American Institute of Mathematical Sciences

Advanced
June  2020, 40(6): 3357-3374. doi: 10.3934/dcds.2020049

Spike solutions for a mass conservation reaction-diffusion system

Shin-Ichiro Ei 1,, and  Shyuh-Yaur Tzeng 2,

1. 

Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
2. 

Department of Mathematics, National Changhua University of Education, Changhua, Taiwan

* Corresponding author: Shin-Ichiro Ei

-dedicated to 70th birthday of Prof. Wei-Ming Ni-

Received  March 2019 Revised  July 2019 Published  October 2019

This article deals with a mass conservation reaction-diffusion system. As a model for studying cell polarity, we are interested in the existence of spike solutions and some properties related to its dynamics. Variational arguments will be employed to investigate the existence questions. The profile of a spike solution looks like a standing pulse. In addition, the motion of such spikes in heterogeneous media will be derived.

Keywords: Reaction-diffusion system, mass conservation, spike solution, cell polarity, localized pattern, standing pulse, invariant manifold.
Mathematics Subject Classification: Primary: 35B25, 35J505, 35K57; Secondary: 37L6.
Citation: Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 394-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

C.-N. Chen and Y. S. Choi, Standing pulse solutions to FitzHugh-Nagumo equations, Arch. Rational Mech. Anal., 206 (2012), 741-777.  doi: 10.1007/s00205-012-0542-3.  Google Scholar

[3]

C.-N. Chen and Y. S. Choi, Traveling pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 1-45.  doi: 10.1007/s00526-014-0776-z.  Google Scholar

[4]

C.-N. Chen, Y. S. Choi and N. Fusco, The $\Gamma$-limit of traveling waves in the FitzHugh-Nagumo system, J. Differential Equations, 267 (2019), 1805-1835.  doi: 10.1016/j.jde.2019.02.023.  Google Scholar

[5]

C.-N. Chen, Y.-S. Choi and X. F. Ren, Bubbles and droplets in a singular limit of the FitzHugh-Nagumo system, Interfaces and Free Boundaries, 20 (2018), 165-210.  doi: 10.4171/IFB/400.  Google Scholar

[6]

C.-N. Chen, Y.-S. Choi, Y. Y. Hu and X. F. Ren, Higher dimensional bubble profiles in a singular limit of the FitzHugh-Nagumo system, SIAM J. Math. Anal., 50 (2018), 5072-5095.  doi: 10.1137/17M1144933.  Google Scholar

[7]

C.-N. Chen, S.-I. Ei and S.-Y. Tzeng, Heterogeneity induced effects for pulse dynamics in FitzHugh-Nagumo type systems, Physica D: Nonlinear Phenomena, 382/383 (2018), 22-32.  doi: 10.1016/j.physd.2018.07.001.  Google Scholar

[8]

C.-N. Chen and X. J. Hu, Maslov index for homoclinic orbits of Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéarie, 24 (2007), 589-603.  doi: 10.1016/j.anihpc.2006.06.002.  Google Scholar

[9]

C.-N. Chen and X. J. Hu, Stability criteria for reaction-diffusion systems with skew-gradient structure, Communications in Partial Differential Equations, 33 (2008), 189-208.  doi: 10.1080/03605300601188755.  Google Scholar

[10]

C.-N. Chen and X. J. Hu, Stability analysis for standing pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 827-845.  doi: 10.1007/s00526-013-0601-0.  Google Scholar

[11]

C.-N. Chen and K. Tanaka, A variational approach for standing waves of FitzHugh-Nagumo type systems, J. Differential Equations, 257 (2014), 109-144.  doi: 10.1016/j.jde.2014.03.013.  Google Scholar

[12]

C.-N. Chen and S.-Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Differential Equations, 177 (2001), 121-145.  doi: 10.1006/jdeq.2000.3996.  Google Scholar

[13]

A. Doelman, P. van Heijster and T. J. Kaper, Pulse dynamics in a three-component system: Existence analysis, J. Dynam. Differential Equations, 21 (2009), 73-115.  doi: 10.1007/s10884-008-9125-2.  Google Scholar

[14]

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

[15]

S.-I. Ei and J. C. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimension, Japan J. Ind. Appl. Math., 19 (2002), 181-226.  doi: 10.1007/BF03167453.  Google Scholar

[16]

S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007), 015203(R). doi: 10.1103/PhysRevE.75.015203.  Google Scholar

[17]

S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021.  Google Scholar

[18]

J. Keener and J. Sneyd, Mathematical Physiology. Vol. I: Cellular Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/I. Springer, New York, 2009. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[19]

M. Kuwamura, S. Seirin-Lee and S.-I. Ei, Dynamics of localized unimodal patterns in reaction-diffusion systems for cell polarization by extracellular signaling, SIAM J. APPL. MATH., 78 (2018), 3238-3257.  doi: 10.1137/18M1163749.  Google Scholar

[20]

E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, J. Math. Anal. Appl., 411 (2014), 107-118.  doi: 10.1016/j.jmaa.2013.09.039.  Google Scholar

[21]

Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007.  Google Scholar

[22]

W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.  Google Scholar

[23]

M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity, PLoS Comput. Biol., 3 (2007), 1040-1054.  doi: 10.1371/journal.pcbi.0030108.  Google Scholar

[24]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[25]

C. Reinecke and G. Sweers, A positive solution on ${\mathbb R}^n$ to a equations of FitzHugh-Nagumo type, J. Differential Equations, 153 (1999), 292-312.  doi: 10.1006/jdeq.1998.3560.  Google Scholar

[26]

T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., 71 (2009), 1329-1349.  doi: 10.1016/j.na.2008.12.007.  Google Scholar

[27]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Localized patterns in a three-component FitzHugh-Nagumo model revisited via an action functional, J. Dynam. Differential Equations, 30 (2018), 521-555.  doi: 10.1007/s10884-016-9557-z.  Google Scholar

[28]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Pinned solutions in a heterogeneous three-component FitzHugh-Nagumo model, J. Dyn. Differ. Equ., 31 (2019), 153-203.  doi: 10.1007/s10884-018-9694-7.  Google Scholar

[29]

E. Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104.  doi: 10.1007/BF00276548.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 394-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

C.-N. Chen and Y. S. Choi, Standing pulse solutions to FitzHugh-Nagumo equations, Arch. Rational Mech. Anal., 206 (2012), 741-777.  doi: 10.1007/s00205-012-0542-3.  Google Scholar

[3]

C.-N. Chen and Y. S. Choi, Traveling pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 1-45.  doi: 10.1007/s00526-014-0776-z.  Google Scholar

[4]

C.-N. Chen, Y. S. Choi and N. Fusco, The $\Gamma$-limit of traveling waves in the FitzHugh-Nagumo system, J. Differential Equations, 267 (2019), 1805-1835.  doi: 10.1016/j.jde.2019.02.023.  Google Scholar

[5]

C.-N. Chen, Y.-S. Choi and X. F. Ren, Bubbles and droplets in a singular limit of the FitzHugh-Nagumo system, Interfaces and Free Boundaries, 20 (2018), 165-210.  doi: 10.4171/IFB/400.  Google Scholar

[6]

C.-N. Chen, Y.-S. Choi, Y. Y. Hu and X. F. Ren, Higher dimensional bubble profiles in a singular limit of the FitzHugh-Nagumo system, SIAM J. Math. Anal., 50 (2018), 5072-5095.  doi: 10.1137/17M1144933.  Google Scholar

[7]

C.-N. Chen, S.-I. Ei and S.-Y. Tzeng, Heterogeneity induced effects for pulse dynamics in FitzHugh-Nagumo type systems, Physica D: Nonlinear Phenomena, 382/383 (2018), 22-32.  doi: 10.1016/j.physd.2018.07.001.  Google Scholar

[8]

C.-N. Chen and X. J. Hu, Maslov index for homoclinic orbits of Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéarie, 24 (2007), 589-603.  doi: 10.1016/j.anihpc.2006.06.002.  Google Scholar

[9]

C.-N. Chen and X. J. Hu, Stability criteria for reaction-diffusion systems with skew-gradient structure, Communications in Partial Differential Equations, 33 (2008), 189-208.  doi: 10.1080/03605300601188755.  Google Scholar

[10]

C.-N. Chen and X. J. Hu, Stability analysis for standing pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 827-845.  doi: 10.1007/s00526-013-0601-0.  Google Scholar

[11]

C.-N. Chen and K. Tanaka, A variational approach for standing waves of FitzHugh-Nagumo type systems, J. Differential Equations, 257 (2014), 109-144.  doi: 10.1016/j.jde.2014.03.013.  Google Scholar

[12]

C.-N. Chen and S.-Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Differential Equations, 177 (2001), 121-145.  doi: 10.1006/jdeq.2000.3996.  Google Scholar

[13]

A. Doelman, P. van Heijster and T. J. Kaper, Pulse dynamics in a three-component system: Existence analysis, J. Dynam. Differential Equations, 21 (2009), 73-115.  doi: 10.1007/s10884-008-9125-2.  Google Scholar

[14]

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

[15]

S.-I. Ei and J. C. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimension, Japan J. Ind. Appl. Math., 19 (2002), 181-226.  doi: 10.1007/BF03167453.  Google Scholar

[16]

S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007), 015203(R). doi: 10.1103/PhysRevE.75.015203.  Google Scholar

[17]

S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021.  Google Scholar

[18]

J. Keener and J. Sneyd, Mathematical Physiology. Vol. I: Cellular Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/I. Springer, New York, 2009. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[19]

M. Kuwamura, S. Seirin-Lee and S.-I. Ei, Dynamics of localized unimodal patterns in reaction-diffusion systems for cell polarization by extracellular signaling, SIAM J. APPL. MATH., 78 (2018), 3238-3257.  doi: 10.1137/18M1163749.  Google Scholar

[20]

E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, J. Math. Anal. Appl., 411 (2014), 107-118.  doi: 10.1016/j.jmaa.2013.09.039.  Google Scholar

[21]

Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007.  Google Scholar

[22]

W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.  Google Scholar

[23]

M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity, PLoS Comput. Biol., 3 (2007), 1040-1054.  doi: 10.1371/journal.pcbi.0030108.  Google Scholar

[24]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[25]

C. Reinecke and G. Sweers, A positive solution on ${\mathbb R}^n$ to a equations of FitzHugh-Nagumo type, J. Differential Equations, 153 (1999), 292-312.  doi: 10.1006/jdeq.1998.3560.  Google Scholar

[26]

T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., 71 (2009), 1329-1349.  doi: 10.1016/j.na.2008.12.007.  Google Scholar

[27]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Localized patterns in a three-component FitzHugh-Nagumo model revisited via an action functional, J. Dynam. Differential Equations, 30 (2018), 521-555.  doi: 10.1007/s10884-016-9557-z.  Google Scholar

[28]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Pinned solutions in a heterogeneous three-component FitzHugh-Nagumo model, J. Dyn. Differ. Equ., 31 (2019), 153-203.  doi: 10.1007/s10884-018-9694-7.  Google Scholar

[29]

E. Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104.  doi: 10.1007/BF00276548.  Google Scholar

[1]

Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191
[2]

Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062
[3]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[4]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[5]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[6]

Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707
[7]

Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255
[8]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[9]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[10]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[11]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
[12]

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
[13]

Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
[14]

Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227
[15]

W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893
[16]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299
[17]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039
[18]

Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673
[19]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817
[20]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (147)
  • HTML views (183)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences