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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

Yasumasa Nishiura 1, , Takashi Teramoto 2, and  Xiaohui Yuan 3,

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.
Keywords: bifurcation, heterogeneous media, Reaction-diffusion equations, traveling spot..
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B3.
Citation: Yasumasa Nishiura, Takashi Teramoto, Xiaohui Yuan. Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 307-338. doi: 10.3934/cpaa.2012.11.307
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

T. Teramoto, Traveling spots through a line of heterogeneity,, unpublished., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[40]

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

[41]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

T. Teramoto, Traveling spots through a line of heterogeneity,, unpublished., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[40]

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

[41]

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

[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.  Google Scholar

[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.  Google Scholar

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