Incorporating two coupling noises into a nonlinear competitive system with saturation effect

In this paper, a novel stochastic two-species competitive system with saturation effect is formulated, in which there exist two noise resources and their coupling mode is relatively complex and every noise source has effect on the intrinsic growth rates of both species. With the help of some suitable Lyapunov functions, sufficient conditions for stochastic permanence are established as exponential extinction, extinction, permanence in time average and asymptotic pathwise estimation of system. The effect of coupling noise on the asymptotic behaviors of the populations is shown.

Dynamics of stochastic Holling II predator-prey under Markovian-switching with jumps

In this paper, a stochastic Holling II predator-prey model under Markovian switching with jumps is investigated. The aim is to find out how the Markovian switching and the jump noise affect the dynamics of this model. Firstly, we study the properties of the solutions, for example, the existence and uniqueness of the global positive solution, the uniform boundedness of the pth moment and the pathwise estimation. Secondly, sufficient criteria for extinction and strong persistence in the mean are established. Results show that jump noise can essentially change the nature of the system, i.e., it can make strongly persistent species extinct and extinct species persistent. We also observe that both the overall extinction and strong persistence in the mean have close relationships with the stationary probability distribution of the Markov chain. Finally, numerical examples are introduced to illustrate the results.

The Stationary Distribution of Competitive Lotka-Volterra Population Systems with Jumps

Dynamics of Lotka-Volterra population with jumps (LVWJ) have recently been established (see Bao et al., 2011, and Bao and Yuan, 2012). They provided some useful criteria on the existence of stationary distribution and some asymptotic properties for LVWJ. However, the uniqueness of stationary distribution forn≥2and asymptotic pathwise estimationlimt→+∞⁡(1/t)∫0t‍|X(s)|pds (p>0)are still unknown for LVWJ. One of our aims in this paper is to show the uniqueness of stationary distribution and asymptotic pathwise estimation for LVWJ. Moreover, some characterizations for stationary distribution are provided.

Dynamics of Nonautonomous Stochastic Gilpin-Ayala Competition Model with Jumps

The nonautonomous stochastic Gilpin-Ayala competition model driven by Lévy noise is considered. First, it is shown that this model has a global positive solution. Then, we discuss the asymptotic behavior of the solution including moment and pathwise estimation. Finally, sufficient conditions for extinction, nonpersistence in the mean, and weak persistence of the solution are established.