Inside Dynamics of Delayed Traveling Waves

Abstract

The notion of inside dynamics of traveling waves has been introduced in the recent paper [14]. Assuming that a traveling 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 ∂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 into two main classes: pulled waves and pushed waves. Using the same framework, we study the pulled/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 a review of the latest results on the existence of traveling 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 non-reproductive and motionless juvenile stage can slightly enhance the genetic diversity of a species colonizing an empty environment.

Speed Selection and Stability of Wavefronts for Delayed Monostable Reaction-Diffusion Equations

“…Corollary 1 Assume that the initial function w 0 satisfies the hypotheses (IC1), (IC2) and (4). Suppose first that λ > λ * and c * > c # , then the solution of (1),…”

“…The latter condition, however, excludes a subclass of equations (1) possessing so called pushed minimal traveling fronts [35,43]. Since pushed wavefronts are quite interesting from both applied [11,33] and mathematical [4,12,15,17,34,35,41,43] points of view, their existence, uniqueness and stability properties in the case of delayed monotone model (1) were recently considered in [17,40,43]. Particularly, the existence of the minimal speed of front propagation c * was proved in [17,43] (if g is neither monotone nor subtangential at 0, the existence of c * is an important open problem).…”

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

“…In consequence, we are interested only in nonstationary wavefronts and will consider speed c = 0. Another well studied particular case of (2) is when the nonlinearity g(u, v) is non-decreasing in v for each fixed u [7,12,14,22,24,27,28,32]. Indeed, this kind of monotonicity allows a successful application of the maximum principle and comparison techniques.…”

“…Since c < 0, this implies that φ(s) ∈ (e 1 , e 2 ), φ(s − cτ ) ∈ (κ, e 2 ) so that sup S =: s 0 ∈ S is finite. After differentiating (7) at s 0 , we obtain the following…”

Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction

“…Under this assumption, all wavefronts of equation (1) are known as 'pulled' fronts (see [5,14,32,33,34,39,47] for further details), model (1) is linearly determined [19,46] and there exists a positive number c * > 0 (called the minimal speed of propagation) separating the positive axis on the set of admissible semi-wavefronts speeds [c * , +∞) and the set [0, c * ) of velocities c for which does not exist any non-constant positive bounded wave solution u(t, x) = φ(x + ct) [15]. Furthermore, the minimal speed c * is determined from the characteristic equation…”