Dynamics of a non-autonomous stochastic Gilpin-Ayala model

This paper is concerned with a stochastic non-autonomous Gilpin-Ayala model. First, it is shown that this model has a global positive solution. Then sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence of the solution are established. The critical number between weak persistence and extinction is obtained. Finally, the lower- and upper-growth rate of the solution are investigated. Several numerical figures are introduced to illustrate the results. Some recent results are improved and generalized.     

Second Kind Chebyshev Wavelet Galerkin Method for Stochastic Itô-Volterra Integral Equations

In this paper, an efficient wavelet Galerkin method based on the stochastic operational matrix of second kind Chebyshev wavelet is proposed for solving stochastic Itô-Volterra integral equations. Convergence and error analysis of the presented wavelets method are investigated. The numerical results are compared with exact solution and those of other existing methods.     

Stochastic control of helicopter suspended load position

Modelling of the motion of a helicopter with a suspended load is described. The mathematical model takes into account stochastic disturbances acting on the load suspension point. The proposed approach allows solution of the problem of optimal control with minimization of oscillation and control power expenditure. The evolution of the system solution with time is investigated for various disturbance intensities. Computer calculation results are presented as a function of the suspension length and the intensity of stochastic disturbances.     

Well-Posedness and Asymptotic Behaviors for a Predator-Prey System with Lévy Noise

In this paper, well-posedness and asymptotic behaviors for a predator-prey system with Lévy noise are studied by using stochastic analytical techniques. Firstly, the existence and uniqueness of positive global solution with positive initial value is proved. Then, stochastic permanence for the system is investigated. Finally, persistence in mean and extinction for the system are discussed and some numerical simulations are provided to support our results.     

Stationary Distribution and Extinction of Stochastic HTLV-I Infection Model with CTL Immune Response under Regime Switching

In this paper, the stochastic HTLV-I infection model with CTL immune response is investigated. Firstly, we show that the stochastic system exists unique positive global solution originating from the positive initial value. Secondly, we obtain that the existence of ergodic stationary distribution of the model by stochastic Lyapunov functions. Thirdly, we establish sufficient conditions for extinction of the infected cells. Finally, numerical simulations are carried out to illustrate the theoretical results.     

Stochastic generalized porous media equations with reflection

A non-negative Markovian solution is constructed for a class of stochastic generalized porous media equations with reflection. To this end, some regularity properties and a comparison theorem are proved for stochastic generalized porous media equations, which are interesting by themselves. Invariant probability measures and ergodicity of the solution are also investigated.     

Asymptotic Behavior of a Stochastic Predator-prey Model with Beddington-DeAngelis Functional Response and L\''{e}vy Jumps

A stochastic two-prey-one-predator model with Beddington-DeAngelis functional response and L\''{e}vy jumps is proposed and investigated in this paper. First of all, we prove the existence and uniqueness of the global positive solution, and stochastic ultimate boundedness of the solution. Next, under a simple assumption, by using It\^{o} formula and other important inequalities, some sufficient conditions are established to ensure the extinction and persistence in the mean of the system. The results show that neither strong white noise nor L\''{e}vy noise is conducive to the persistence of the population. Finally, the theoretical results are verified by numerical simulations.     

Stochastic parameterization for large eddy simulation of geophysical flows

Recently, stochastic, as opposed to deterministic, parameterizations are being investigated to model the effects of unresolved subgrid scales (SGS) in large eddy simulations (LES) of geophysical flows. We analyse such a stochastic approach in the barotropic vorticity equation to show that (i) if the stochastic parameterization approximates the actual SGS stresses, then the solution of the stochastic LES approximates the ``true" solution at appropriate scale sizes; and that (ii) when the filter scale size approaches zero, the solution of the stochastic LES approaches the true solution.

     

Heat equation with a general stochastic measure on nested fractals

A stochastic heat equation on an unbounded nested fractal driven by a general stochastic measure is investigated. Existence, uniqueness and continuity of the mild solution are proved provided that the spectral dimension of the fractal is less than 4/3.     

Stochastic hybrid system with non-homogeneous jumps

A study of a class of stochastic hybrid dynamic processes is investigated. The hybrid dynamic process is composed of both continuous and discrete time states. In this work we assume that its continuous time state is driven by the Brownian motion process, while the transitions of its discrete time state are governed by either a non-homogeneous Poisson process or by hitting the boundaries. Under this formulation we develop an infinitesimal generator of the stochastic hybrid dynamic process. Moreover we obtain results concerning the quantitative properties of the solution process. A few illustrative examples are presented.