scholarly journals Dynamic Behaviors of a General Discrete Nonautonomous System of Plankton Allelopathy with Delays

2008 ◽  
Vol 2008 ◽  
pp. 1-22 ◽  
Author(s):  
Yaoping Chen ◽  
Fengde Chen ◽  
Zhong Li
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Nonautonomous System ◽  
Suitable Assumption ◽  
Dynamic Behaviors ◽  
Plankton Allelopathy

We study the dynamic behaviors of a general discrete nonautonomous system of plankton allelopathy with delays. We first show that under some suitable assumption, the system is permanent. Next, by constructing a suitable Lyapunov functional, we obtain a set of sufficient conditions which guarantee the global attractivity of the two species. After that, by constructing an extinction-type Lyapunov functional, we show that under some suitable assumptions, one species will be driven to extinction. Finally, two examples together with their numerical simulations show the feasibility of the main results.

scholarly journals Permanence and Global Attractivity of a Discrete Two-Prey One-Predator Model with Infinite Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Zhixiang Yu ◽  
Zhong Li
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Predator Model

A discrete two-prey one-predator model with infinite delay is proposed. A set of sufficient conditions which guarantee the permanence of the system is obtained. By constructing a suitable Lyapunov functional, we also obtain sufficient conditions ensuring the global attractivity of the system. An example together with its numerical simulation shows the feasibility of the main results.

scholarly journals Dynamics of the Almost Periodic Discrete Mackey–Glass Model

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 333 ◽  
Author(s):  
Zhijian Yao ◽  
Jehad Alzabut ◽  
Debaldev Jana
Keyword(s):  
Numerical Simulations ◽  
Blood Cells ◽  
Lyapunov Functional ◽  
Contraction Mapping ◽  
Exponential Convergence ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Almost Periodic ◽  
Discrete Lyapunov Functional ◽  
Glass Model

This paper is concerned with a class of the discrete Mackey–Glass model that describes the process of the production of blood cells. Prior to proceeding to the main results, we prove the boundedness and extinction of its solutions. By means of the contraction mapping principle and under appropriate assumptions, we prove the existence of almost periodic positive solutions. Furthermore and by the implementation of the discrete Lyapunov functional, sufficient conditions are established for the exponential convergence of the almost periodic positive solution. Examples, as well as numerical simulations are illustrated to demonstrate the effectiveness of the theoretical findings of the paper. Our results are new and generalize some previously-reported results in the literature.

scholarly journals Dynamic Analysis of a Model for Spruce Budworm Populations with Delay

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ahmadjan Muhammadhaji ◽  
Azhar Halik
Keyword(s):  
Numerical Simulation ◽  
Dynamic Analysis ◽  
Periodic Solutions ◽  
Population Model ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Spruce Budworm ◽  
Positive Periodic Solutions ◽  
Inequality Techniques

A class of delayed spruce budworm population model is considered. Compared with previous studies, both autonomous and nonautonomous delayed spruce budworm population models are considered. By using the inequality techniques, continuation theorem, and the construction of suitable Lyapunov functional, we establish a set of easily verifiable sufficient conditions on the permanence, existence, and global attractivity of positive periodic solutions for the considered system. Finally, an example and its numerical simulation are given to illustrate our main results.

Global attractivity of positive periodic solution for a predator–prey model with modified Leslie–Gower Holling-type II schemes and a deviating argument

2014 ◽  
Vol 07 (06) ◽  
pp. 1450071 ◽  
Author(s):  
Kai Wang ◽  
Yanling Zhu
Keyword(s):  
Lyapunov Functional ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Uniform Persistence ◽  
Type Ii ◽  
Predator Prey ◽  
Deviating Argument ◽  
Predator Prey Model ◽  
Prey Model

In this paper, by utilizing the comparison theorem and constructing a suitable Lyapunov functional the predator–prey model with modified Leslie–Gower Holling-type II schemes and a deviating argument is studied. Some sufficient conditions are obtained for uniform persistence and global attractivity of positive periodic solutions for this model. Furthermore, an example shows that the obtained criteria are easily verifiable.

PERSISTENCE, EXTINCTION AND STABILITY FOR NONLINEAR PLANKTON ALLELOPATHY MODEL WITH DELAYED NEGATIVE FEEDBACKS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250027 ◽  
Author(s):  
YAN-PING LIU ◽  
ZHI-XUE LUO
Keyword(s):  
Numerical Simulations ◽  
Sufficient Conditions ◽  
Lyapunov Functionals ◽  
Analysis Method ◽  
Average Value ◽  
Competitive Systems ◽  
Comparison Argument ◽  
Negative Feedbacks ◽  
Plankton Allelopathy ◽  
The Stability

By the techniques of comparison argument and Lyapunov-like functionals, some criteria about persistence and extinction of the species are obtained. And then, with the help of constructing Lyapunov functionals and some new analysis method, sufficient conditions, provided with the form of average value of a function, to guarantee the stability of the system are derived. Finally, some examples together with their numerical simulations show the feasibility of these main results. Our conclusions are different from many existing forms for nonlinear competitive systems.

scholarly journals On the Dynamics of a Nonautonomous Predator-Prey Model with Hassell-Varley Type Functional Response

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Yan Zhang ◽  
Shujing Gao ◽  
Kuangang Fan
Keyword(s):  
Lyapunov Function ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Incidence Rate ◽  
Functional Response ◽  
Global Attractivity ◽  
Migratory Birds ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model

The dynamic behaviors of a nonautonomous system for migratory birds with Hassell-Varley type functional response and the saturation incidence rate are studied. Under quite weak assumptions, some sufficient conditions are obtained for the permanence and extinction of the disease. Moreover, the global attractivity of the model is discussed by constructing a Lyapunov function. Numerical simulations which support our theoretical analysis are also given.

scholarly journals Periodic Solutions for n-Species Lotka-Volterra Competitive Systems with Pure Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Ahmadjan Muhammadhaji ◽  
Rouzimaimaiti Mahemuti ◽  
Zhidong Teng
Keyword(s):  
Periodic Solutions ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Competitive Systems ◽  
Volterra Systems ◽  
Special Cases

We study a class of periodic general n-species competitive Lotka-Volterra systems with pure delays. Based on the continuation theorem of the coincidence degree theory and Lyapunov functional, some new sufficient conditions on the existence and global attractivity of positive periodic solutions for the n-species competitive Lotka-Volterra systems are established. As an application, we also examine some special cases of the system, which have been studied extensively in the literature.

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 Dynamic Behaviors of a Competitive System with Beddington-DeAngelis Functional Response

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Shengbin Yu ◽  
Fengde Chen
Keyword(s):  
Periodic Solution ◽  
Numerical Simulations ◽  
Functional Response ◽  
Recent Literature ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic ◽  
Competitive System ◽  
Dynamic Behaviors ◽  
Partial Extinction

This article studies a competitive system with Beddington-DeAngelis functional response and establishes sufficient conditions on permanence, partial extinction, and the existence of a unique almost periodic solution for the system. The results supplement and generalize the main conclusions in recent literature. Numerical simulations have been presented to validate the analytical results.

A Markov-switching predator–prey model with Allee effect for preys

2020 ◽  
Vol 13 (03) ◽  
pp. 2050018
Author(s):  
Xiaoxia Guo ◽  
Zhiming Guo
Keyword(s):  
Numerical Simulations ◽  
Allee Effect ◽  
Global Attractivity ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results ◽  
Prey Populations

This paper concerns with a Markov-switching predator–prey model with Allee effect for preys. The conditions under which extinction of predator and prey populations occur have been established. Sufficient conditions are also given for persistence and global attractivity in mean. In addition, stability in the distribution of the system under consideration is derived under some assumptions. Finally, numerical simulations are carried out to illustrate theoretical results.

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