We consider a Lotka-Volterra system of two competing species subject to multiplicative α-stable Lévy noise. The interaction parameter between the species is a random process which obeys a stochastic differential equation with a generalized bistable potential in the presence both of a periodic driving term and an additive α-stable Lévy noise. We study the species dynamics, which is characterized by two different regimes, exclusion of one species and coexistence of both. We find quasiperiodic oscillations and stochastic resonance phenomenon in the dynamics of the competing species, analyzing the role of the Lévy noise sources.