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 υi ≥ 0
(i ∈ I ⊂ N), the inside dynamics of the wave is then
given by the spatio-temporal evolution of the densities of the components
υi. 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.