SAND99-8801J, December, 1999.

[40]

V. K. Vanag and I. R. Epstein, Localized patterns in reaction-diffusion systems, Chaos, 17 (2007), 037110. doi: 10.1063/1.2752494.

[41]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

[42]

X. Yuan, T. Teramoto and Y. Nishiura, Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system, Phys. Rev. E, 75 (2007), 036220. doi: 10.1103/PhysRevE.75.036220.

[43]

A. M. Zhabotinsky, M. D. Eager and I. R. Epstein, Refraction and reflection of chemical waves, Phys. Rev. Lett., 71 (1993), 1526-1529. doi: 10.1103/PhysRevLett.71.1526.

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