Hostname: page-component-7b9c58cd5d-6tpvb Total loading time: 0 Render date: 2025-03-15T12:12:13.952Z Has data issue: false hasContentIssue false
  • English
  • Français

Travelling waves for diffusive and strongly competitive systems: Relative motility and invasion speed

Published online by Cambridge University Press:  08 May 2015

LÉO GIRARDIN  and
GRÉGOIRE NADIN
LÉO GIRARDIN
Affiliation:
École Normale Supérieure de Cachan, France email: leo.girardin@ens-cachan.fr
GRÉGOIRE NADIN
Affiliation:
Laboratoire Jacques-Louis Lions, CNRS, Université Paris 6, France email: nadin@ann.jussieu.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our interest here is to find the invader in a two species, diffusive and competitive Lotka–Volterra system in the particular case of travelling wave solutions. We investigate the role of diffusion in homogeneous domains. We might expect a priori two different cases: strong interspecific competition and weak interspecific competition. In this paper, we study the first one and obtain a clear conclusion: the invading species is, up to a fixed multiplicative constant, the more diffusive one.

Keywords

travelling wavesreaction-diffusion systemwave speedcompetitionsegregation phenomena
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 4 , August 2015 , pp. 521 - 534
Copyright
Copyright © Cambridge University Press 2015 

References

[1]Berestycki, H. (1981) Le nombre de solutions de certains problèmes semi-linéaires elliptiques. J. Funct. Anal. 40 (1), 1–29.CrossRefGoogle Scholar
[2]Conti, M., Verzini, G. & Terracini, S. (2005) A regularity theory for optimal partition problems. In: SPT 2004—Symmetry and Perturbation Theory, World Sci. Publ., Hackensack, pp. 91–98.CrossRefGoogle Scholar
[3]Conti, M., Terracini, S. & Verzini, G. (2005) A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. 54 (3), 779–815.CrossRefGoogle Scholar
[4]Crooks, E. C. M., Dancer, E. N., Hilhorst, D., Mimura, M. & Ninomiya, H. (2004) Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions. Nonlinear Anal. Real World Appl. 5 (4), 645–665.CrossRefGoogle Scholar
[5]Dancer, E. N., Hilhorst, D., Mimura, M. & Peletier, L. A. (1999) Spatial segregation limit of a competition-diffusion system. Eur. J. Appl. Math. 10 (2), 97–115.CrossRefGoogle Scholar
[6]Dockery, J., Hutson, V., Mischaikow, K. & Pernarowski, M. (1998) The evolution of slow dispersal rates: A reaction diffusion model. J. Math. Biol. 37 (1), 61–83.CrossRefGoogle Scholar
[7]Du, Y. & Lin, Z. (2010) Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377–405.CrossRefGoogle Scholar
[8]Gardner, R. A. (1982) Existence and stability of travelling wave solutions of competition models: A degree theoretic approach. J. Differ. Equ. 44 (3), 343–364.CrossRefGoogle Scholar
[9]Huang, W. & Han, M. (2011) Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model. J. Differ. Equ. 251 (6), 1549–1561.CrossRefGoogle Scholar
[10]Kan-On, Y. (1995) Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. SIAM J. Math. Anal. 26 (2), 340–363.CrossRefGoogle Scholar
[11]Kolmogorov, A., Petrovsky, I. & Piscounov, N. (1937) Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin Université d'Etat à Moscou (Bjul. Moskovskogo Gos. Univ.) 1 (1), 1–26.Google Scholar
[12]Lewis, M. A., Li, B. & Weinberger, H. F. (2002) Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45 (3), 219–233.CrossRefGoogle ScholarPubMed
[13]Nakashima, K. & Wakasa, T. (2007) Generation of interfaces for Lotka–Volterra competition–diffusion system with large interaction rates. J. Differ. Equ. 235 (2), 586–608.CrossRefGoogle Scholar
[14]Quitalo, V. (2013) A free boundary problem arising from segregation of populations with high competition. Arch. Ration. Mech. Anal. 210 (3), 857–908.CrossRefGoogle Scholar