scholarly journals The Extinction of a Non-Autonomous Allelopathic Phytoplankton Model with Nonlinear Inter-Inhibition Terms and Feedback Controls

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 173 ◽  
Author(s):  
Liang Zhao ◽  
Fengde Chen ◽  
Saixi Song ◽  
Guizhen Xuan
Keyword(s):  
Differential Equation ◽  
Comparison Theorem ◽  
Difference Equations ◽  
Crucial Role ◽  
Sufficient Conditions ◽  
Toxic Substances ◽  
Competitive System ◽  
Feedback Controls

A non-autonomous allelopathic phytoplankton model with nonlinear inter-inhibition terms and feedback controls is studied in this paper. Based on the comparison theorem of differential equation, some sufficient conditions for the permanence of the system are obtained. We study the extinction of one of the species by using some suitable Lyapunov type extinction function. Our analyses extend those of Xie et al. (Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton. Advances in Difference Equations, 2016, 2016, 258) and show that the feedback controls and toxic substances have no effect on the permanence of the system but play a crucial role on the extinction of the system. Some known results are extended.

scholarly journals Dynamics of a Discrete Allelopathic Phytoplankton Model with Infinite Delays and Feedback Controls

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Liang Zhao ◽  
Bin Qin ◽  
Fengde Chen
Keyword(s):  
Comparison Theorem ◽  
Crucial Role ◽  
Sufficient Conditions ◽  
Lyapunov Functionals ◽  
Toxic Substances ◽  
Feedback Controls ◽  
Infinite Delays

A discrete allelopathic phytoplankton model with infinite delays and feedback controls is studied in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantees the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the extinction of the system are obtained. Our results extend and supplement some known results and show that the feedback controls and toxic substances play a crucial role on the permanence and extinction of the system.

Extinction in a discrete Lotka–Volterra competitive system with the effect of toxic substances and feedback controls

2015 ◽  
Vol 08 (01) ◽  
pp. 1550012 ◽  
Author(s):  
Lijuan Chen ◽  
Fengde Chen
Keyword(s):  
Feedback Control ◽  
Lyapunov Function ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
The Other ◽  
Toxic Substances ◽  
Two Dimensional ◽  
Competitive System ◽  
Feedback Controls ◽  
Analysis Technique

In this paper, we consider a discrete Lotka–Volterra competitive system with the effect of toxic substances and feedback controls. By using the method of discrete Lyapunov function and by developing a new analysis technique, we obtain the sufficient conditions which guarantee that one of the two species will be driven to extinction while the other will be permanent. We improve the corresponding results of Li and Chen [Extinction in two-dimensional discrete Lotka–Volterra competitive system with the effect of toxic substances, Dynam. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 15 (2008) 165–178]. Also, an example together with their numerical simulations shows the feasibility of our main results. It is shown that toxic substances and feedback control variables play an important role in the dynamics of the system.

scholarly journals Almost periodic solution of a discrete competitive system with delays and feedback controls

2019 ◽  
Vol 17 (1) ◽  
pp. 120-130 ◽  
Author(s):  
Yalong Xue ◽  
Xiangdong Xie ◽  
Qifa Lin
Keyword(s):  
Numerical Simulation ◽  
Periodic Solution ◽  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Competitive System ◽  
Feedback Controls ◽  
Uniformly Asymptotic Stability ◽  
Positive Almost Periodic Solution

Abstract A discrete nonlinear almost periodic multispecies competitive system with delays and feedback controls is proposed and investigated. We obtain sufficient conditions to ensure the permanence of the system. Also, we establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. In additional, an example together with its numerical simulation are presented to illustrate the feasibility of the main result.

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 Extinction in Two-Species Nonlinear Discrete Competitive System

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Liqiong Pu ◽  
Xiangdong Xie ◽  
Fengde Chen ◽  
Zhanshuai Miao
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Discrete System ◽  
Sufficient Conditions ◽  
The Other ◽  
Discrete Equation ◽  
Toxic Substances ◽  
Competitive System ◽  
Globally Attractive ◽  
Nonlinear Discrete System

We propose a nonlinear discrete system of two species with the effect of toxic substances. By constructing a suitable Lyapunov-type function, we obtain the sufficient conditions which guarantee that one of the components will be driven to extinction while the other will be globally attractive with any positive solution of a discrete equation. Two examples together with their numerical simulations illustrate the feasibility of our main results. The results not only improve but also complement some known results.

scholarly journals Extinction of a two species competitive stage-structured system with the effect of toxic substance and harvesting

2019 ◽  
Vol 17 (1) ◽  
pp. 856-873 ◽  
Author(s):  
Xiaoyan Huang ◽  
Fengde Chen ◽  
Xiangdong Xie ◽  
Liang Zhao
Keyword(s):  
Toxic Substance ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Volterra Model ◽  
Toxic Substances ◽  
Competitive System ◽  
Structured System ◽  
Extinction Property ◽  
Appl Math

Abstract The extinction property of a two species competitive stage-structured phytoplankton system with harvesting is studied in this paper. Several sets of sufficient conditions which ensure that one of the components will be driven to extinction are established. Our results supplement and complement the results of Li and Chen [Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances, J. Comput. Appl. Math., 2009, 231(1), 143-153] and Liu, Chen, Luo et al. [Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 2002, 274(2), 667-684].

scholarly journals Extinction in Nonautonomous Discrete Lotka-Volterra Competitive System with Pure Delays and Feedback Controls

2009 ◽  
Vol 2009 ◽  
pp. 1-19 ◽  
Author(s):  
Ling Zhang ◽  
Zhidong Teng ◽  
Tailei Zhang ◽  
Shujing Gao
Keyword(s):  
Discrete Time ◽  
Sufficient Conditions ◽  
Lyapunov Functionals ◽  
Competitive System ◽  
Feedback Controls

The paper discusses a nonautonomous discrete time Lotka-Volterra competitive system with pure delays and feedback controls. New sufficient conditions for which a part of then-species is driven to extinction are established by using the method of multiple discrete Lyapunov functionals.

PERMANENCE OF A DISCRETE n-SPECIES LOTKA–VOLTERRA TYPE FOOD-CHAIN SYSTEM WITH FEEDBACK CONTROLS AND TIME DELAYS

2008 ◽  
Vol 01 (03) ◽  
pp. 299-311 ◽  
Author(s):  
XUMING HUANG ◽  
WENSHENG YANG ◽  
XUEPENG LI
Keyword(s):  
Difference Equation ◽  
Comparison Theorem ◽  
Food Chain ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Feedback Controls ◽  
Volterra Type ◽  
Chain System

In this paper, a discrete n-species Lotka–Volterra type food-chain system with time delays and feedback controls is proposed. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system.

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.

Sign in / Sign up

Export Citation Format

Share Document