scholarly journals Stochastic Logistic Systems with Jumps

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ruihua Wu ◽  
Ke Wang
Keyword(s):  
Positive Solutions ◽  
Logistic Model ◽  
Ultimate Boundedness ◽  
Stochastically Ultimate Boundedness ◽  
Jump Noise

This paper is concerned with a stochastic nonautonomous logistic model with jumps. In the model, the martingale and jump noise are taken into account. This model is new and more feasible and applicable. Sufficient criteria for the existence of global positive solutions are obtained; then asymptotic boundedness inpth moment, stochastically ultimate boundedness, and asymptotic pathwise behavior are to be considered.

Dynamics behaviors of a delayed competitive system in a random environment

2015 ◽  
Vol 08 (05) ◽  
pp. 1550062 ◽  
Author(s):  
Ronghua Tan ◽  
Huili Xiang ◽  
Yiping Chen ◽  
Zhijun Liu
Keyword(s):  
Global Existence ◽  
Random Environment ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Ultimate Boundedness ◽  
Unique Positive Solution ◽  
Competitive System ◽  
Stochastic Perturbations ◽  
Stochastically Ultimate Boundedness ◽  
White Noises

In the real world, the population systems are often subject to white noises and a system with such stochastic perturbations tends to be suitably modeled by stochastic differential equations. This paper is concerned with the dynamic behaviors of a delay stochastic competitive system. We first obtain the global existence of a unique positive solution of system. Later, we show that the solution of system will be stochastically ultimate boundedness. However, large noises may make the system extinct exponentially with probability one. Also, sufficient conditions for the global attractivity of system are established. Finally, illustrated examples are given to show the effectiveness of the proposed criteria.

scholarly journals Stochastic Delay Logistic Model under Regime Switching

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Zheng Wu ◽  
Hao Huang ◽  
Lianglong Wang
Keyword(s):  
Regime Switching ◽  
Logistic Model ◽  
Sufficient Conditions ◽  
Function Technique ◽  
Stochastic Permanence ◽  
Stochastic Delay ◽  
Gronwall’S Inequality ◽  
Diffusion In Random Environment ◽  
Stochastically Ultimate Boundedness ◽  
Generalized Itô Formula

This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue ofV-function technique,M-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.

Dynamics of a stochastic periodic SIRS model with time delay

2020 ◽  
pp. 2050072
Author(s):  
Xiangyun Shi ◽  
Yimeng Cao
Keyword(s):  
Time Delay ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Lyapunov Functions ◽  
Ultimate Boundedness ◽  
Sirs Epidemic Model ◽  
Dynamical Behaviors ◽  
Sirs Model ◽  
Global Positive Solution ◽  
Stochastically Ultimate Boundedness

Dynamical behaviors of a stochastic periodic SIRS epidemic model with time delay are investigated. By constructing suitable Lyapunov functions and applying Itô’s formula, the existence of the global positive solution and the property of stochastically ultimate boundedness of model (1.1) are proved. Moreover, the extinction and the persistence of the disease are established. The results are verified by numerical simulations.

scholarly journals Stochastically Ultimate Boundedness and Global Attraction of Positive Solution for a Stochastic Competitive System

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Shengliang Guo ◽  
Zhijun Liu ◽  
Huili Xiang
Keyword(s):  
Lyapunov Function ◽  
Positive Solution ◽  
Finite Time ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Ultimate Boundedness ◽  
Competitive System ◽  
Global Attraction ◽  
Stochastically Ultimate Boundedness ◽  
Theoretical Results

A stochastic competitive system is investigated. We first show that the positive solution of the above system does not explode to infinity in a finite time, and the existence and uniqueness of positive solution are discussed. Later, sufficient conditions for the stochastically ultimate boundedness of positive solution are derived. Also, with the help of Lyapunov function, sufficient conditions for the global attraction of positive solution are established. Finally, numerical simulations are presented to justify our theoretical results.

scholarly journals Dynamics of a single predator multiple prey model with stochastic perturbation and seasonal variation

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 535-549
Author(s):  
Hong-Wen Hui ◽  
Lin-Fei Nie
Keyword(s):  
Seasonal Variation ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Analysis Method ◽  
Ultimate Boundedness ◽  
Persistence In The Mean ◽  
Global Positive Solution ◽  
The Mean ◽  
Stochastically Ultimate Boundedness ◽  
Theoretical Results

Considering various factors are stochastic rather than deterministic in the evolution of populations growth, in this paper, we propose a single predator multiple prey stochastic model with seasonal variation. By using the method of solving an explicit solution, the existence of global positive solution of this model are obtained. The method is more convenient than Lyapunov analysis method for some population models. Moreover, the stochastically ultimate boundedness are considered by using the comparison theorem of stochastic differential equation. Further, some sufficient conditions for the extinction and strong persistence in the mean of populations are discussed, respectively. In addition, by constructing some suitable Lyapunov functions, we show that this model admits at least one periodic solution. Finally, numerical simulations clearly illustrate the main theoretical results and the effects of white noise and seasonal variation for the persistence and extinction of populations.

scholarly journals Dynamical Behavior of a Stochastic Ratio-Dependent Predator-Prey System

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Zheng Wu ◽  
Hao Huang ◽  
Lianglong Wang
Keyword(s):  
Dynamical Behavior ◽  
Ultimate Boundedness ◽  
Unique Positive Solution ◽  
Stochastic Permanence ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Dependent Model ◽  
Ratio Dependent ◽  
Stochastically Ultimate Boundedness

This paper is concerned with a stochastic ratio-dependent predator-prey model with varible coefficients. By the comparison theorem of stochastic equations and the Itô formula, the global existence of a unique positive solution of the ratio-dependent model is obtained. Besides, some results are established such as the stochastically ultimate boundedness and stochastic permanence for this model.

scholarly journals Extinction Analysis of Stochastic Predator–Prey System with Stage Structure and Crowley–Martin Functional Response

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 252 ◽  
Author(s):  
Conghui Xu ◽  
Guojian Ren ◽  
Yongguang Yu
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Ultimate Boundedness ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Global Positive Solution ◽  
Stochastically Ultimate Boundedness ◽  
Stochastic Extinction

In this paper, we researched some dynamical behaviors of a stochastic predator–prey system, which is considered under the combination of Crowley–Martin functional response and stage structure. First, we obtained the existence and uniqueness of the global positive solution of the system. Then, we studied the stochastically ultimate boundedness of the solution. Furthermore, we established two sufficient conditions, which are separately given to ensure the stochastic extinction of the prey and predator populations. In the end, we carried out the numerical simulations to explain some cases.

scholarly journals Permanence and Global Attractivity of a Discrete Logistic Model with Impulses

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Chunyu Gao ◽  
Qingxi Guo
Keyword(s):  
Dynamic System ◽  
Numerical Simulations ◽  
Positive Solutions ◽  
Continuous Time ◽  
Logistic Model ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Euler Method ◽  
Logistic Equation ◽  
Time Dynamic

By piecewise Euler method, we construct a discrete logistic equation with impulses. The constructed model is more easily implemented at computer and is a better analogue of the continuous-time dynamic system. The dynamic behaviors of the constructed model are investigated. Sufficient conditions which guarantee the permanence and the global attractivity of positive solutions of the model are obtained. Numerical simulations show the feasibility of the main results.

scholarly journals Stochastic Delay Population Dynamics under Regime Switching: Global Solutions and Extinction

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Zheng Wu ◽  
Hao Huang ◽  
Lianglong Wang
Keyword(s):  
Regime Switching ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Volterra Model ◽  
Ultimate Boundedness ◽  
Stochastic Delay ◽  
Switching Diffusion ◽  
Diffusion In Random Environment ◽  
Stochastically Ultimate Boundedness ◽  
Generalized Itô Formula

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Finally, an example is given to illustrate the main results.

Effects of Toxicants on a Non-Autonomous Predator-Prey Model with Random Perturbation

2015 ◽  
Vol 737 ◽  
pp. 487-490
Author(s):  
Yan Zhang ◽  
Kuan Gang Fan ◽  
Qing Yun Wang
Keyword(s):  
Random Perturbation ◽  
Stochastic Perturbation ◽  
Ultimate Boundedness ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Persistence In The Mean ◽  
Global Positive Solution ◽  
The Mean ◽  
Stochastically Ultimate Boundedness

A new non-autonomous predator-prey model in a polluted environment with stochastic perturbation is considered in this paper. The existence of a global positive solution and stochastically ultimate boundedness are derived. Furthermore, some sufficient and necessary criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean are obtained. At last, a series of numerical simulations to illustrate our mathematical findings are presented.

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