scholarly journals Dynamics of Nonautonomous Stochastic Gilpin-Ayala Competition Model with Jumps

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yanchao Zang ◽  
Junping Li ◽  
Jiangang Liu
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Lévy Noise ◽  
Model Driven ◽  
Global Positive Solution ◽  
The Mean ◽  
Pathwise Estimation ◽  
Levy Noise

The nonautonomous stochastic Gilpin-Ayala competition model driven by Lévy noise is considered. First, it is shown that this model has a global positive solution. Then, we discuss the asymptotic behavior of the solution including moment and pathwise estimation. Finally, sufficient conditions for extinction, nonpersistence in the mean, and weak persistence of the solution are established.

A stochastic vaccinated epidemic model incorporating Lévy processes with a general awareness-induced incidence

2021 ◽  
pp. 2150044
Author(s):  
Khadija Akdim ◽  
Adil Ez-Zetouni ◽  
Mehdi Zahid
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Dynamic Behavior ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Lévy Noise ◽  
Global Positive Solution ◽  
Levy Noise ◽  
Disease Free

In this paper, we investigate a stochastic vaccinated epidemic model with a general awareness-induced incidence perturbed by Lévy noise. First, we show that this model has a unique global positive solution. Therefore, we establish the dynamic behavior of the solution around both disease-free and endemic equilibrium points. Furthermore, when [Formula: see text], we give sufficient conditions for the existence of an ergodic stationary distribution to the model when the jump part in the Lévy noise is null. Finally, we present some examples to illustrate the analytical results by numerical simulations.

Analysis of a stochastic predator–prey model with prey subject to disease and Lévy noise

2019 ◽  
Vol 19 (05) ◽  
pp. 1950038
Author(s):  
Meihong Qiao ◽  
Shenglan Yuan
Keyword(s):  
Positive Solution ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Lévy Noise ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Boundedness ◽  
Prey Model ◽  
Global Positive Solution ◽  
Levy Noise

We consider a non-autonomous predator–prey model, with prey subject to the disease and Lévy noise. We show the existence of global positive solution and stochastic boundedness. Then, we examine the asymptotic properties of the solution. Finally, we offer sufficient conditions for persistence and extinction.

scholarly journals Asymptotic Behavior of a Stochastic Two-Species Competition Model under the Effect of Disease

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Rong Liu ◽  
Guirong Liu
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Deterministic Model ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Species Competition ◽  
Martingale Inequality ◽  
Interior Equilibrium ◽  
Exponential Martingale ◽  
Global Positive Solution

This paper is concerned with a stochastic two-species competition model under the effect of disease. It is assumed that one of the competing populations is vulnerable to an infections disease. By the comparison theorem of stochastic differential equations, we prove the existence and uniqueness of global positive solution of the model. Then, the asymptotic pathwise behavior of the model is given via the exponential martingale inequality and Borel-Cantelli lemma. Next, we find a new method to prove the boundedness of the pth moment of the global positive solution. Then, sufficient conditions for extinction and persistence in mean are obtained. Furthermore, by constructing a suitable Lyapunov function, we investigate the asymptotic behavior of the stochastic model around the interior equilibrium of the deterministic model. At last, some numerical simulations are introduced to justify the analytical results. The results in this paper extend the previous related results.

Dynamics of an autonomous Gilpin–Ayala competition model with random perturbation

2021 ◽  
pp. 2050043
Author(s):  
Jing Fu ◽  
Qixing Han ◽  
Daqing Jiang ◽  
Yanyan Yang
Keyword(s):  
White Noise ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Necessary Conditions ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Interacting Species ◽  
Sufficient And Necessary Conditions ◽  
Global Positive Solution

This paper discusses the dynamics of a Gilpin–Ayala competition model of two interacting species perturbed by white noise. We obtain the existence of a unique global positive solution of the system and the solution is bounded in [Formula: see text]th moment. Then, we establish sufficient and necessary conditions for persistence and the existence of an ergodic stationary distribution of the model. We also establish sufficient conditions for extinction of the model. Moreover, numerical simulations are carried out for further support of present research.

ASYMPTOTIC BEHAVIOR OF A STOCHASTIC DELAYED CHEMOSTAT MODEL WITH NUTRIENT STORAGE

2018 ◽  
Vol 26 (02) ◽  
pp. 225-246 ◽  
Author(s):  
SHULIN SUN ◽  
XIAOFENG ZHANG
Keyword(s):  
Asymptotic Behavior ◽  
Time Delay ◽  
Function Analysis ◽  
Sufficient Conditions ◽  
Classical Approach ◽  
Nutrient Storage ◽  
Chemostat Model ◽  
Global Positive Solution ◽  
The Mean ◽  
Simulation Results

In this paper, a stochastic delayed chemostat model with nutrient storage is proposed and investigated. First, we state that there is a unique global positive solution for this stochastic system. Second, using the classical approach of Lyapunov function analysis, this stochastic delayed chemostat model is discussed in detail. We establish some sufficient conditions for the extinction of the microorganism, furthermore, we prove that the microorganism will become persistent in the mean in the chemostat under some conditions. Finally, the obtained results are illustrated by computer simulations, and simulation results reveal the effects of time delay on the persistence and extinction of the microorganism.

Study on a susceptible–infected–vaccinated model with delay and proportional vaccination

2018 ◽  
Vol 11 (08) ◽  
pp. 1850102 ◽  
Author(s):  
Shuqi Gan ◽  
Fengying Wei
Keyword(s):  
Positive Solution ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Stochastic Epidemic ◽  
Persistence In The Mean ◽  
Global Positive Solution ◽  
The Mean ◽  
Stochastic Epidemic Model

A susceptible–infected–vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper [Formula: see text]-function therewith. Then a sufficient condition that guarantees the disappearances of diseases is derived when the indicator [Formula: see text]. Further, if [Formula: see text], then we obtain that the solution is weakly permanent with probability one. We also derived the sufficient conditions of the persistence in the mean for the susceptible and infected when another indicator [Formula: see text].

scholarly journals Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

2015 ◽  
Vol 15 (02) ◽  
pp. 1550011
Author(s):  
Gabriel Deugoué ◽  
Mamadou Sango
Keyword(s):  
Strong Solution ◽  
Strong Solutions ◽  
Lévy Noise ◽  
Mean Square ◽  
Stability Of Solutions ◽  
Model Driven ◽  
The Mean ◽  
The Stability ◽  
Non Gaussian ◽  
Levy Noise

We establish the existence, uniqueness and approximation of the strong solutions for the stochastic 3D LANS-α model driven by a non-Gaussian Lévy noise. Moreover, we also study the stability of solutions. In particular, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution.

scholarly journals Dynamics of a stochastic SIQR epidemic model with saturated incidence rate

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5239-5253 ◽  
Author(s):  
Li-Li Wang ◽  
Nan-Jing Huang ◽  
Donal O’Regan
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Global Positive Solution ◽  
The Mean

The purpose of this paper is to propose and investigate a stochastic SIQR epidemic model with saturated incidence rate. Firstly, we give some conditions to guarantee the stochastic SIQR epidemic model has a unique global positive solution. Then we verify that the disease in this model will die out exponentially if Rs 0 < 1, while the disease will be persistent in the mean if Rs 0 > 1. Moreover, by constructing suitable Lyapunov functions, we establish some sufficient conditions for the existence of an ergodic stationary distribution for the model. Finally, we provide some numerical simulations to illustrate the analytical results.

Adaptive state estimation of Markov switched neural networks driven by Lévy noise

2019 ◽  
Vol 42 (2) ◽  
pp. 330-336
Author(s):  
Dongbing Tong ◽  
Qiaoyu Chen ◽  
Wuneng Zhou ◽  
Yuhua Xu
Keyword(s):  
Neural Networks ◽  
State Estimation ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Lévy Noise ◽  
Linear Matrix ◽  
Delayed Neural Networks ◽  
Switched Neural Networks ◽  
Inequality Technique ◽  
Levy Noise

This paper proposes the [Formula: see text]-matrix method to achieve state estimation in Markov switched neural networks with Lévy noise, and the method is very distinct from the linear matrix inequality technique. Meanwhile, in light of the Lyapunov stability theory, some sufficient conditions of the exponential stability are derived for delayed neural networks, and the adaptive update law is obtained. An example verifies the condition of state estimation and confirms the effectiveness of results.

scholarly journals A Class of Stochastic Nonlinear Delay System with Jumps

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ling Bai ◽  
Kai Zhang ◽  
Wenju Zhao
Keyword(s):  
Sufficient Conditions ◽  
Delay System ◽  
Lévy Noise ◽  
Local Martingale ◽  
Martingale Inequality ◽  
Strong Law ◽  
Delay Differential System ◽  
Delay Differential ◽  
Levy Noise ◽  
Nonlinear Delay

We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.

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