  In this paper we prove a weak comparison principle for a reaction-diffusion
system without uniqueness of solutions. We apply the abstract results to the
Lotka-Volterra system with diffusion, a generalized logistic equation and to a
model of fractional-order chemical autocatalysis with decay. Morever, in the
case of the Lotka-Volterra system a weak maximum principle is given, and a
suitable estimate in the space of essentially bounded functions $L^{\infty}$ is
proved for at least one solution of the problem.
