A viscosity solution method for the spreading speed formula in slowly varying media

International audienceIn this paper, we consider reaction-diffusion-advection equations in slowly periodi-cally oscillating media. We prove the existence of and we give explicit expressions of the asymptotic spreading speeds of invasion of the unstable state 0 in any direction, when the period of the invaded medium becomes infinitely large. The limiting sprea-ding speeds involve families of 1-periodic Hamilton-Jacobi equations. In the case of one-dimensional reaction-diffusion equations, we analyze the relative effects of small perturbations of the diffusion and the reaction coefficients, and we compare the spreading speeds in slowly oscillating media to the homogenized spreading speeds in rapidly oscillating media

A KPP road-field system with spatially periodic exchange terms

We take interest in a reaction-diffusion system which has been recently proposed [11] as a model for the effect of a road on propagation phenomena arising in epidemiology and ecology. This system consists in coupling a classical Fisher-KPP equation in a half-plane with a line with fast diffusion accounting for a straight road. The effect of the line on spreading properties of solutions (with compactly supported initial data) was investigated in a series of works starting from [11]. We recover these earlier results in a more general spatially periodic framework by exhibiting a threshold for road diffusion above which the propagation is driven by the road and the global speed is accelerated. We also discuss further applications of our approach, which will rely on the construction of a suitable generalized principal eigenvalue, and investigate in particular the spreading of solutions with exponentially decaying initial data.Comment: Updated version, minor typos and details fixe

Existence of multi-dimensional pulsating fronts for KPP equations: a new formulation approach

This paper is concerned with the existence of pulsating travelling fronts for a KPP reaction-diffusion equation posed in a multi-dimensional periodic medium. We provide an alternative proof of the classic existence result. Our proof relies largely on the construction of a wave profile under a moving frame, which avoids many technical difficulties in dealing with degenerate elliptic equations. Intriguingly, our analysis also yields that the profile of the front propagating along each rational direction in $\mathbb{S}^{N-1}$ is periodic in time