On some positive-feedback systems with different boundary conditions

A two-dimensional, positive-feedback system with diffusion is investigated, the two components being subject to different boundary conditions. For the onset of a non-negative, non-trivial stationary solution (which is proved to be unique) a necessary and sufficient condition is established, which is expressed in terms of a certain parameter. While the latter are, in general, difficult to compute exactly, in the one-dimensional case an explicit computation of the critical value of the parameter is provided.