  We consider an interacting particle system which models the sterile insect
technique. It is the superposition of a generalized contact process with
exchanges of particles on a finite cylinder with open boundaries (see Kuoch et
al., 2017). We show that when the system is in contact with reservoirs at
different slowdown rates, the hydrodynamic limit is a set of coupled non linear
reaction-diffusion equations with mixed boundary conditions. We also prove the
hydrostatic limit when the macroscopic equations exhibit a unique attractor.
