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Inside Dynamics of Delayed Traveling Waves

Published online by Cambridge University Press:  12 June 2013

O. Bonnefon ,
J. Garnier ,
F. Hamel  and
L. Roques
O. Bonnefon
Affiliation:
UR 546 Biostatistique et Processus Spatiaux, INRA, 84000 Avignon, France
J. Garnier
Affiliation:
UR 546 Biostatistique et Processus Spatiaux, INRA, 84000 Avignon, France Aix Marseille Université, CNRS, Centrale Marseille, LATP, UMR 7353 , 13453 Marseille, France
F. Hamel
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, LATP, UMR 7353 , 13453 Marseille, France Institut Universitaire de France
L. Roques*
Affiliation:
UR 546 Biostatistique et Processus Spatiaux, INRA, 84000 Avignon, France
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

The notion of inside dynamics of traveling waves has been introduced in the recent paper[14]. Assuming that a traveling waveu(t,x) = U(x − c   t)is made of several components υi ≥ 0(i ∈ I ⊂ N), the inside dynamics of the wave is thengiven by the spatio-temporal evolution of the densities of the componentsυi. For reaction-diffusion equations of theform∂tu(t,x) = ∂xxu(t,x) + f(u(t,x)),where f is of monostable or bistable type, the results in [14] show that traveling waves can be classified intotwo main classes: pulled waves and pushed waves. Using the same framework, we study thepulled/pushed nature of the traveling wave solutions of delay equations

            ∂tu(t,x) = ∂xxu(t,x) + F(u(t −τ,x),u(t,x))

We begin with areview of the latest results on the existence of traveling wave solutions of suchequations, for several classical reaction terms. Then, we give analytical and numericalresults which describe the inside dynamics of these waves. From a point of view ofpopulation ecology, our study shows that the existence of a non-reproductive andmotionless juvenile stage can slightly enhance the genetic diversity of a speciescolonizing an empty environment.

Keywords

traveling wavesdelayed equationpulled and pushed solutionsKobayashi’s equationHutchinson’s equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 42 - 59
Copyright
© EDP Sciences, 2013

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