Travelling Waves in Near-Degenerate Bistable Competition Models

E.O. Alzahrani; F.A. Davidson; N. Dodds

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 5, page 13-35
  • ISSN: 0973-5348

Abstract

top
We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.

How to cite

top

Alzahrani, E.O., Davidson, F.A., and Dodds, N.. "Travelling Waves in Near-Degenerate Bistable Competition Models." Mathematical Modelling of Natural Phenomena 5.5 (2010): 13-35. <http://eudml.org/doc/197697>.

@article{Alzahrani2010,
abstract = {We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.},
author = {Alzahrani, E.O., Davidson, F.A., Dodds, N.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {competition; reaction-diffusion; free energy; bistable; travelling waves; asymptotic boundary conditions; wave direction},
language = {eng},
month = {7},
number = {5},
pages = {13-35},
publisher = {EDP Sciences},
title = {Travelling Waves in Near-Degenerate Bistable Competition Models},
url = {http://eudml.org/doc/197697},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Alzahrani, E.O.
AU - Davidson, F.A.
AU - Dodds, N.
TI - Travelling Waves in Near-Degenerate Bistable Competition Models
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/7//
PB - EDP Sciences
VL - 5
IS - 5
SP - 13
EP - 35
AB - We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.
LA - eng
KW - competition; reaction-diffusion; free energy; bistable; travelling waves; asymptotic boundary conditions; wave direction
UR - http://eudml.org/doc/197697
ER -

References

top
  1. R. S. Cantrell, C. Cosner. Spatial Ecology via Reaction-Diffusion Equations. John Wiley and Sons Ltd, New York, 2003.  
  2. L.C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, 2010.  
  3. P. Grindrod. The Theory and Applications of Reaction-Diffusion Equations: Patterns and Waves. Clarendon Press, Oxford, 1996.  
  4. J.S. Guo, J. Tsai. The asymptotic behavior of solutions of the buffered bistable system. J. Math. Biol., 53 (2006), No. 1, 179–213. 
  5. S. Heinze, B. Schweizer. Creeping fronts in degenerate reaction-diffusion systems. Nonlinearity, 18 (2005), No. 6, 2455–2476. 
  6. S. Heinze, B. Schweizer, H. Schwetlick.Existence of front solutions in degenerate reaction diffusion systems. Preprint 2004-03, SFB 359, University of Heidelberg, 2004.  
  7. Y. Hosono. 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.  
  8. Y. Hosono, M. Mimura. Singular perturbation approach to traveling waves in competing and diffusing species models. J. Math. Kyoto University, 22 (1982), No. 3, 435–461. 
  9. B. Kazmierczak, V. Volpert. Travelling waves in partially degenerate reaction-diffusion systems. Mathematical Modelling of Natural Phenomena, 2 (2007), No. 2, 106–125. 
  10. B. Kazmierczak, V. Volpert. Calcium waves in systems with immobile buffers as a limit of waves for systems with nonzero diffusion. Nonlinearity, 21 (2008), No. 1, 71–96. 
  11. B. Kazmierczak, V. Volpert. Mechano-chemical calcium waves in systems with immobile buffers. Archives of Mechanics, 60 (2008), No. 1, 3–22. 
  12. D.J. B. Lloyd, B. Sandstede, D. Avitabile, A.R. Champneys. Localized hexagon patterns of the planar Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 7 (2008), No. 3, 1049–1100. 
  13. J.D. Murray. Mathematical Biology, II: Spatial Models and Biomedical Applications, volume 2. Springer-Verlag, Berlin, 2003.  
  14. A. Okubo, S.A. Levin. Diffusion and Ecological Problems: Modern Perspectives. Springer-Verlag, New York, 2001.  
  15. J.A. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, Berlin, 1994.  
  16. J.C. Tsai, J. Sneyd. Existence and stability of traveling waves in buffered systems. SIAM J. Applied Math., 66 (2005), No. 1, 237–265. 
  17. 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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.