Permanence and Extinction of Regime-Switching Predator-Prey Models

Bao, Jianhai; Shao, Jinghai

doi:10.1137/15m1024512

Cited by 90 publications

(48 citation statements)

References 29 publications

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“…By utilizing a comparison theorem, one observes that 0 < y(t) ≤ ψ(t) for all t ≥ 0 a.s. The following result is taken from [4], we cite it as a lemma.…”

Section: Moment Boundednessmentioning

confidence: 99%

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

Li¹,

Song²,

Xia³

et al. 2019

SIAM J. Appl. Math.

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This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs) and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.

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“…By utilizing a comparison theorem, one observes that 0 < y(t) ≤ ψ(t) for all t ≥ 0 a.s. The following result is taken from [4], we cite it as a lemma.…”

Section: Moment Boundednessmentioning

confidence: 99%

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

Li¹,

Song²,

Xia³

et al. 2019

SIAM J. Appl. Math.

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show abstract

“…An exception to this trend can be found in [19], where the authors proposed instability conditions for discrete-time Markov jump linear systems without such identification of system matrices. We also remark that, although almost sure stability has been often studied in the literature [12][13][14][15][16][17][18], the exponential mean stability is more commonly employed in the systems and control theory [2,3].…”

Section: Introductionmentioning

confidence: 99%

Kronecker weights for instability analysis of Markov jump linear systems

Mei

Ogura

2019

IET Control Theory & Applications

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In this paper, we analyze the instability of continuous-time Markov jump linear systems. Although there exist several effective criteria for the stability of Markov jump linear systems, there is a lack of methodologies for verifying their instability. In this paper, we present a novel criterion for the exponential mean instability of Markov jump linear systems. The main tool of our analysis is an auxiliary Markov jump linear system, which results from taking the Kronecker products of the given system matrices and a set of appropriate matrix weights.We furthermore show that the problem of finding matrix weights for tighter instability analysis can be transformed to the spectral optimization of an affine matrix family, which can be efficiently performed by gradient-based non-smooth optimization algorithms. We confirm the effectiveness of the proposed methods by numerical examples.

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“…The stochastic differential equation with Markovian switching (SDEwMS) have found several important applications in real life situations, see for example, [2,10,14,15,19,20] and references therein. Often, explicit solutions of such equations are not known and hence it becomes necessary to find their approximate solutions.…”

Section: Introductionmentioning

confidence: 99%

On explicit tamed Milstein-type scheme for stochastic differential equation with Markovian switching

Kumar

2020

Journal of Computational and Applied Mathematics

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We propose a new tamed Milstein-type scheme for stochastic differential equation with Markovian switching when drift coefficient is assumed to grow super-linearly. The strong rate of convergence is shown to be equal to 1.0 under mild regularity (e.g. once differentiability) requirements on drift and diffusion coefficients. Novel techniques are developed to tackle two-fold difficulties arising due to jumps of the Markov chain and the reduction of regularity requirements on the coefficients.

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Permanence and Extinction of Regime-Switching Predator-Prey Models

Cited by 90 publications

References 29 publications

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

Kronecker weights for instability analysis of Markov jump linear systems

On explicit tamed Milstein-type scheme for stochastic differential equation with Markovian switching

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