Inside Dynamics of Delayed Traveling Waves

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 ∂ t u ( t,x ) =  ∂ xx u ( 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 ∂ 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 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.