  The amplification of an external signal is a key step in direction sensing of
biological cells. We consider a simple model for the response to a
time-depending signal, which was previously proposed by the last three authors.
The model consists of a bulk-surface reaction-diffusion model. We prove that in
a suitable asymptotic limit the system converges to a bulk-surface parabolic
obstacle type problem. For this model and a reduction to a nonlocal surface
equation we show an L1-contraction property and, in the case of time-constant
signals, the stability of stationary states.
