We are interested in the dynamical behaviour of the solution set to a two component reaction–diffusion system posed on non coincident spatial do- mains. The underlying biological problem is a predator–prey system featuring a non local numerical response to predation involving an integral kernel. Quite interesting while complex dynamics emerge from preliminary numerical simu- lations, driven both by diffusivities and by the parametric form or shape of the integral kernel. We consider a simplified version of this problem, with constant coefficients, and give some hints on the large time dynamics of solutions.