This thesis focuses on the demonstration of the existence of and analysis of phenomena related to the discontinuous non-equilibrium phase transition between an active (or reactive) state and a poisoned (or extinguished) state occurring in a stochastic lattice-gas realization of Schloegl's second model for autocatalysis. This realization, also known as the Quadratic Contact Process, involves spontaneous annihilation, autocatalytic creation, and diffusion or hopping of particles on a square lattice, where creation at empty sites requires a suitable nearby pair of particles. The poisoned state exists for all annihilation rates p>0 and is an absorbing particle-free "vacuum" state. The populated active steady state exists just for p below a critical value, pe. Our initial studies without particle hopping demonstrated a dependence on orientation or slope, S, of the equistability or stationary point p=peq(S) for a planar interface separating active and poisoned states. They also showed that pe = peq(S=1) for diagonal interfaces. This orientation dependence was shown to extend to non-zero hop rates h>0, but to quickly weaken with increasing h. If pf denotes the critical value below which a finite population can survive, then we show that pf =peq(S=0) and that pf < pe. This strict inequality contrasts a postulate of Durrett, and is a direct consequence of the occurrence of coexisting stable active and poisoned states for a finite range pf y p y pe. Although this so-called generic two-phase coexistence (2PC) contrasts behavior in thermodynamic systems. However, one still finds metastability and nucleation phenomena similar to those in discontinuous equilibrium transitions. We also provide a theoretical framework for analysis of such metastability phenomena. Extensions of the basic model are considered to different lattices and to introduce tricritical behavior. Most precise analysis was performed with kinetic Monte Carlo simulation. However, we also developed exact hierarchical master equations and performed approximate truncation analysis of these equations.