A generalized parabolic partial differential equation is studied with systems of impulse functions as the coefficients of the nonlinear term (varying medium). The latter are interpreted as moving point catalysts in a reaction-diffusion system. The existence of a unique solution in an appropriate space of continuous functions is established (initial value problem). The particular case where catalysts are given by the weighted atoms of a stable random measure and migrate according to one-dimensional stable stochastic processes is discussed in detail. Solutions to this nonlinear equation are constructed in a random medium that are homogeneous (in distribution) and ergodic with respect to the spatial shift