Effective transmission conditions for reaction-diffusion processes in domains separated by thin channels

We consider a reaction–diffusion equation in a domain consisting of two bulk regions connected via small channels periodically distributed within a thin layer. The height and the thickness of the channels are of order $ϵ$, and the equation inside the layer depends on the parameter $ϵ$ and an additional parameter $\gamma \in [-1,1)$, which describes the size of the diffusion in the layer. We derive effective models for the limit $\epsilon \to 0 $, when the channel-domain is replaced by an interface between the two bulk-domains.