We consider an interacting particle system which models the sterile insect technique on a finite cylinder. When in contact with slow reservoirs , the hydrodynamic limit of the latter system is a set of coupled reaction diffusion equations with mixed boundary conditions, boundary conditions which depends on the slow-down rates of the reservoirs. The goal of this talk is to prove the existence of a hydrostatic limit for a class of parameters. In other words, starting from a microscopic equilibrium, the system will evolve  and converge to a macroscopic equilibrium, corresponding to the unique stationary solution of the hydrodynamic equation.