Schloegl's second model (also known as the quadratic contact process) on a lattice involves spontaneous particle annihilation at rate p and autocatalytic particle creation at empty sites with n≥2 occupied neighbors. The particle creation rate for exactly n occupied neighbors is selected here as n(n−1)/[z(z−1)] for lattice coordination number z. We analyze this model on a Bethe lattice. Precise behavior for stochastic models on regular periodic infinite lattices is usually surmised from kinetic Monte Carlo simulation on a finite lattice with periodic boundary conditions. However, the persistence of boundary effects for a Bethe lattice complicates this process, e.g., by inducing spatially heterogenous states. This motivates the exploration of various boundary conditions and unconventional simulation ensembles on the Bethe lattice to predict behavior for infinite size. We focus on z=3, and predict a discontinuous transition to the vacuum state on the infinite lattice when p exceeds a threshold value of around 0.053.