October  2015, 8(5): 847-856. doi: 10.3934/dcdss.2015.8.847

Reduced model from a reaction-diffusion system of collective motion of camphor boats

1. 

Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, 060-0810

2. 

Department of Mathematical Sciences Based on Modeling and Analysis, Meiji University, Nakano-ku, Tokyo, 164-8525

3. 

Research Institute for Electronic Science, Hokkaido University / JST CREST, Sapporo, 060-0812, Japan

4. 

Faculty of Engineering, Musashino University / JST CREST, Koto-ku, Tokyo, 135-8181, Japan

Received  December 2013 Revised  July 2014 Published  July 2015

Various motions of camphor boats in the water channel exhibit both a homogeneous and an inhomogeneous state, depending on the number of boats, when unidirectional motion along an annular water channel can be observed even with only one camphor boat. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Since the experimental results described above are thought of as a kind of bifurcation phenomena, we would like to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.
Citation: Shin-Ichiro Ei, Kota Ikeda, Masaharu Nagayama, Akiyasu Tomoeda. Reduced model from a reaction-diffusion system of collective motion of camphor boats. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 847-856. doi: 10.3934/dcdss.2015.8.847
References:
[1]

M. K. Chaudhury and G. M. Whitesides, How to make water run uphill, Science, 256 (1992), 1539-1541. doi: 10.1126/science.256.5063.1539.  Google Scholar

[2]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137. doi: 10.1023/A:1012980128575.  Google Scholar

[3]

S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems, Physica D, 165 (2002), 176-198. doi: 10.1016/S0167-2789(02)00379-2.  Google Scholar

[4]

E. Heisler, N. J. Suematsu, A. Awazu and H. Hiraku, Swarming of self-propelled camphor boats, Physical Review E, 85 (2012), 055201. doi: 10.1103/PhysRevE.85.055201.  Google Scholar

[5]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. Google Scholar

[6]

S.-I. Ei, K. Ikeda, M. Nagayama and A. Tomoeda, Application of a center manifold theory to a reaction-diffusion system of collective motion of camphor disks and boats, Math. Bohem., 139 (2014), 363-371. Google Scholar

[7]

M. Inaba, H. Yamanaka and S. Kondo, Pigment pattern formation by contact-dependent depolarization, Science, 335 (2012), 677. doi: 10.1126/science.1212821.  Google Scholar

[8]

T. Miura and R. Tanaka, In Vitro vasculogenesis models revisited-measurement of VEGF diffusion in matrigel, Mathematical Modelling of Natural Phenomena, 4 (2009), 118-130. doi: 10.1051/mmnp/20094404.  Google Scholar

[9]

M. Nagayama, S. Nakata, Y. Doi and Y. Hayashima, A theoretical and experimental study on the unidirectional motion of a camphor disk, Physica D: Nonlinear Phenomena, 194 (2004), 151-165. doi: 10.1016/j.physd.2004.02.003.  Google Scholar

[10]

S. Nakata, Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii and K. Yoshikawa, Self-rotation of a camphor scraping on water: New insight into the old problem, Langmuir, 13 (1997), 4454-4458. doi: 10.1021/la970196p.  Google Scholar

[11]

N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats, Physical Review E, 81 (2010), 056210. doi: 10.1103/PhysRevE.81.056210.  Google Scholar

[12]

A. Tomoeda, K. Nishinari. D. Chowdhury and A. Schadschneider, An information-based traffic control in a public conveyance system: Reduced clustering and enhanced efficiency, Physica A: Statistical Mechanics and its Applications, 384 (2007), 600-612. doi: 10.1016/j.physa.2007.05.047.  Google Scholar

[13]

A. Tomoeda, D. Yanagisawa, T. Imamura and K. Nishinari, Propagation speed of a starting wave in a queue of pedestrians, Physical Review E, 86 (2012), 036113. doi: 10.1103/PhysRevE.86.036113.  Google Scholar

show all references

References:
[1]

M. K. Chaudhury and G. M. Whitesides, How to make water run uphill, Science, 256 (1992), 1539-1541. doi: 10.1126/science.256.5063.1539.  Google Scholar

[2]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137. doi: 10.1023/A:1012980128575.  Google Scholar

[3]

S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems, Physica D, 165 (2002), 176-198. doi: 10.1016/S0167-2789(02)00379-2.  Google Scholar

[4]

E. Heisler, N. J. Suematsu, A. Awazu and H. Hiraku, Swarming of self-propelled camphor boats, Physical Review E, 85 (2012), 055201. doi: 10.1103/PhysRevE.85.055201.  Google Scholar

[5]

D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic, Nature, 407 (2000), 487-490. Google Scholar

[6]

S.-I. Ei, K. Ikeda, M. Nagayama and A. Tomoeda, Application of a center manifold theory to a reaction-diffusion system of collective motion of camphor disks and boats, Math. Bohem., 139 (2014), 363-371. Google Scholar

[7]

M. Inaba, H. Yamanaka and S. Kondo, Pigment pattern formation by contact-dependent depolarization, Science, 335 (2012), 677. doi: 10.1126/science.1212821.  Google Scholar

[8]

T. Miura and R. Tanaka, In Vitro vasculogenesis models revisited-measurement of VEGF diffusion in matrigel, Mathematical Modelling of Natural Phenomena, 4 (2009), 118-130. doi: 10.1051/mmnp/20094404.  Google Scholar

[9]

M. Nagayama, S. Nakata, Y. Doi and Y. Hayashima, A theoretical and experimental study on the unidirectional motion of a camphor disk, Physica D: Nonlinear Phenomena, 194 (2004), 151-165. doi: 10.1016/j.physd.2004.02.003.  Google Scholar

[10]

S. Nakata, Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii and K. Yoshikawa, Self-rotation of a camphor scraping on water: New insight into the old problem, Langmuir, 13 (1997), 4454-4458. doi: 10.1021/la970196p.  Google Scholar

[11]

N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats, Physical Review E, 81 (2010), 056210. doi: 10.1103/PhysRevE.81.056210.  Google Scholar

[12]

A. Tomoeda, K. Nishinari. D. Chowdhury and A. Schadschneider, An information-based traffic control in a public conveyance system: Reduced clustering and enhanced efficiency, Physica A: Statistical Mechanics and its Applications, 384 (2007), 600-612. doi: 10.1016/j.physa.2007.05.047.  Google Scholar

[13]

A. Tomoeda, D. Yanagisawa, T. Imamura and K. Nishinari, Propagation speed of a starting wave in a queue of pedestrians, Physical Review E, 86 (2012), 036113. doi: 10.1103/PhysRevE.86.036113.  Google Scholar

[1]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249

[2]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839

[3]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213

[4]

Josselin Garnier, George Papanicolaou, Tzu-Wei Yang. Mean field model for collective motion bistability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 851-879. doi: 10.3934/dcdsb.2018210

[5]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129

[6]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873

[7]

Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161

[8]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[9]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159

[10]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

[11]

Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature. Mathematical Biosciences & Engineering, 2017, 14 (2) : 437-453. doi: 10.3934/mbe.2017027

[12]

Ezio Di Costanzo, Marta Menci, Eleonora Messina, Roberto Natalini, Antonia Vecchio. A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 443-472. doi: 10.3934/dcdsb.2019189

[13]

A. Carati. Center manifold of unstable periodic orbits of helium atom: numerical evidence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 97-104. doi: 10.3934/dcdsb.2003.3.97

[14]

Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075

[15]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, 2021, 29 (3) : 2325-2358. doi: 10.3934/era.2020118

[16]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157

[17]

Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173

[18]

Jingli Ren, Gail S. K. Wolkowicz. Preface: Recent advances in bifurcation theory and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : i-ii. doi: 10.3934/dcdss.2020417

[19]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387

[20]

Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (142)
  • HTML views (0)
  • Cited by (1)

[Back to Top]