Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-12T23:34:35.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave-like Solutions for Nonlocal Reaction-diffusion Equations: a Toy Model

Published online by Cambridge University Press:  12 June 2013

G. Nadin ,
L. Rossi ,
L. Ryzhik  and
B. Perthame
G. Nadin*
Affiliation:
Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 6 and CNRS UMR 7598, F-75005, Paris
L. Rossi
Affiliation:
Dipartimento di Matematica, Università degli Studi di Padova
L. Ryzhik
Affiliation:
Department of Mathematics, Stanford University, Stanford CA 94305
B. Perthame
Affiliation:
Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 6 and CNRS UMR 7598, F-75005, Paris
*
⋆ Corresponding author. E-mail: nadin@ann.jussieu.fr
Get access

Abstract

Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviour than for the usual Fisher equation. A striking numerical observation is that a traveling wave with minimal speed can connect a dynamically unstable steady state 0 to a Turing unstable steady state 1, see [12]. This is proved in [1, 6] in the case where the speed is far from minimal, where we expect the wave to be monotone.

Here we introduce a simplified nonlocal Fisher equation for which we can build simple analytical traveling wave solutions that exhibit various behaviours. These traveling waves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connect these two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. The latter exist in a regime where time dynamics converges to another object observed in [3, 8]: a wave that connects 0 to a pulsating wave around 1.

Keywords

traveling wavewavetrainsnonlocal elliptic equations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 33 - 41
Copyright
© EDP Sciences, 2013

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.)

References

Alfaro, M., Coville, J.. Rapid traveling waves in the nonlocal Fisher equation connect two unstable states. Appl. Math. Lett., 25:2095–2099, 2012. CrossRefGoogle Scholar
Apreutesei, N., Bessonov, N., Volpert, V., Vougalter, V.. Spatial structures and generalized travelling waves for an integro-differential equation. Disc. Cont. Dyn. Syst. B, 13(3):537–557, 2010. CrossRefGoogle Scholar
Berestycki, H., Nadin, G., Perthame, B., Ryzhik, L.. The non-local Fisher-KPP equation: traveling waves and steady states. Nonlinearity, 22(12):2813–2844, 2009. CrossRefGoogle Scholar
Britton, N.. Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math., 50(6):1663–1688, 1990. CrossRefGoogle Scholar
Doelman, A., Sandstede, B., Scheel, A., Schneider, G.. The dynamics of modulated wave trains. Mem. Amer. Math. Soc., 199(934), 2009. Google Scholar
Fang, J., Zhao, X-Q.. Monotone wavefronts of the nonlocal Fisher-KPP equation. Nonlinearity, 24(11):3043–3054, 2011. CrossRefGoogle Scholar
Furter, J-É, Grinfeld, M.. Local vs. nonlocal interactions in population dynamics. J. Math. Biol., 27(1):65–80, 1989. CrossRefGoogle Scholar
Genieys, S., Volpert, V., Auger, P.. Pattern and waves for a model in population dynamics with nonlocal consumption of resources. Math. Modelling Nat. Phenom., 1:65–82, 2006. Google Scholar
Gomez, A., Trofimchuk, S.. Monotone traveling wavefronts of the KPP-Fisher delayed equation. J. Diff. Eq., 250(4):1767–1787, 2011. CrossRefGoogle Scholar
Gourley, S.. Traveling front solutions of a nonlocal Fisher equation. J. Math. Biol., 41(3):272–284, 2000. CrossRefGoogle Scholar
Kwong, M.K., Ou, C.. Existence and nonexistence of monotone traveling waves for the delayed Fisher equation. J. Diff. Eq., 249(3):728–745, 2010. CrossRefGoogle Scholar
Nadin, G., Perthame, B., Tang, M.. Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. C. R. Math. Acad. Sci. Paris, 349(9-10):553–557, 2011. CrossRefGoogle Scholar
Turing, A.. The chemical basis of morphogenesis. Phil. Trans. Royal Soc. London. Serie B, Biol. Sc., 237(641):37–72, 1952. CrossRefGoogle Scholar