Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type

Arnaud Ducrot (1) , Thomas Giletti (2) , Hiroshi Matano (3)

1 LMAH - Laboratoire de Mathématiques Appliquées du Havre
2 IECL - Institut Élie Cartan de Lorraine
3 MIMS - Meiji Institute for Advanced Study of Mathematical Sciences

We investigate spreading properties of solutions of a large class of two-component 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 predator—in some situations.

Mots clés: Long time behaviour; spreading speeds; reaction-diffusion; prey-predator systems

Domaines: Equations aux dérivées partielles [math.AP]

Arnaud Ducrot, Thomas Giletti, Hiroshi Matano. Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type. Calculus of Variations and Partial Differential Equations, 2019, 58 (4), pp.137. ⟨10.1007/s00526-019-1576-2⟩. ⟨hal-02169246⟩