Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay
Author:
Je-Chiang Tsai
Journal:
Quart. Appl. Math. 69 (2011), 123-146
MSC (2000):
Primary 34A34, 34A12, 35K57
DOI:
https://doi.org/10.1090/S0033-569X-2010-01236-7
Published electronically:
December 29, 2010
MathSciNet review:
2807981
Full-text PDF Free Access
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Abstract: The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the $(m+1)$th order ($A+mB\rightarrow (m+1)B$) and a decay step of the same order ($B\rightarrow C$) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor.
References
- Shangbing Ai and Wenzhang Huang, Travelling waves for a reaction-diffusion system in population dynamics and epidemiology, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 4, 663–675. MR 2173333, DOI https://doi.org/10.1017/S0308210500004054
- Shangbing Ai and Wenzhang Huang, Travelling wavefronts in combustion and chemical reaction models, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 4, 671–700. MR 2345776, DOI https://doi.org/10.1017/S0308210505001095
- Norman T. J. Bailey, The mathematical theory of infectious diseases and its applications, 2nd ed., Hafner Press [Macmillan Publishing Co., Inc.] New York, 1975. MR 0452809
- Mary Ballyk, Le Dung, Don A. Jones, and Hal L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math. 59 (1999), no. 2, 573–596. MR 1654407, DOI https://doi.org/10.1137/S0036139997325345
- J. Billingham and D. J. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves, Philos. Trans. Roy. Soc. London Ser. A 334 (1991), no. 1633, 1–24. MR 1155096, DOI https://doi.org/10.1098/rsta.1991.0001
- J. Billingham and D. J. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. II. An initial value problem with an immobilized or nearly immobilized autocatalyst, Philos. Trans. Roy. Soc. London Ser. A 336 (1991), no. 1644, 497–539. MR 1133118, DOI https://doi.org/10.1098/rsta.1991.0098
- Jack Carr, Applications of centre manifold theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, New York-Berlin, 1981. MR 635782
- Xinfu Chen and Yuanwei Qi, Sharp estimates on minimum travelling wave speed of reaction diffusion systems modelling autocatalysis, SIAM J. Math. Anal. 39 (2007), no. 2, 437–448. MR 2338414, DOI https://doi.org/10.1137/060665749
- Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369.
- Gowland R. J. and Stedman, G. (1983) A novel moving boundary reaction involving hydroxylamine and nitric acid. J. Chem. Soc., Chem. Commun. 10, 1038–1039.
- Gray, P. (1988) Instabilities and oscillations in chemical reactions in closed and open systems. Proc. R. Soc. A 415, 1–34.
- Jong-Shenq Guo and Je-Chiang Tsai, Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models, Quart. Appl. Math. 67 (2009), no. 3, 559–578. MR 2547640, DOI https://doi.org/10.1090/S0033-569X-09-01153-9
- Hodgkin, A. L. and Huxley, A. F. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (London) 117, 500–544.
- Yuzo Hosono and Bilal Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World 1 (1994), no. 3, 277–290. MR 1303097
- Yuzo Hosono and Bilal Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci. 5 (1995), no. 7, 935–966. MR 1359215, DOI https://doi.org/10.1142/S0218202595000504
- Yuzo Hosono, Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 1, 115–125. MR 2300326, DOI https://doi.org/10.3934/dcdsb.2007.8.115
- Wenzhang Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations 16 (2004), no. 3, 745–765. MR 2109164, DOI https://doi.org/10.1007/s10884-004-6115-x
- Wenzhang Huang, Uniqueness of traveling wave solutions for a biological reaction-diffusion equation, J. Math. Anal. Appl. 316 (2006), no. 1, 42–59. MR 2201748, DOI https://doi.org/10.1016/j.jmaa.2005.04.084
- Anders Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal. 8 (1984), no. 8, 851–856. MR 753763, DOI https://doi.org/10.1016/0362-546X%2884%2990107-X
- A. L. Kay, D. J. Needham, and J. A. Leach, Travelling waves for a coupled, singular reaction-diffusion system arising from a model of fractional order autocatalysis with decay. I. Permanent form travelling waves, Nonlinearity 16 (2003), no. 2, 735–770. MR 1959108, DOI https://doi.org/10.1088/0951-7715/16/2/322
- C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol. 42 (1980), no. 3, 397–429. MR 661329, DOI https://doi.org/10.1016/S0092-8240%2880%2980057-7
- Kermack, W. O. and McKendric, A. G. (1927) Contribution to the mathematical theory of epidemic. Proc. R. Soc. A 115, 700–721.
- Kolmogorov, A. N., Petrovsky, I. G. and Piskunov, N. S. (1937) Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Université d’État à Moscou, Ser. Int., Sect. A. 1, 1–25.
- Martine Marion, Qualitative properties of a nonlinear system for laminar flames without ignition temperature, Nonlinear Anal. 9 (1985), no. 11, 1269–1292. MR 813658, DOI https://doi.org/10.1016/0362-546X%2885%2990035-5
- J. H. Merkin and D. J. Needham, Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system, J. Engrg. Math. 23 (1989), no. 4, 343–356. MR 1029938, DOI https://doi.org/10.1007/BF00128907
- J. H. Merkin and D. J. Needham, The development of travelling waves in a simple isothermal chemical system. II. Cubic autocatalysis with quadratic and linear decay, Proc. Roy. Soc. London Ser. A 430 (1990), no. 1879, 315–345. MR 1068302, DOI https://doi.org/10.1098/rspa.1990.0093
- Merkin, J. H. and Needham, D. J. (1991) The development of travelling waves in a simple isothermal chemical system. IV. Quadratic autocatalysis with quadratic decay. Proc. R. Soc. A 434, 531–554.
- J. H. Merkin and D. J. Needham, Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension, Z. Angew. Math. Phys. 44 (1993), no. 4, 707–721. MR 1239888, DOI https://doi.org/10.1007/BF00948484
- D. J. Needham and J. H. Merkin, The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay, Philos. Trans. Roy. Soc. London Ser. A 337 (1991), no. 1646, 261–274. MR 1143726, DOI https://doi.org/10.1098/rsta.1991.0122
- Yuanwei Qi, The development of travelling waves in cubic auto-catalysis with different rates of diffusion, Phys. D 226 (2007), no. 2, 129–135. MR 2296235, DOI https://doi.org/10.1016/j.physd.2006.11.010
- Saul, A. and Showalter, K. (1984) Propagating reaction-diffusion fronts. In Oscillations and Traveling waves in chemical systems (ed. R. J. Field and M. Burger), Wiley, New York.
- Hal L. Smith and Xiao-Qiang Zhao, Traveling waves in a bio-reactor model, Nonlinear Anal. Real World Appl. 5 (2004), no. 5, 895–909. MR 2085700, DOI https://doi.org/10.1016/j.nonrwa.2004.05.001
- Voronkov, V. G. and Semenov, N. N. (1939) Zh. Fiz. Khim. 13, 1695.
- Zaikin, A. N. and Zhabotinskii, A. M. (1970) Concentration wave propagation in two-dimensional liquid-phase self-oscillating system. Nature 225, 535–537.
References
- Ai, S. and Huang, W. (2005) Traveling waves for a reaction-diffusion system in population dynamics and epidemiology. Proc. Roy. Soc. Edinburgh Sect. A 135A, 663–675. MR 2173333 (2006e:35188)
- Ai, S. and Huang, W. (2007) Traveling wavefronts in combustion and chemical reaction models. Proc. Roy. Soc. Edinburgh Sect. A 137A, 671–700. MR 2345776 (2008g:35106)
- Bailey, N. T. J. (1975) The mathematical theory of infectious diseases and its applications. 2nd edition, Hafner Press. MR 0452809 (56:11084)
- Ballyk, M., Dung, L., Jones, D. A. and Smith, H. L. (1999) Effects of random motility on microbial growth and competition in a flow reactor. SIAM J. Appl. Math. 59, 573–596. MR 1654407 (2001a:92027)
- Billingham, J. and Needham, D. J. (1991a) The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves. Philos. Trans. R. Soc. Ser. A 334, 1–24. MR 1155096 (92m:80020)
- Billingham, J. and Needham, D. J. (1991b) The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. II. An initial value problem with immobilised or nearly immobilised autocatalyst. Philos. Trans. R. Soc. Ser. A 336, 497–539. MR 1133118 (93a:80012)
- Carr, J. (1981) Applications of centre manifold theory. Springer-Verlag, New York. MR 635782 (83g:34039)
- Chen, X. and Qi, Y. (2007) Sharp estimates on minimum traveling wave speed of reaction diffusion systems modelling autocatalysis. SIAM J. Math. Anal. 39, 437–448. MR 2338414 (2008h:34083)
- Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics 7, 353–369.
- Gowland R. J. and Stedman, G. (1983) A novel moving boundary reaction involving hydroxylamine and nitric acid. J. Chem. Soc., Chem. Commun. 10, 1038–1039.
- Gray, P. (1988) Instabilities and oscillations in chemical reactions in closed and open systems. Proc. R. Soc. A 415, 1–34.
- Guo, J.-S. and Tsai, J.-C. (2009) Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models. Quart. Appl. Math. 67, 559–578. MR 2547640
- Hodgkin, A. L. and Huxley, A. F. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (London) 117, 500–544.
- Hosono, Y. and Ilyas, B. (1994) Existence of traveling waves with any positive speed for a diffusive epidemic model. Nonlin. World 1, 277–290. MR 1303097 (95k:92021)
- Hosono, Y. and Ilyas, B. (1995) Travelling waves for a simple diffusive epidemic model. Math. Models Meth. Appl. Sci. 5, 935–966. MR 1359215 (96j:35248)
- Hosono, Y. (2007) Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete Contin. Dyn. Syst. Ser. B 8, 115–125. MR 2300326 (2008c:35148)
- Huang, W. (2004) Travelling waves for a biological reaction-diffusion model. J. Dynam. Diff. Eqns. 16, 745–765. MR 2109164 (2005g:35175)
- Huang, W. (2006) Uniqueness of traveling wave solutions for a biological reaction-diffusion equation. J. Math. Anal. Appl. 316, 42–59. MR 2201748 (2006k:35149)
- Källén, A. (1984) Thresholds and traveling waves in an epidemic model for rabies. Nonlinear Anal. TMA 8, 851–856. MR 753763 (86h:92042)
- Kay, A. L., Needham, D. J. and Leach, J. A. (2003) Travelling waves for a coupled, singular reaction-diffusion system arising from a model of fractional order autocatalysis with decay: I. Permanent form travelling waves. Nonlinearity 16, 735–770. MR 1959108 (2004c:35215)
- Kennedy, C. R. and Aris, R. (1980) Travelling waves in a simple population model involving growth and death. Bull. Math. Biol. 42, 397–429. MR 661329 (84f:92041)
- Kermack, W. O. and McKendric, A. G. (1927) Contribution to the mathematical theory of epidemic. Proc. R. Soc. A 115, 700–721.
- Kolmogorov, A. N., Petrovsky, I. G. and Piskunov, N. S. (1937) Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Université d’État à Moscou, Ser. Int., Sect. A. 1, 1–25.
- Marion, M. (1985) Qualitative properties of a nonlinear system for laminar flames without ignition temperature. Nonlin. Analysis 9, 1269–1292. MR 813658 (87j:80016)
- Merkin, J. H. and Needham, D. J. (1989) Propagation reaction-diffusion waves in a simple isothermal quadratic chemical system. J. Eng. Math. 23, 343–356. MR 1029938 (90k:80024)
- Merkin, J. H. and Needham, D. J. (1990) The development of travelling waves in a simple isothermal chemical system. II. Cubic autocatalysis with quadratic and linear decay. Proc. R. Soc. A 430, 315–345. MR 1068302 (91i:80008)
- Merkin, J. H. and Needham, D. J. (1991) The development of travelling waves in a simple isothermal chemical system. IV. Quadratic autocatalysis with quadratic decay. Proc. R. Soc. A 434, 531–554.
- Merkin, J. H. and Needham, D. J. (1993) Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension. J. Appl. Math. Phys. (ZAMP) 44, 707–721. MR 1239888 (94f:35067)
- Needham, D. J. and Merkin, J. H. (1991) The development of travelling waves in a simple isothermal chemical system with general orders of autocatalysis and decay. Philos. Trans. Roy. Soc. London Ser. A 337, 261–274. MR 1143726 (93a:80013)
- Qi, Y. (2007) The development of traveling waves in cubic auto-catalysis with different rates of diffusion. Physica D. 226, 129–135. MR 2296235 (2007k:35270)
- Saul, A. and Showalter, K. (1984) Propagating reaction-diffusion fronts. In Oscillations and Traveling waves in chemical systems (ed. R. J. Field and M. Burger), Wiley, New York.
- Smith, H. L. and Zhao., X. Q. (2004) Travelling waves in a bio-reactor model. Nonlin. Analysis Real World Applic. 5, 895–909. MR 2085700 (2005g:35161)
- Voronkov, V. G. and Semenov, N. N. (1939) Zh. Fiz. Khim. 13, 1695.
- Zaikin, A. N. and Zhabotinskii, A. M. (1970) Concentration wave propagation in two-dimensional liquid-phase self-oscillating system. Nature 225, 535–537.
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Additional Information
Je-Chiang Tsai
Affiliation:
Department of Mathematics, National Chung Cheng University, 168, University Road, Min-Hsiung, Chia-Yi 621, Taiwan
Email:
tsaijc@math.ccu.edu.tw
Keywords:
Isothermal autocatalytic chemical reaction,
traveling waves,
reaction-diffusion systems,
centre manifold
Received by editor(s):
July 29, 2009
Published electronically:
December 29, 2010
Additional Notes:
The author is supported in part by the National Science Council of Taiwan
Article copyright:
© Copyright 2010
Brown University