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Phytoplankton is an important indicator organism to evaluate the quality of water environment, which may reflect the nutritional level of the sea area. Conversely, environmental conditions can directly affect the community structure of phytoplankton. A stochastic phytoplankton–zooplankton model considering non-degenerate and degenerate diffusions is formulated in this paper. What’s more, in both systems, we obtain sufficient conditions of extinction and ergodicity. Our results demonstrate that the weaker white noise can ensure the permanence of zooplankton and the stronger white noise can lead to the disappearance of zooplankton. Moreover, the threshold value of extinction and persistence can serve as a theoretical basis for controlling phytoplankton and zooplankton. Numerical examples are performed on the analysis results of the two cases to confirm our theoretical results. In addition, we also provide a real-life case study, the validity of the model is verified based on experimental data, and it is shown that fluctuation of external environment and the consumption of phytoplankton by zooplankton will affect the growth and number of phytoplankton. Effectively controlling the quantity of phytoplankton can delay the occurrence of water bloom or red tide. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study a stochastic predator-prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce a low concentration trimolecular stochastic chemical reaction model. We will prove that the solution of the system is positive and global. And then we draw a conclusion that there is a stationary distribution for the stochastic system and it has ergodicity under appropriate conditions. Finally, we test our theory conclusion by simulations. It is interesting that no matter what states the unique equilibrium of ordinary differential equation model appears, and regardless of whether the limit cycle exists, our stochastic model is always ergodic. [ABSTRACT FROM AUTHOR]

In this paper, we study a three-dimensional stochastic vegetation–water model in arid ecosystems, where the soil water and the surface water are considered. First, for the deterministic model, the possible equilibria and the related local asymptotic stability are studied. Then, for the stochastic model, by constructing some suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution ϖ (·) . In a biological interpretation, the existence of the distribution ϖ (·) implies the long-term persistence of vegetation under certain conditions. Taking the stochasticity into account, a quasi-positive equilibrium D ¯ ∗ related to the vegetation-positive equilibrium of the deterministic model is defined. By solving the relevant Fokker–Planck equation, we obtain the approximate expression of the distribution ϖ (·) around the equilibrium D ¯ ∗ . In addition, we obtain sufficient condition R 0 E < 1 for vegetation extinction. For practical application, we further estimate the probability of vegetation extinction at a given time. Finally, based on some actual vegetation data from Wuwei in China and Sahel, some numerical simulations are provided to verify our theoretical results and study the impact of stochastic noise on vegetation dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce stochasticity into a multigroup SIS model. We present the sufficient condition for the exponential extinction of the disease and prove that the noises significantly raise the threshold of a deterministic system. In the case of persistence, we prove that there exists an invariant distribution which is ergodic. [ABSTRACT FROM AUTHOR]