# American Institute of Mathematical Sciences

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. doi: 10.1126/science.256.5063.1539. [2] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85. doi: 10.1023/A:1012980128575. [3] S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems,, Physica D, 165 (2002), 176. doi: 10.1016/S0167-2789(02)00379-2. [4] E. Heisler, N. J. Suematsu, A. Awazu and H. Hiraku, Swarming of self-propelled camphor boats,, Physical Review E, 85 (2012). doi: 10.1103/PhysRevE.85.055201. [5] D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic,, Nature, 407 (2000), 487. [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. [7] M. Inaba, H. Yamanaka and S. Kondo, Pigment pattern formation by contact-dependent depolarization,, Science, 335 (2012). doi: 10.1126/science.1212821. [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. doi: 10.1051/mmnp/20094404. [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. doi: 10.1016/j.physd.2004.02.003. [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. doi: 10.1021/la970196p. [11] N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats,, Physical Review E, 81 (2010). doi: 10.1103/PhysRevE.81.056210. [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. doi: 10.1016/j.physa.2007.05.047. [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). doi: 10.1103/PhysRevE.86.036113.

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##### References:
 [1] M. K. Chaudhury and G. M. Whitesides, How to make water run uphill,, Science, 256 (1992), 1539. doi: 10.1126/science.256.5063.1539. [2] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85. doi: 10.1023/A:1012980128575. [3] S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems,, Physica D, 165 (2002), 176. doi: 10.1016/S0167-2789(02)00379-2. [4] E. Heisler, N. J. Suematsu, A. Awazu and H. Hiraku, Swarming of self-propelled camphor boats,, Physical Review E, 85 (2012). doi: 10.1103/PhysRevE.85.055201. [5] D. Helbing, I. Farkas and T. Vicsek, Simulating dynamical features of escape panic,, Nature, 407 (2000), 487. [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. [7] M. Inaba, H. Yamanaka and S. Kondo, Pigment pattern formation by contact-dependent depolarization,, Science, 335 (2012). doi: 10.1126/science.1212821. [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. doi: 10.1051/mmnp/20094404. [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. doi: 10.1016/j.physd.2004.02.003. [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. doi: 10.1021/la970196p. [11] N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats,, Physical Review E, 81 (2010). doi: 10.1103/PhysRevE.81.056210. [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. doi: 10.1016/j.physa.2007.05.047. [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). doi: 10.1103/PhysRevE.86.036113.
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