Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-31T03:46:22.802Z Has data issue: false hasContentIssue false
  • English
  • Français

Travelling Waves in Near-Degenerate Bistable CompetitionModels

Published online by Cambridge University Press:  27 July 2010

E.O. Alzahrani ,
F.A. Davidson  and
N. Dodds
Get access

Abstract

We study a class of bistable reaction-diffusion systems used to model two competingspecies. Systems in this class possess two uniform stable steady states representingsemi-trivial solutions. Principally, we are interested in the case where the ratio of thediffusion coefficients is small, i.e. in thenear-degenerate case. First, limiting arguments are presented to relatesolutions to such systems to those of the degenerate case where one species is assumed notto diffuse. We then consider travelling wave solutions that connect the two stablesemi-trivial states of the non-degenerate system. Next, a general energy function for thefull system is introduced. Using this and the limiting arguments, we are able to determinethe wave direction for small diffusion coefficient ratios. The results obtained onlyrequire knowledge of the system kinetics.

Keywords

competitionreaction-diffusionfree energybistabletravelling waves

Information

Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 5: Reaction-diffusion waves , 2010 , pp. 13 - 35
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

R. S. Cantrell, C. Cosner. Spatial Ecology via Reaction-Diffusion Equations. John Wiley and Sons Ltd, New York, 2003.
L.C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, 2010.
P. Grindrod. The Theory and Applications of Reaction-Diffusion Equations: Patterns and Waves. Clarendon Press, Oxford, 1996.
Guo, J.S., Tsai, J.. The asymptotic behavior of solutions of the buffered bistable system. J. Math. Biol., 53 (2006), No. 1, 179–213.CrossRefGoogle ScholarPubMed
Heinze, S., Schweizer, B.. Creeping fronts in degenerate reaction-diffusion systems. Nonlinearity, 18 (2005), No. 6, 2455–2476.CrossRefGoogle Scholar
S. Heinze, B. Schweizer, H. Schwetlick.Existence of front solutions in degenerate reaction diffusion systems. Preprint 2004-03, SFB 359, University of Heidelberg, 2004.
Hosono, Y.. Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competition models. In Numerical and Applied Mathematics (Paris1989) IMACS Ann. Comput. Appl. Math., (1989), No 2., 687–692. Google Scholar
Hosono, Y., Mimura, M.. Singular perturbation approach to traveling waves in competing and diffusing species models. J. Math. Kyoto University, 22 (1982), No. 3, 435–461.CrossRefGoogle Scholar
Kazmierczak, B., Volpert, V.. Travelling waves in partially degenerate reaction-diffusion systems. Mathematical Modelling of Natural Phenomena, 2 (2007), No. 2, 106–125.CrossRefGoogle Scholar
Kazmierczak, B., Volpert, V.. Calcium waves in systems with immobile buffers as a limit of waves for systems with nonzero diffusion. Nonlinearity, 21 (2008), No. 1, 71–96.CrossRefGoogle Scholar
Kazmierczak, B., Volpert, V.. Mechano-chemical calcium waves in systems with immobile buffers. Archives of Mechanics, 60 (2008), No. 1, 3–22.Google Scholar
Lloyd, D.J. B., Sandstede, B., Avitabile, D., Champneys, A.R.. Localized hexagon patterns of the planar Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 7 (2008), No. 3, 1049–1100.CrossRefGoogle Scholar
J.D. Murray. Mathematical Biology, II: Spatial Models and Biomedical Applications, volume 2. Springer-Verlag, Berlin, 2003.
A. Okubo, S.A. Levin. Diffusion and Ecological Problems: Modern Perspectives. Springer-Verlag, New York, 2001.
J.A. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, Berlin, 1994.
Tsai, J.C., Sneyd, J.. Existence and stability of traveling waves in buffered systems. SIAM J. Applied Math., 66 (2005), No. 1, 237–265.CrossRefGoogle Scholar
A.I. Volpert, V.A. Volpert, V.A. Volpert. Traveling Wave Solutions of Parabolic Systems: Translations of Mathematical Monographs, volume 140. American Mathematical Society, Providence, R.I., 1994.