Analysis on stochastic delay Lotka–Volterra systems driven by Lévy noise

In this paper, two types of stochastic Lotka–Volterra systems (i.e., competition system and predator–prey system) with time delays and Lévy noise are discussed. For each model, we establish sufficient and necessary criteria for stability in time average and extinction of each population. The thresholds between stability in time average and extinction of each model are obtained. Some recent results are improved and extended significantly. These results point out that: (i) time delays are harmless for stability in time average and extinction of our model; (ii) both stability in time average and extinction have close relationships with Lévy noise; (iii) the interaction rates play significant roles in determining the stability in time average and extinction of the populations.