I am currently working on the diffusive aspect of the reaction-diffusion equations on the field-road space ℝ^{N-1}×ℝ_{+} with N≥2 (introduced by H.Berestycki, J.-M.Roquejoffre and L.Rossi in 2012). More precisely, I am interesting in the solution (u,v) = ( u(t,x) , v(t,x,y) ) of the following problem:

starting from an initial datum (u_{0},v_{0}) = ( u_{0}(x) , v_{0}(x,y) ) bounded and integrable (a step for example).

One calls "field the domain ℝ^{N-1}×ℝ_{+} and "road" the field's boundary, ie. ℝ^{N-1}×{0}.

The function v stands for the individuals density on the field while u represents the individuals density on the road.

Individuals diffuse with a coefficient d on the field and with a coefficient D on the road with, typically, D>d.

Second line of the previous system is called exchange condition. This boundary condition is of sourced-Robin type and is in the core of the model since it allows to manage the behavior of individuals which arrive at the boundary of the field (bouncing or exiting towards the road) and immigration from the road.

Taking the associated linear system, ie. f≡0, one may show that the mass of the population is preserved over time; we thus get a purely diffusive system.

Knowing that the action of the diffusion in the whole space ℝ^{N} on a reasonable datum (bounded and integrable) provokes the extinction of the population by spreading individuals (local uniform convergence of the solution toward 0 as t→+∞), one wonders to what extend the inclusion of the road (ie. a fast diffusion channel) may impact the rate of population extinction.

The simulations here based on microscopic indivual motions are a good introduction to well understand the dynamic of the field-road model.