Asymptotic Properties of Hybrid Diffusion Systems

Zhu, Chao; Yin, George

doi:10.1137/060649343

Cited by 381 publications

(127 citation statements)

References 18 publications

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“…which a path issuing from x reaches the set U n0 is finite for any x ∈ U \U n0 , and sup x∈U0 E x τ < +∞ for every compact subset U 0 ⊂ U (see p. 1163 of Zhu and Yin [38]). Then by adopting the conclusion of Theorem…”

Section: Which Can Guarantee That the Smallest Eigenvalue Of The Diffmentioning

confidence: 99%

Stationary distribution and global asymptotic stability of a three-species stochastic food-chain system

Qian¹,

Deng²

2017

Turk J Math

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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.

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Section: Which Can Guarantee That the Smallest Eigenvalue Of The Diffmentioning

confidence: 99%

Stationary distribution and global asymptotic stability of a three-species stochastic food-chain system

Qian¹,

Deng²

2017

Turk J Math

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“…More recently, some improved sufficient conditions on persistence in mean or extinction for stochastic competitive Lotka-Volterra systems are provided by Liu and Shen [8] based on the space decomposition method. More details can be referred to [9,10], and [11]. However, the approaches proposed by these authors (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Persistence in mean and extinction on stochastic competitive Gilpin-Ayala systems with regime switching

Liu

Zhu

2017

Adv Differ Equ

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We are interested in the persistence in mean and extinction for a stochastic competitive Gilpin-Ayala system with regime switching. Based on the stochastic LaSalle theorem and the space-decomposition method, we derive generalized sufficient criteria on persistence in mean and extinction. By constructing a novel Lyapunov function we establish sufficient criteria on partial persistence in mean and partial extinction for the system. Finally, we provide two examples to demonstrate the feasibility and validity of our proposed methods.

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“…The following lemma is used to prove the theorem related to the stationary distribution for SDE (3) (see [14] …”

Section: Preliminariesmentioning

confidence: 99%

Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs

Ansari

Myr

Omari

2017

Discrete Dynamics in Nature and Society

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We investigate the various conditions that control the extinction and stability of a nonlinear mathematical spread model with stochastic perturbations. This model describes the spread of viruses into an infected computer network which is powered by a system of antivirus software. The system is analyzed by using the stability theory of stochastic differential equations and the computer simulations. First, we study the global stability of the virus-free equilibrium state and the virus-epidemic equilibrium state. Furthermore, we use the Itô formula and some other theoretical theorems of stochastic differential equation to discuss the extinction and the stationary distribution of our system. The analysis gives a sufficient condition for the infection to be extinct (i.e., the number of viruses tends exponentially to zero). The ergodicity of the solution and the stationary distribution can be obtained if the basic reproduction number R is bigger than 1, and the intensities of stochastic fluctuations are small enough. Numerical simulations are carried out to illustrate the theoretical results.

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Asymptotic Properties of Hybrid Diffusion Systems

Cited by 381 publications

References 18 publications

Stationary distribution and global asymptotic stability of a three-species stochastic food-chain system

Stationary distribution and global asymptotic stability of a three-species stochastic food-chain system

Persistence in mean and extinction on stochastic competitive Gilpin-Ayala systems with regime switching

Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs

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