Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models
Authors:
Jong-Shenq Guo and Je-Chiang Tsai
Journal:
Quart. Appl. Math. 67 (2009), 559-578
MSC (2000):
Primary 34A34, 34A12; Secondary 35K57
DOI:
https://doi.org/10.1090/S0033-569X-09-01153-9
Published electronically:
May 6, 2009
MathSciNet review:
2547640
Full-text PDF Free Access
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Abstract: We study the existence and uniqueness of traveling wave solutions for a class of two-component reaction-diffusion systems with one species being immobile. Such a system has a variety of applications in epidemiology, bio-reactor models, and isothermal autocatalytic chemical reaction systems. Our result not only generalizes earlier results of Ai and Huang (Proceedings of the Royal Society of Edinburgh 2005; 135A:663–675), but also establishes the existence and uniqueness of traveling wave solutions to the reaction-diffusion system for an isothermal autocatalytic chemical reaction of any order in which the autocatalyst is assumed to decay to the inert product at a rate of the same order.
References
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References
- S. Ai and W. Huang, Traveling waves for a reaction-diffusion system in population dynamics and epidemiology, Proc. Roy. Soc. Edinburgh Sect. A 135A (2005), 663–675. MR 2173333 (2006e:35188)
- S. Ai and W. Huang, Traveling wavefronts in combustion and chemical reaction models, Proc. Roy. Soc. Edinburgh Sect. A 137A (2007), 671–700. MR 2345776 (2008g:35106)
- R. Aris, P. Gray and S. K. Scott, Modeling of cubic autocatalysis by successive bimolecular steps, Chem. Eng. Sci. 43 (1988), 207–211.
- 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. Ser. A 334 (1991), 1–24. MR 1155096 (92m:80020)
- 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. Ser. A 336 (1991), 497–539. MR 1133118 (93a:80012)
- N.T.J. Bailey, The mathematical theory of infectious diseases, Griddin, London, 1975.
- M. Ballyk, L. Dung, D. A. Jones, and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math. 59 (1999), 573–596. MR 1654407 (2001a:92027)
- J. Carr, Applications of centre manifold theory, Springer-Verlag, New York, 1981. MR 635782 (83g:34039)
- X. Chen and Y. Qi, Sharp estimates on minimum traveling wave speed of reaction diffusion systems modelling autocatalysis, SIAM J. Math. Anal. 39 (2007), 437–448. MR 2338414 (2008h:34083)
- S. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol. 17 (1983), 11–32. MR 707221 (84j:92031)
- P. Gray, Instabilities and oscillations in chemical reactions in closed and open systems, Proc. Roy. Soc. London Ser. A 415 (1988), 1–34.
- Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlin. World 1 (1994), 277–290. MR 1303097 (95k:92021)
- Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model, Math. Models Meth. Appl. Sci. 5 (1995), 935–966. MR 1359215 (96j:35248)
- Y. Hosono, Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), 115–125. MR 2300326 (2008c:35148)
- W. Huang, Travelling waves for a biological reaction-diffusion model, J. Dynam. Diff. Eqns. 16 (2004), 745–765. MR 2109164 (2005g:35175)
- A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal. TMA 8 (1984), 851–856. MR 753763 (86h:92042)
- W. O. Kermack and A. G. McKendric, Contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A 115 (1927), 700–721.
- C. R. Kennedy and R. Aris, Travelling waves in a simple population model involving growth and death, Bull. Math. Biol. 42 (1980), 397–429. MR 661329 (84f:92041)
- 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), 315–345. MR 1068302 (91i:80008)
- J. H. Merkin and D. J. Needham, The development of travelling waves in a simple isothermal chemical system. IV. Quadratic autocatalysis with quadratic decay, Proc. Roy. Soc. London Ser. A 434 (1991), 531–554.
- J. H. Merkin and D. J. Needham, 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), 261–274. MR 1143726 (93a:80013)
- J. D. Murray, Mathematical biology. I: An introduction, Springer-Verlag, New York, 2004. MR 1908418 (2004b:92003)
- Y. Qi, The development of travelling waves in cubic auto-catalysis with different rates of diffusion, Physica D. 226 (2007), 129–135. MR 2296235 (2007k:35270)
- A. Saul and K. Showalter, Propagating reaction-diffusion fronts, in: R.J. Field and M. Burger (Eds.), Oscillations and Traveling waves in chemical systems, Wiley, New York, 1984.
- H. L. Smith and X. Q. Zhao, Travelling waves in a bio-reactor model, Nonlinear Anal. Real World Appl. 5 (2004), 895–909. MR 2085700 (2005g:35161)
- V. G. Voronkov and N. N. Semenov, Zh. Fiz. Khim. 13 (1939), 1695.
- J. Xin, Front propagation in heterogeneous media, SIAM Rev. 42 (2000), 161–230. MR 1778352 (2001i:35184)
- A. N. Zaikin and A. M. Zhabotinskii, Concentration wave propagation in two-dimensional liquid-phase self-oscillating system, Nature 225 (1970), 535–537.
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Additional Information
Jong-Shenq Guo
Affiliation:
Department of Mathematics, National Taiwan Normal University, 88, Section 4, Ting Chou Road, Taipei 116, Taiwan
Email:
jsguo@math.ntnu.edu.tw
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:
Traveling waves,
reaction-diffusion systems,
centre manifold
Received by editor(s):
March 13, 2008
Published electronically:
May 6, 2009
Additional Notes:
The first author was supported in part by the National Science Council of the Republic of China under the contracts NSC 96-2119-M-003-001.
The second author is the corresponding author and was partially supported by the National Science Council of the Republic of China under the contracts NSC 96-2115-M-194-003-MY3.
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.