Stochastic Nicholson-type delay differential system

Wang, Wentao; Shi, Ce; Chén, Wěi

doi:10.1080/00207179.2019.1651941

Cited by 16 publications

(15 citation statements)

References 33 publications

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“…Stochastic models are proposed to capture the uncertainty and variations in the transmission of the diseases. Many recent works dedicated to stochastic epidemiological and ecological models can be found in [37,38,40,41,50,52,[58][59][60][61]. There is still considerable uncertainty with regard to the study of stability, therefore we have addressed the stochastic global exponential and practical stability.…”

Section: Resultsmentioning

confidence: 99%

“…Then σ (ω) 2 4 < -2(a + A(ω) 2 ) is the asymptotic mean square stability condition of the zero solution of (37). And the condition of mean square stability in case of the independence between A(ω), σ (ω), and x(t, ω) is σ (ω) 2 2 < -2(a + A(ω) ).…”

Section: Illustrative Examplesmentioning

confidence: 98%

“…The species become extinct due to the global stability of the zero solution of (5), see [31]. Many authors have studied the dynamics of this equation [31][32][33][34][35], and stochastically [36][37][38].…”

Section: Assumptionmentioning

confidence: 99%

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Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

et al. 2021

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This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

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Section: Resultsmentioning

confidence: 99%

Section: Illustrative Examplesmentioning

confidence: 98%

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Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

et al. 2021

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“…The species become extinct due to the global stability of the zero solution of (4), see [30]. Many authors have studied the dynamics of this equation in the deterministic sense [23,26,[30][31][32], and stochastically [18,19,24,25,33]. This system is exposed to stochastic perturbation in the form of white noise which is assumed to be proportional to the deviation of the current state of the system from the zero solution, therefore, the stochastic version of ( 4) is in the form…”

Section: Nicholson's Blowflies Modelmentioning

confidence: 99%

“…Constructing a suitable Lyapunov functional is a good approach for investigating the necessary and sufficient conditions of stability of the equilibrium points [10,[14][15][16]. Dedicated to the study of stability of stochastic delay differential equations (SDDEs), there are many works, including, but not limited to, use this class of equations to control the stabilization problem of the controlled inverted pendulum, model the infectious diseases and model the population dynamics of the Australian sheep-blowfly known as Nicholson's blowfly [17][18][19]. By stochastic neutral differential equations (SNDEs), the stochastic neural networks and stochastic cellular neural networks have been used to model many of the human activities in science [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Mean Square Stability of the Zero Equilibrium of the Nonlinear Delay Differential Equation: Nicholson's Blowflies Application

El-Metwally

Sohaly

Elbaz

2021

Preprint

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We are concerned about the stochastic nonlinear delay differential equation. The stochasticity arises from the white Gaussian noise which is the time derivative of the standard Brownian motion. The main objective of this paper is to introduce a new technique using Lyapunov functional for the study of stability of the zero solution of the stochastic delay differential system. Constructing a new appropriate deterministic system in the neighborhood of the origin is an effective way to investigate the necessary and sufficient conditions of stability in the sense of the mean square. Nicholson's blowflies equation is one of the major problems in ecology, necessary conditions for the possible extinction of the Nicholson's blowflies population are investigated. We support our theoretical results by providing areas of stability and some numerical simulations of the solution of the system using the Euler-Maruyama scheme which is mean square stable \cite{Maruyama1955,Cao2004}.

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Stationary distribution and periodic solution of a stochastic Nicholson's blowflies model with distributed delay

Jiang

Hayat

et al. 2021

Math Methods in App Sciences

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Due to the complex set of abiotic and biotic interaction variables in the ecosystems, the dynamics of animal populations are generally considered to be driven in astate-dependent manner. This paper presents an analysis of a stochastic Nicholson's blowflies model with distributed delay, which consists of the existence and uniqueness of globally positive solution, extinction, existence of positive T-periodic solution and positive recurrence. Our results suggest that a smaller white noise is necessary to cause the existence of positive T-periodic solution and positive recurrence of the stochastic model, while a larger white noise will accelerate the extinction of the population. In addition, numerical simulation is used to support our main results.

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Stochastic Nicholson-type delay differential system

Cited by 16 publications

References 33 publications

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

Mean Square Stability of the Zero Equilibrium of the Nonlinear Delay Differential Equation: Nicholson's Blowflies Application

Stationary distribution and periodic solution of a stochastic Nicholson's blowflies model with distributed delay

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