Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise

Dang, Nguyen Hai; Du, Nguyen Huu; Yin, Gang

doi:10.1016/j.jde.2014.05.029

Cited by 47 publications

(30 citation statements)

References 8 publications

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“…Remark 5. In [12], it is easy to see that if the condition (4.4) is valid, then the condition (H) holds. In (b) of Theorem 7 and Theorem 9, we therefore have weakened the conditions ensuring the global attractivity of the Ω-limit set of the system and the convergence in total variation of the instantaneous measure to the stationary measure.…”

Section: Global Convergence Of the Distribution In Total Variation Normmentioning

confidence: 99%

“…Remark 6. The method used in the proof of Theorem 8 is similar to that given in Theorem 3.1 of [12]. In the proceeding of the proof, it is most important to find a compact set O ⊂ X satisfying (3.28).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Letting p k = min e∈M {p e k } for each k ∈ N, we can get for any z ∈ [k −1 , k] 3 ∩ K, e ∈ M and any Borel set B ⊂ U , where we here use the fact that in the proof of Step 1 in this theorem, we can assume that the statep i = 1 without loss of generality. As in the proof of Theorem 4.2 in [12], we construct a lattice distribution a(nK) = 2 −n , n ∈ N, and the corresponding Markov transition function K a (z, e, A) = ∞ n=1 2 −n P(nK, z, e, A) for any (z, e) ∈ X and A ∈ B(X). Moreover, the kernel K a has an everywhere non-trivial continuous component T : X × B(X) → [0, +∞) defined by T (z, e, A) = 2 −(n k+1 +1) p k+1 d 4 · m A ∩ (U × {1}) when z ∈ (((k + 1) −1 , k + 1) 3 \(k −1 , k) 3 ) ∩K, k ∈ N. Thus, the process ((S(t), I(t), R(t)), r(t)) is a T -process.…”

Section: )mentioning

confidence: 99%

“…It is worth mentioning that the modified method can be applied to the case where all parameters of the model considered in [7,15,16] were derived by the stochastic process r(t), however, here we only allow the transmission rate β of model (1.2) to be disturbed because in reality it may be more sensitive to environmental fluctuations than other parameters of model (1.1) for human populations. In particular, by the new method, under weaker conditions we can directly extend the results in [11,12,15] from two environmental states to any finite ones. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

“…In the case of the disease persistence, we will analyze the ergodic behavior of system (1.2). By skilled techniques, authors in [11,12] applied the stochastic stability theory of Markov processes in [22,23,24,25] to two-dimensional Kolmogorov systems of competitive type switching between two environmental states. They obtained the sufficient conditions for the global attractivity of the Ω-limit set of the system and the convergence of the instantaneous measure to the stationary measure in total variation.…”

Section: Introductionmentioning

confidence: 99%

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Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Dan

Liu

Cui

2017

Journal of Differential Equations

116

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This paper studies the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and varying population size, which is formulated as a piecewise deterministic Markov process. A threshold dynamic determined by the basic reproduction number R 0 is established: the disease can be eradicated almost surely if R 0 < 1, while the disease persists almost surely if R 0 > 1. The existing method for analyzing ergodic behavior of population systems has been generalized. The modified method weakens the required conditions and has no limitations for both the number of environmental regimes and the dimension of the considered system. When R 0 > 1, the existence of a stationary probability measure is obtained. Furthermore, with the modified method, the global attractivity of the Ω-limit set of the system and the convergence in total variation to the stationary measure are both demonstrated under a mild extra condition.

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Section: Global Convergence Of the Distribution In Total Variation Normmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Dan

Liu

Cui

2017

Journal of Differential Equations

116

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Extinction and positive recurrence of a regime‐switching HIV/AIDS model with treatment and standard incidence

Gao

Jiang

Hayat

et al. 2022

Math Methods in App Sciences

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In this paper, we investigate the dynamics of a high-dimensional human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) model with treatment and standard incidence, which is perturbed by telegraph noise. The switching is formulated by a continuous time Markov chain. Firstly, we show the existence and uniqueness of the global positive solution. Then, conditions for extinction of the disease are established. Moreover, through constructing suitable Lyapunov functions, we derive sufficient conditions for the existence of positive recurrence of the solutions. Positive recurrence implies all the individuals can coexist and persist in the long term. Finally, numerical experiments are performed for supporting the theoretical results.

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Stochastic Lotka–Volterra food chains

Hening

Nguyen

2017

J. Math. Biol.

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Abstract. We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and n − 1 predators. The jth predator eats the j − 1th species and is eaten by the j + 1th predator; this way each species only interacts with at most two other species -the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting.As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction.Another novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.

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Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise

Cited by 47 publications

References 8 publications

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Extinction and positive recurrence of a regime‐switching HIV/AIDS model with treatment and standard incidence

Stochastic Lotka–Volterra food chains

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