The success of fast reaction: a discrete reaction-diffusion model
Author:
Matthias Büger
Journal:
Quart. Appl. Math. 62 (2004), 623-641
MSC:
Primary 35K57; Secondary 34C11, 35Q80, 37N25, 91B28, 92D15
DOI:
https://doi.org/10.1090/qam/2104265
MathSciNet review:
MR2104265
Full-text PDF Free Access
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Abstract: We discuss the dynamics of a system of $2n$ ordinary differential equations that can be looked at as the discrete version of a system of two reaction—diffusion equations, which differ only in their sensitivity to the reaction term. Such reaction—diffusion systems occur in evolutionary models from biology. It is known that only the fastest reacting species survives in generic situations. We prove similar results for the related discrete system and give an interpretation of the results in terms of mathematical finance.
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N. F. Britton, Reaction-diffusion equations and their applications to biology, Academic Press, London 1986.
M. Büger, The evolution of fast reaction: A reaction-diffusion model, Nonlin. Anal. Real World Appl. 3 (2002), 543–554.
R. Courant, D. Hilbert, Methods of mathematical physics, Vols. 1, 2, Wiley Interscience, New York 1962.
J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol. 37 (1998), 61–83.
P. C. Fife, Lecture notes in biomathematics: Mathematical aspects of reacting and diffusing systems, Springer-Verlag, Berlin—Heidelberg—New York 1979.
D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin—Heidelberg 1981.
S. Hsu, H. Smith, P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348 (1996), 4083–4094.
J. C. Hull, Options, futures and other derivatives, 4$^{th}$ edition, Prentice Hall 2000.
V. Hutson, J. Lopez-Gomez, K. Mischaikow, G. Vickers, Limit behaviour for a competing species problem with diffusion, in R. P. Agarwal (editor), Dynamical Systems and Applications, World Scientific, Singapore 1995, 343–358.
S. N. Neftei, An introduction to the Mathematics of financial derivatives, Academic Press 1996.
P. Wilmott, S. Howison Jr., The Mathematics of financial derivatives, Cambridge Univ. Press 1995.
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© Copyright 2004
American Mathematical Society
