  Our study focuses on analyzing the behavior of a stochastic predator-prey
model with a time delay and logistic growth of prey, influenced by L\'{e}vy
noise. Initially, we establish the existence, uniqueness, and boundedness of a
positive solution that spans globally. Subsequently, we explore the conditions
under which extinction occurs, and identify adequate criteria for persistence.
Finally, we validate our theoretical findings through numerical simulations,
which also helps illustrate the dynamics of the stochastic delayed
predator-prey model based on different criteria.

Dynamical Systems;Probability