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Journal Articles Math. Mod. Nat. Phen. Year : 2013

Inside dynamics of delayed travelling waves

Abstract

The notion of inside dynamics of travelling waves has been introduced in the recent paper of Garnier et al. (2012). Assuming that a travelling wave u(t, x) = U (x − c t) is made of several components υ i ≥ 0 (i ∈ I ⊂ N), the inside dynamics of the wave is then given by the spatio-temporal evolution of the densities of the components υ i . For reaction-diffusion equations of the form ∂t u(t, x) = ∂xx u(t, x) + f (u(t, x)), where f is of monostable or bistable type, the results in (Garnier et al., 2012) show that travelling waves can be classified into two main classes: pulled waves and pushed waves. Using the same framework, we study the pulled/pushed nature of the travelling wave solutions of delay equations ∂t u(t, x) = ∂xx u(t, x) + F (u(t − τ, x), u(t, x)). We begin with a review of the latest results on the existence of travelling wave solutions of such equations, for several classical reaction terms. Then, we give analytical and numerical results which describe the inside dynamics of these waves. From a point of view of population ecology, our study shows that the existence of a juvenile stage can slightly enhance the genetic diversity of a species colonizing an empty environment.
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Dates and versions

hal-00788675 , version 1 (15-02-2013)

Identifiers

  • HAL Id : hal-00788675 , version 1

Cite

Olivier Bonnefon, Jimmy Garnier, Francois Hamel, Lionel Roques. Inside dynamics of delayed travelling waves. Math. Mod. Nat. Phen., 2013, 8, pp.44--61. ⟨hal-00788675⟩
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