This paper intends to study some dynamical properties of a stochastic three-dimensional Lotka–Volterra system. Under some mild assumptions, we first introduce a simple method to show that the model has a global and positive solution almost surely. Secondly, we prove that this model has a stationary distribution. Then we study the global asymptotic stability of the positive solution. Finally, some numerical simulations are introduced to illustrate the theoretical results. | Turk J Math (2017) 41: 1292 – 1307 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Stationary distribution and global asymptotic stability of a three-species stochastic food-chain system Hong QIU∗, Wenmin DENG College of Science, Civil Aviation University of China, Tianjin, . China Received: • Accepted/Published Online: • Final Version: Abstract: This paper intends to study some dynamical properties of a stochastic three-dimensional Lotka–Volterra system. Under some mild assumptions, we first introduce a simple method to show that the model has a global and positive solution almost surely. Secondly, we prove that this model has a stationary distribution. Then we study the global asymptotic stability of the positive solution. Finally, some numerical simulations are introduced to illustrate the theoretical results. Key words: Stochastic food-chain model, stationary distribution, global asymptotic stability 1. Introduction When species interact with each other the population dynamics of each species is affected, and the predator– prey interaction for the predation of one species by another is one of the important ecological phenomena. The first predator–prey model is two species, which was proposed by Volterra and Lotka in the mid 1920s to explain the oscillation of certain fish catches in the Adriatic (see [33]). The population dynamics become more complex when the interacting species are three than when there are two. Nonetheless, such models attracted considerable attention; for example, Pande [34] considered the coexistence of three species with one prey and two predators. Krikorian [19] studied the global asymptotic stability and global boundedness of the classical Volterra equations modeling with three-species predator–prey interactions. Farkas and Freedman [7] gave the stability criterion for a system of a three-dimensional case when two .
