The number and stability of stationary and periodic solutions in a semi- phenomenological reaction mechanism that qualitatively models the Zhabotinskii- Belousov reaction are studied as functions of parameters in the model. Both multiple stationary solutions and multiple periodic solutions can exist simultaneously. Periodic solutions bifurcate in one of three ways: at zero amplitude as predicted from the Hopf theorem; pairwise at finite amplitude, the so-called "hard" bifurcation; or by coalescense with separatrix loops. Detailed computations for several cases reveal the dependence of the period and amplitude of the periodic solutions on the parameters. The results show that the model exhibits all the qualitative features that might be expected in intracellular reactions and so can serve as a model system for theoretical studies of pattern formation in developing systems.