Optimal Harvesting for a Stochastic Predator-prey Model with S-type Distributed Time Delays

Global dynamics and optimal harvesting in a stochastic two-predators one-prey system with distributed delays and Lévy noise

“…As is well-known, two predators compete with each other for one-prey are more beneficial than predator-prey models (see [1,14,22,32,34]) or competitive models (see [9,10,13,27]). Besides, considering the influence of distributed delays, white noise and Lévy noise, in this paper we propose a stochastic two-predators one-prey model with Lévy noise and distributed delays.…”

“…From (32) and Lemma 2.3 we obtain that lim t→∞ x 3 (t) = 0 if b 3 ≤ 0 and lim sup t→∞ x 3 (t) ≤ (b 3 /a 33 ) if b 3 > 0. Thus, we obtain that condition (i) holds.…”

“…This lemma can be proved easily by using Lemma 2.3, Lemma 4.2 and a similar argument as in [17,32]. Hence, we here omit it.…”

“…To investigate the optimal harvesting problem of model (2), we further introduce the following assumption and the asymptotic stability in the distribution for model (2). This lemma can be proved easily by using a standard argument as in [31,32]. Hence, we here omit it.…”

“…Based on the study of ecological populations, it is crucial to take time delays into consideration since delays are omnipresent in the real world. Since May [26] found that time delays not only disturb the stability of logistic model's positive steady state but also cause periodic vibration, an increasing number of scholars study the effects of delays [14,18,22,[32][33][34]. Therefore, it is interesting and important to consider the following delayed stochastic two-predators one-prey model with white noise and Lévy noise dx…”