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August  2014, 19(6): 1769-1781. doi: 10.3934/dcdsb.2014.19.1769

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, Chile

3. 

Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  June 2013 Revised  April 2014 Published  June 2014

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).
Citation: Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769
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.  doi: 10.1016/j.jde.2008.01.004.  Google Scholar

[2]

M. S. P. Eastham, The Asymptotic Solution of Linear Differential Systems,, Clarendon Press, (1989).   Google Scholar

[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.  doi: 10.1063/1.461481.  Google Scholar

[4]

J. Fang and J. Wu, Monotone travelling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst., 32 (2012), 3043.  doi: 10.3934/dcds.2012.32.3043.  Google Scholar

[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.  doi: 10.1021/ja00814a003.  Google Scholar

[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.  doi: 10.3934/dcdsb.2011.16.189.  Google Scholar

[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.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

[8]

J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, Discrete Contin. Dyn. Syst., 9 (2003), 925.  doi: 10.3934/dcds.2003.9.925.  Google Scholar

[9]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model,, J. Dynam. Differential Equations, 22 (2010), 285.  doi: 10.1007/s10884-010-9159-0.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.05.012.  Google Scholar

[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.   Google Scholar

[12]

B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759.  doi: 10.1088/0951-7715/24/6/004.  Google Scholar

[13]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[14]

G. Lin and W.-T. Li, Travelling wavefronts of Belousov-Zhabotinskii system with diffusion and delay,, Appl. Math. Letters, 22 (2009), 341.  doi: 10.1016/j.aml.2008.04.006.  Google Scholar

[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.  Google Scholar

[16]

G. Lv and M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra systems,, Nonlinear Anal. Real World Appl., 11 (2010), 1323.  doi: 10.1016/j.nonrwa.2009.02.020.  Google Scholar

[17]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[18]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1.  doi: 10.1023/A:1021889401235.  Google Scholar

[19]

J. D. Murray, On traveling wave solutions in a model for Belousov-Zhabotinskii reaction,, J. Theor. Biol., 56 (1976), 329.  doi: 10.1016/S0022-5193(76)80078-1.  Google Scholar

[20]

J. D. Murray, Lectures on Nonlinear Differential Equations, Models in biology,, Clarendon Press, (1977).   Google Scholar

[21]

M. R. Roussel, The use of delay differential equations in chemical kinetics,, J. Phys. Chem., 100 (1996), 8323.  doi: 10.1021/jp9600672.  Google Scholar

[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.  doi: 10.3934/dcds.2013.33.2169.  Google Scholar

[23]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction,, J. Differential Equations, 254 (2013), 3690.  doi: 10.1016/j.jde.2013.02.005.  Google Scholar

[24]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems,, Amer. Math. Soc., (1994).   Google Scholar

[25]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

[26]

Q. Ye and M. Wang, Traveling wave front solutions of Noyes-Field System for Belousov-Zhabotinskii reaction,, Nonlinear Anal., 11 (1987), 1289.  doi: 10.1016/0362-546X(87)90046-0.  Google Scholar

show all references

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.  doi: 10.1016/j.jde.2008.01.004.  Google Scholar

[2]

M. S. P. Eastham, The Asymptotic Solution of Linear Differential Systems,, Clarendon Press, (1989).   Google Scholar

[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.  doi: 10.1063/1.461481.  Google Scholar

[4]

J. Fang and J. Wu, Monotone travelling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst., 32 (2012), 3043.  doi: 10.3934/dcds.2012.32.3043.  Google Scholar

[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.  doi: 10.1021/ja00814a003.  Google Scholar

[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.  doi: 10.3934/dcdsb.2011.16.189.  Google Scholar

[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.  doi: 10.1007/s10884-011-9214-5.  Google Scholar

[8]

J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, Discrete Contin. Dyn. Syst., 9 (2003), 925.  doi: 10.3934/dcds.2003.9.925.  Google Scholar

[9]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model,, J. Dynam. Differential Equations, 22 (2010), 285.  doi: 10.1007/s10884-010-9159-0.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.05.012.  Google Scholar

[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.   Google Scholar

[12]

B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759.  doi: 10.1088/0951-7715/24/6/004.  Google Scholar

[13]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[14]

G. Lin and W.-T. Li, Travelling wavefronts of Belousov-Zhabotinskii system with diffusion and delay,, Appl. Math. Letters, 22 (2009), 341.  doi: 10.1016/j.aml.2008.04.006.  Google Scholar

[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.  Google Scholar

[16]

G. Lv and M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra systems,, Nonlinear Anal. Real World Appl., 11 (2010), 1323.  doi: 10.1016/j.nonrwa.2009.02.020.  Google Scholar

[17]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[18]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1.  doi: 10.1023/A:1021889401235.  Google Scholar

[19]

J. D. Murray, On traveling wave solutions in a model for Belousov-Zhabotinskii reaction,, J. Theor. Biol., 56 (1976), 329.  doi: 10.1016/S0022-5193(76)80078-1.  Google Scholar

[20]

J. D. Murray, Lectures on Nonlinear Differential Equations, Models in biology,, Clarendon Press, (1977).   Google Scholar

[21]

M. R. Roussel, The use of delay differential equations in chemical kinetics,, J. Phys. Chem., 100 (1996), 8323.  doi: 10.1021/jp9600672.  Google Scholar

[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.  doi: 10.3934/dcds.2013.33.2169.  Google Scholar

[23]

E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction,, J. Differential Equations, 254 (2013), 3690.  doi: 10.1016/j.jde.2013.02.005.  Google Scholar

[24]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems,, Amer. Math. Soc., (1994).   Google Scholar

[25]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

[26]

Q. Ye and M. Wang, Traveling wave front solutions of Noyes-Field System for Belousov-Zhabotinskii reaction,, Nonlinear Anal., 11 (1987), 1289.  doi: 10.1016/0362-546X(87)90046-0.  Google Scholar

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