Asymptotic behaviour of a nonlinear coupled system for uptake processes with metabolic competition phenomena

In this paper, we analyze the steady state uptake and coupled diffusion-reaction of two metabolites such as, for instance, glucose and galactose, within a single enterocyte. The considered mathematical model consists of a non-linear coupled system posed in a domain with a highly oscillating boundary, whose boundary oscillation has a ε-periodicity in its first variable but fixed amplitude. At the microscopic scale, the boundary conditions encode a competitive interaction between the two metabolites for the same “carrier” influenced by the oscillation amplitude through a parameter k ≥ 1. Using the periodic unfolding operator in a comb-shaped domain and some monotonicity techniques, we study the asymptotic behaviour of the inhibition phenomena that occur in the uptake process, when ε tends to zero. Depending on the values of the parameter k in the surface reaction term, two different limit regimes arise: for k > 1, the system asymptotically decouples into separate systems of equations for each metabolite, without any trace of their competition. The most relevant case is k=1, where the limit problem remains coupled through a lower-order term describing the effect of competition/inhibition between the two metabolites.