<p>This work is to analyze the effects of massive and sudden environmental fluctuations and seasonal variation in the evolution of predator–prey species, which is modeled by a stochastic predator–prey model with Lévy noises and time-periodic coefficients. Firstly, the unique global positive solution of the model is established, and the stochastically ultimate boundedness of the solution is studied. Secondly, the threshold conditions for the time-average persistence in probability and extinction of the populations are derived by comparing the solution near the boundary and some stochastic estimates. Moreover, by using the properties of periodic Markov processes and Krylov–Bogolyubov’s method, we show that there exists a periodic measure in the model provided that it is time-average persistent in probability. To ensure the uniqueness of the periodic measure, the strong Feller property and irreducibility for the two-parameter Markov semigroup are proved under the non-degeneracy assumption. Finally, we give several examples and numerical simulations to verify the effectiveness of the theoretical results and investigate the influences of small and large jump noises on the dynamics of the model.</p>