During the operation, the pulse detonation engine (PDE) may have to work with different fuel/air equivalence ratios (from a lean to rich fuel mixture) as in the cold start-up operation, which would strongly affect the engine performance characteristics and the outputs of interest. For a better operational control, it is necessary to gain understanding of the effect of the equivalence ratio on the dynamics of the processes of PDE. Thus, in this study, numerical simulations are performed for different ethylene fuel/air equivalence ratios to study its effect on the dynamics of the deflagration-to-detonation transition (DDT) and detonation processes. In particular, the density-based solver with a shock-capturing scheme is employed to solve for the viscous, compressible, and reacting flows governed by reacting Navier-Stokes equations. The computed flame propagation speed, run-up distance, and Chapman-Jouguet detonation velocity are comparable to the current experimental results. In addition, the numerical results show that the minimum values of the run-up distance and run-up time, as well as the maximum value of the detonation velocity occur at the equivalence ratio of about 1.1. Analysis of the computed results associate these findings to the firm correlation of the flame speed with equivalence ratio, which is in turn function of the temperature, pressure, and mixture composition. The shifting of the outputs of interest to the richer fuel side from the stoichiometric point can be attributed to the combustion product dissociation, mixture heat capacity, and oxidizer enrichment.