Analysis of a stochastic predator–prey population model with Allee effect and jumps

• Main Authors: Rong Liu, Guirong Liu
• Format: Article
• Language: English
• Published: SpringerOpen 2019-03-01
• Series: Journal of Inequalities and Applications
• Subjects: Allee effect; Lévy noise; Exponential martingale inequality; Chebyshev’s inequality; Predator–prey
• Online Access: http://link.springer.com/article/10.1186/s13660-019-2014-x

Abstract This paper is concerned with a stochastic predator–prey model with Allee effect and Lévy noise. First, by the comparison theorem of stochastic differential equations, we prove that the model has a unique global positive solution starting from the positive initial value. Then we investigate the asymptotic pathwise behavior of the model by the generalized exponential martingale inequality and the Borel–Cantelli lemma. Next, we establish the conditions under which predator and prey populations are extinct. Furthermore, we show that the global positive solution is stochastically ultimate bounded under some conditions by using the Bernoulli equation and Chebyshev’s inequality. At last, we introduce some numerical simulations to support the main results obtained. The results in this paper generalize and improve the previous related results.