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 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 Permanence and Positive Periodic Solutions of a Discrete Delay Competitive System

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Wenjie Qin ◽  
Zhijun Liu
Keyword(s):  
Periodic Solutions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Discrete Delay ◽  
Positive Periodic Solutions ◽  
Competitive System ◽  
Discrete Function ◽  
Species Population

A discrete time non-autonomous two-species competitive system with delays is proposed, which involves the influence of many generations on the density of species population. Sufficient conditions for permanence of the system are given. When the system is periodic, by using the continuous theorem of coincidence degree theory and constructing a suitable Lyapunov discrete function, sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. As an application, examples and their numerical simulations are presented to illustrate the feasibility of our main results.

scholarly journals Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances

2016 ◽  
Vol 14 (1) ◽  
pp. 1157-1173 ◽  
Author(s):  
Fengde Chen ◽  
Xiaoxing Chen ◽  
Shouying Huang
Keyword(s):  
Functional Response ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
The Other ◽  
Toxic Substances ◽  
Two Dimensional ◽  
Competitive System ◽  
Volterra Systems ◽  
Appl Math

AbstractA two species non-autonomous competitive phytoplankton system with Beddington-DeAngelis functional response and the effect of toxic substances is proposed and studied in this paper. Sufficient conditions which guarantee the extinction of a species and global attractivity of the other one are obtained. The results obtained here generalize the main results of Li and Chen [Extinction in two dimensional nonautonomous Lotka-Volterra systems with the effect of toxic substances, Appl. Math. Comput. 182(2006)684-690]. Numeric simulations are carried out to show the feasibility of our results.

scholarly journals Dynamic Behaviors of a Discrete Lotka-Volterra Competition System with Infinite Delays and Single Feedback Control

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Liang Zhao ◽  
Xiangdong Xie ◽  
Liya Yang ◽  
Fengde Chen
Keyword(s):  
Biological Control ◽  
Feedback Control ◽  
Comparison Theorem ◽  
Global Attractivity ◽  
Control Variable ◽  
Sufficient Conditions ◽  
Lyapunov Functionals ◽  
Competitive System ◽  
Competition System ◽  
Infinite Delays

A nonautonomous discrete two-species Lotka-Volterra competition system with infinite delays and single feedback control is considered in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that if the the discrete Lotka-Volterra competitive system with infinite delays and without feedback control is permanent, then, by choosing some suitable feedback control variable, the permanent species will be driven to extinction. That is, the feedback control variable, which represents the biological control or some harvesting procedure, is the unstable factor of the system. Such a finding overturns the previous scholars’ recognition on feedback control variables.

scholarly journals Existence and Global Attractivity of Positive Periodic Solutions for a Two-Species Competitive System with Stage Structure and Impulse

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Changjin Xu ◽  
Daxue Chen
Keyword(s):  
Periodic Solution ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Stage Structure ◽  
Positive Periodic Solution ◽  
Positive Periodic Solutions ◽  
Competitive System

A class of nonautonomous two-species competitive system with stage structure and impulse is considered. By using the continuation theorem of coincidence degree theory, we derive a set of easily verifiable sufficient conditions that guarantee the existence of at least a positive periodic solution, and, by constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Finally, an illustrative example is given to demonstrate the correctness of the obtained 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.

A generalized stochastic competitive system with Ornstein–Uhlenbeck process

2020 ◽  
pp. 2150001
Author(s):  
Baodan Tian ◽  
Liu Yang ◽  
Xingzhi Chen ◽  
Yong Zhang
Keyword(s):  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Competitive System ◽  
Uhlenbeck Process ◽  
Stochastic Perturbations ◽  
Stochastic Permanence ◽  
Persistence In The Mean ◽  
Ornstein Uhlenbeck Process ◽  
Global Positive Solution ◽  
The Mean

A generalized competitive system with stochastic perturbations is proposed in this paper, in which the stochastic disturbances are described by the famous Ornstein–Uhlenbeck process. By theories of stochastic differential equations, such as comparison theorem, Itô’s integration formula, Chebyshev’s inequality, martingale’s properties, etc., the existence and the uniqueness of global positive solution of the system are obtained. Then sufficient conditions for the extinction of the species almost surely, persistence in the mean and the stochastic permanence for the system are derived, respectively. Finally, by a series of numerical examples, the feasibility and correctness of the theoretical analysis results are verified intuitively. Moreover, the effects of the intensity of the stochastic perturbations and the speed of the reverse in the Ornstein–Uhlenbeck process to the dynamical behavior of the system are also discussed.

scholarly journals Global Attractivity and Extinction of a Discrete Competitive System with Infinite Delays and Single Feedback Control

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Yalong Xue ◽  
Xiangdong Xie ◽  
Qifa Lin ◽  
Fengde Chen
Keyword(s):  
Feedback Control ◽  
Comparison Theorem ◽  
Global Attractivity ◽  
Control Variable ◽  
Sufficient Conditions ◽  
Lyapunov Functionals ◽  
Species Competition ◽  
Competitive System ◽  
Competition System ◽  
Infinite Delays

A nonautonomous discrete two-species competition system with infinite delays and single feedback control is considered in this paper. Based on the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Then, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that, by choosing some suitable feedback control variable, one of two species will be driven to extinction.

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.

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