On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction

Elena Trofimchuk; Manuel Pinto; Sergei Trofimchuk

doi:10.3934/dcdsb.2014.19.1769

Article Contents

2014, Volume 19, Issue 6: 1769-1781. Doi: 10.3934/dcdsb.2014.19.1769

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On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction

1.
Department of Differential Equations, National Technical University, Kyiv
2.
Facultad de Ciencias, Universidad de Chile, Santiago
3.
Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received: May 31, 2013

Revised: March 31, 2014

Published: July 2014

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Abstract

In this paper, we answer the question about the existence of the minimal speed of front propagation in a delayed version of the Murray model of the Belousov-Zhabotinsky (BZ) chemical reaction. It is assumed that the key parameter $r$ of this model satisfies $0< r \leq 1$ that makes it formally monostable. By proving that the set of all admissible speeds of propagation has the form $[c_*,+\infty)$, we show here that the BZ system with $r \in (0,1]$ is actually of the monostable type (in general, $c_*$ is not linearly determined). We also establish the monotonicity of wavefronts and present the principal terms of their asymptotic expansions at infinity (in the critical case $r=1$ inclusive).

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Mathematics Subject Classification: Primary: 34K12, 35K57; Secondary: 92E20.

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References

[1]	A. Boumenir and V. M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570.doi: 10.1016/j.jde.2008.01.004.
[2]	M. S. P. Eastham, The Asymptotic Solution of Linear Differential Systems, Clarendon Press, Oxford, 1989.
[3]	I. R. Epstein and Y. Luo, Differential delay equations in chemical kinetics. Nonlinear models: the cross-shaped phase diagram and the Oregonator, J. Chem. Phys., 95 (1991), 244-254.doi: 10.1063/1.461481.
[4]	J. Fang and J. Wu, Monotone travelling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst., 32 (2012), 3043-3058.doi: 10.3934/dcds.2012.32.3043.
[5]	R. J. Field and R. M. Noyes, Oscillations in chemical systems. V. Quantitative explanation of band migration in the Belousov-Zhabotinskii reaction, J. Am. Chem. Soc., 96 (1974), 2001-2006.doi: 10.1021/ja00814a003.
[6]	S.-C. Fu, Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction, Discrete Contin. Dyn. Syst. B, 16 (2011), 189-196.doi: 10.3934/dcdsb.2011.16.189.
[7]	J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.doi: 10.1007/s10884-011-9214-5.
[8]	J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9 (2003), 925-936.doi: 10.3934/dcds.2003.9.925.
[9]	W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dynam. Differential Equations, 22 (2010), 285-297.doi: 10.1007/s10884-010-9159-0.
[10]	W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differential Equations, 251 (2011), 1549-1561.doi: 10.1016/j.jde.2011.05.012.
[11]	Ya. I. Kanel, The existence of a solution of traveling wave type for the Belousov-Zhabotinskii system of equations. II, Siberian Math. J., 32 (1991), 390-400.
[12]	B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity, 24 (2011), 1759-1776.doi: 10.1088/0951-7715/24/6/004.
[13]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems J. Functional Anal., 259 (2010), 857-903.doi: 10.1016/j.jfa.2010.04.018.
[14]	G. Lin and W.-T. Li, Travelling wavefronts of Belousov-Zhabotinskii system with diffusion and delay, Appl. Math. Letters, 22 (2009), 341-346.doi: 10.1016/j.aml.2008.04.006.
[15]	G. Lin and S. Ruan, Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays, J. Dynam. Differential Equations, 2014.doi: 10.1007/s10884-014-9355-4.
[16]	G. Lv and M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra systems, Nonlinear Anal. Real World Appl., 11 (2010), 1323-1329.doi: 10.1016/j.nonrwa.2009.02.020.
[17]	S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.doi: 10.1006/jdeq.2000.3846.
[18]	J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-47.doi: 10.1023/A:1021889401235.
[19]	J. D. Murray, On traveling wave solutions in a model for Belousov-Zhabotinskii reaction, J. Theor. Biol., 56 (1976), 329-353.doi: 10.1016/S0022-5193(76)80078-1.
[20]	J. D. Murray, Lectures on Nonlinear Differential Equations, Models in biology, Clarendon Press, Oxford, 1977.
[21]	M. R. Roussel, The use of delay differential equations in chemical kinetics, J. Phys. Chem., 100 (1996), 8323-8330.doi: 10.1021/jp9600672.
[22]	E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction, Discrete Contin. Dyn. Syst., 33 (2013), 2169-2187.doi: 10.3934/dcds.2013.33.2169.
[23]	E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction, J. Differential Equations, 254 (2013), 3690-3714.doi: 10.1016/j.jde.2013.02.005.
[24]	A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Amer. Math. Soc., Providence, 1994.
[25]	J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.doi: 10.1023/A:1016690424892.
[26]	Q. Ye and M. Wang, Traveling wave front solutions of Noyes-Field System for Belousov-Zhabotinskii reaction, Nonlinear Anal., 11 (1987), 1289-1302.doi: 10.1016/0362-546X(87)90046-0.