International audienc

We investigate the inside structure of one-dimensional reaction-diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established. RésuméOn s'intéresse à la structure interne des fronts progressifs de réaction-diffusion en dimension 1. Les termes de réaction sont du type monostable, bistable ou ignition. Les fronts étant décomposés en une somme de composantes ayant des taux de diffusion et de croissance identiques, nous analysons dans cet article les propriétés d'expansion de chaque composante. Dans le cas monostable, il est connu que les fronts peuvent être classés en fronts tirés et poussés, suivant leur vitesse de propagation. On montre que chaque composante localisée d'un front tiré converge vers 0 en temps grand localement dans le repère du front, alors que chaque composante d'un front poussé converge vers une proportion strictement positive du front dans le repère mobile. Ces résultats fournissent une interprétation nouvelle et plus complète de la terminologie « fronts tirés -fronts poussés », qui étend les définitions antérieures au cas de fronts généralisés de transition. Pour des non-linéarités du type bistable ou ignition, on démontre que les fronts sont poussés, au sens qu'ils vérifient les mêmes propriétés que les fronts monostables critiques poussés. On établit également des résultats de convergence uniforme et des estimations précisées des vitesses d'expansion à gauche et à droite des composantes des fronts tirés et poussés.

We investigate spreading properties of solutions of a large class of twocomponent reaction-diffusion systems, including prey-predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known -at least theoretically -about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution -one for the prey and the other for the predatorin some situations.

We consider a bistable (0<θ<1 being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in +∞. Combining refined a priori estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in −∞ to ‘something’ which is strictly above the unstable equilibrium θ in +∞. Furthermore, we present situations (additional bound on the non‐linearity or small delay) where the wave converges to 1 in +∞, whereas the wave is shown to oscillate around 1 in +∞ when, typically, the delay is large.