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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
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Abstract

Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviourthan for the usual Fisher equation. A striking numerical observation is that a travelingwave with minimal speed can connect a dynamically unstable steady state 0 to a Turingunstable steady state 1, see [12]. This is provedin [1, 6] inthe 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 simpleanalytical traveling wave solutions that exhibit various behaviours. These travelingwaves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connectthese two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. Thelatter exist in a regime where time dynamics converges to another object observed in[3, 8]: awave 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

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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