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2015, 2015(special): 861-877. doi: 10.3934/proc.2015.0861

Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization

Tatsuki Mori 1, , Kousuke Kuto 2, , Masaharu Nagayama 3, , Tohru Tsujikawa 4, and  Shoji Yotsutani 5,

1. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194, Japan
2. 

Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585
3. 

Research Institute for Electronic Science, Hokkaido University, CREST, Japan Science and Technology Agency, N12W7, Kita-Ward, Sapporo, 060-0812, Japan
4. 

Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192
5. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194

Received  September 2014 Revised  August 2015 Published  November 2015

We are interested in wave-pinning in a reaction-diffusion model for cell polarization proposed by Y.Mori, A.Jilkine and L.Edelstein-Keshet. They showed interesting bifurcation diagrams and stability results for stationary solutions for a limiting equation by numerical computations. Kuto and Tsujikawa showed several mathematical bifurcation results of stationary solutions of this problem. We show exact expressions of all the solution by using the Jacobi elliptic functions and complete elliptic integrals. Moreover, we construct a bifurcation sheet which gives bifurcation diagram. Furthermore, we show numerical results of the stability of stationary solutions.
Keywords: Exact solution, Bifurcation diagram, Elliptic integral., Reaction diffusion model.
Mathematics Subject Classification: 35B32, 35K5.
Citation: Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861
References:
[1]

J. Carr, M. E. Gurtin, and M. Semrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351.

[2]

J. Smoller and A.Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.

[3]

J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer, 1994.

[4]

K. Kuto and T. TsujikawaE, Bifurcation structure of steady-states for bistable equations with nonlocal constraint, Discrete and Continuous Dynamical Systems Supplement, 2013C467-476D.

[5]

S.KosugiE Y. Morita, and S. YotsutaniE, Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals, Discrete and Continuous Dynamical Systems, 19(2007), 609-629D.

[6]

Y.Lou, W-M.Ni and S.Yotsutani, On a limiting system in the Lotka-Voltera competition with cross diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.

[7]

Y.Mori, A.Jilkine and L.Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization SIAM JE ApplE Math., 71 (2011), C1401-1427D.

show all references

References:
[1]

J. Carr, M. E. Gurtin, and M. Semrod, Structured phase transitions on a finite interval, Arch. Rational Mech. Anal., 86 (1984), 317-351.

[2]

J. Smoller and A.Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.

[3]

J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer, 1994.

[4]

K. Kuto and T. TsujikawaE, Bifurcation structure of steady-states for bistable equations with nonlocal constraint, Discrete and Continuous Dynamical Systems Supplement, 2013C467-476D.

[5]

S.KosugiE Y. Morita, and S. YotsutaniE, Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals, Discrete and Continuous Dynamical Systems, 19(2007), 609-629D.

[6]

Y.Lou, W-M.Ni and S.Yotsutani, On a limiting system in the Lotka-Voltera competition with cross diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.

[7]

Y.Mori, A.Jilkine and L.Edelstein-Keshet, Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization SIAM JE ApplE Math., 71 (2011), C1401-1427D.

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