The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. III. Large time development in quadratic autocatalysis
Authors:
J. Billingham and D. J. Needham
Journal:
Quart. Appl. Math. 50 (1992), 343-372
MSC:
Primary 92E20; Secondary 80A32
DOI:
https://doi.org/10.1090/qam/1162280
MathSciNet review:
MR1162280
Full-text PDF Free Access
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Abstract: We study the isothermal, quadratic autocatalytic reaction scheme $A + \\ B \to 2B$ , where $A$ is a reactant and $B$ is an autocatalyst. We consider the situation when a quantity of $B$ is introduced locally into a uniform expanse of $A$ in one-dimensional slab geometry. By analysing the large time asymptotic solution of the resulting initial value problem we show that either a permanent form travelling wave or a phase wave develops. In the case when a travelling wave evolves we determine the first two terms in the large time asymptotic expansion of the propagation speed, which is dependent upon the asymptotic form of the initial input distribution of $B$ far ahead of the wavefront.
N. J. T. Bailey, The Mathematical Theory of Infectious Diseases, Griffen, London, 1975
- 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
- Maury D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978), no. 5, 531–581. MR 494541, DOI https://doi.org/10.1002/cpa.3160310502
R. J. Field and R. M. Noyes, Oscillations in chemical systems. IV Limit cycle behaviour in a model of a real chemical reaction, J. Chem. Phys. 60, 1877–1884 (1974)
R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics 7, 355–369 (1937)
P. Gray, J. F. Griffiths, and S. K. Scott, Oscillation, glow and ignition in carbon monoxide oxidation in an open system, Proc. Roy. Soc. London Ser. A 397, 21–44 (1984)
A. Hanna, A. Saul, and K. Showalter, Detailed studies of propagating fronts in the iodate oxidation of arsenous acid, J. Amer. Chem. Soc. 104, 3838–3844 (1982)
A. Kolmogorov, I. Petrovskii, and N. Piskounov, Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Math. Bull. 1, 1–25 (1937)
- D. A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math. 34 (1978), no. 1, 93–103. MR 482838, DOI https://doi.org/10.1137/0134008
- H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28 (1975), no. 3, 323–331. MR 400428, DOI https://doi.org/10.1002/cpa.3160280302
- 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, D. J. Needham, and S. K. Scott, A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system, Proc. Roy. Soc. London Ser. A 398, 81–116 (1985)
E. E. Sel’kov, Self-oscillation in glycolysis. I A simple kinetic model, European J. Biochem. 4, 79–86 (1968)
- L. J. Slater, Confluent hypergeometric functions, Cambridge University Press, New York, 1960. MR 0107026
A. N. Zaikin and A. M. Zhabotinskii, Concentration wave propagation in a two-dimensional liquid phase self-oscillatory system, Nature 225, 535–536 (1970)
N. J. T. Bailey, The Mathematical Theory of Infectious Diseases, Griffen, London, 1975
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, 1–25 (1991).
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, to appear in Philos. Trans. Roy. Soc. London Ser A 336.
M. D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31, 531–581 (1978)
R. J. Field and R. M. Noyes, Oscillations in chemical systems. IV Limit cycle behaviour in a model of a real chemical reaction, J. Chem. Phys. 60, 1877–1884 (1974)
R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics 7, 355–369 (1937)
P. Gray, J. F. Griffiths, and S. K. Scott, Oscillation, glow and ignition in carbon monoxide oxidation in an open system, Proc. Roy. Soc. London Ser. A 397, 21–44 (1984)
A. Hanna, A. Saul, and K. Showalter, Detailed studies of propagating fronts in the iodate oxidation of arsenous acid, J. Amer. Chem. Soc. 104, 3838–3844 (1982)
A. Kolmogorov, I. Petrovskii, and N. Piskounov, Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Math. Bull. 1, 1–25 (1937)
D. A. Larson, Transient bounds and time-asymptotic behaviour of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math. 34, 93–103 (1978)
H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskounov, Comm. Pure Appl. Math. 28, 323–331 (1975)
J. H. Merkin and D. J. Needham, Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system, J. Engrg. Math. 23, 343–356 (1989)
J. H. Merkin, D. J. Needham, and S. K. Scott, A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system, Proc. Roy. Soc. London Ser. A 398, 81–116 (1985)
E. E. Sel’kov, Self-oscillation in glycolysis. I A simple kinetic model, European J. Biochem. 4, 79–86 (1968)
L. J. Slater, Confluent Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1960
A. N. Zaikin and A. M. Zhabotinskii, Concentration wave propagation in a two-dimensional liquid phase self-oscillatory system, Nature 225, 535–536 (1970)
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Article copyright:
© Copyright 1992
American Mathematical Society
