scholarly journals Dynamical Behaviors of Stochastic Delayed One-Predator and Two-Competing-Prey Systems with Holling Type IV and Crowley-Martin Type Functional Responses

2016 ◽  
Vol 2016 ◽  
pp. 1-16
Author(s):  
Changjian Wang ◽  
Zuoliang Xiong ◽  
Rensheng He ◽  
Hongwei Yin
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Functional Responses ◽  
Stochastic Permanence ◽  
Numerical Examples ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
Stochastic Boundedness ◽  
Globally Positive Solution

This paper is devoted to stochastic delayed one-predator and two-competing-prey systems with two kinds of different functional responses. By establishing appropriate Lyapunov functions, the globally positive solution and stochastic boundedness are investigated. In some case, the stochastic permanence and extinction are also obtained. Moreover, sufficient conditions of the global asymptotic stability of the system are established. Finally, some numerical examples are provided to explain our conclusions.

scholarly journals Dynamical Behavior of a Stochastic Delayed One-Predator and Two-Mutualistic-Prey Model with Markovian Switching and Different Functional Responses

2016 ◽  
Vol 2016 ◽  
pp. 1-16
Author(s):  
Rensheng He ◽  
Zuoliang Xiong ◽  
Desheng Hong
Keyword(s):  
Lyapunov Functions ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Matrix Analysis ◽  
Stationary Probability ◽  
Functional Responses ◽  
Stationary Probability Distribution ◽  
Stochastic Permanence ◽  
Prey Model ◽  
Telegraph Noise

We propose a stochastic delayed one-predator and two-mutualistic-prey model perturbed by white noise and telegraph noise. By theM-matrix analysis and Lyapunov functions, sufficient conditions of stochastic permanence and extinction are established, respectively. These conditions are all dependent on the subsystems’ parameters and the stationary probability distribution of the Markov chain. We also investigate another asymptotic property and finally give two examples and numerical simulations to illustrate main results.

scholarly journals Positive 2D Discrete-Time Linear Lyapunov Systems

2009 ◽  
Vol 19 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Przemysław Przyborowski ◽  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Numerical Examples ◽  
Necessary And Sufficient ◽  
Time Linear

Positive 2D Discrete-Time Linear Lyapunov SystemsTwo models of positive 2D discrete-time linear Lyapunov systems are introduced. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. The discussion is illustrated with numerical examples.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

Discontinuous Lyapunov functions for a class of piecewise affine systems

2018 ◽  
Vol 41 (3) ◽  
pp. 729-736 ◽  
Author(s):  
Farideh Cheraghi-Shami ◽  
Ali-Akbar Gharaveisi ◽  
Malihe M Farsangi ◽  
Mohsen Mohammadian
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Monotonicity Condition ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Local Asymptotic Stability

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

scholarly journals On Stability Analysis for Generalized Neural Networks with Time-Varying Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
M. J. Park ◽  
O. M. Kwon ◽  
E. J. Cha
Keyword(s):  
Neural Networks ◽  
Stability Analysis ◽  
Asymptotic Stability ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Time Varying ◽  
Matrix Inequalities ◽  
Stability Criteria ◽  
Numerical Examples ◽  
Generalized Neural Networks

This paper deals with the problem of stability analysis for generalized neural networks with time-varying delays. With a suitable Lyapunov-Krasovskii functional (LKF) and Wirtinger-based integral inequality, sufficient conditions for guaranteeing the asymptotic stability of the concerned networks are derived in terms of linear matrix inequalities (LMIs). By applying the proposed methods to two numerical examples which have been utilized in many works for checking the conservatism of stability criteria, it is shown that the obtained results are significantly improved comparing with the previous ones published in other literature.

scholarly journals Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model

2016 ◽  
Vol 26 (4) ◽  
pp. 441-452 ◽  
Author(s):  
Andrzej Ruszewski
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Numerical Examples ◽  
New Model ◽  
The Stability ◽  
Partition Method ◽  
Necessary And Sufficient ◽  
Time Linear ◽  
Linear Scalar

Abstract The stability problems of fractional discrete-time linear scalar systems described by the new model are considered. Using the classical D-partition method, the necessary and sufficient conditions for practical stability and asymptotic stability are given. The considerations are il-lustrated by numerical examples.

scholarly journals Dynamical Analysis of a Stochastic Predator-Prey Model with an Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Feng Rao
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Allee Effect ◽  
Dynamical Analysis ◽  
Type Ii ◽  
Qualitative Properties ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Boundedness ◽  
Prey Model

We present and analyze a modified Holling type-II predator-prey model that includes some important factors such as Allee effect, density-dependence, and environmental noise. By constructing suitable Lyapunov functions and applying Itô formula, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. A series of numerical simulations to illustrate these mathematical findings are presented.

scholarly journals Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 331 ◽  
Author(s):  
Peng Liu ◽  
Xinzhu Meng ◽  
Haokun Qi
Keyword(s):  
Stochastic Model ◽  
Stationary Distribution ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Markov Semigroup ◽  
Threshold Analysis ◽  
Numerical Examples ◽  
Stochastic Properties ◽  
Symmetric Method ◽  
Temporary Immunity

In this paper, a stochastic model with relapse and temporary immunity is formulated. The main purpose of this model is to investigate the stochastic properties. For two incidence rate terms, we apply the ideas of a symmetric method to obtain the results. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the extinction and persistence of this system. Then, we investigate the existence of a stationary distribution for this model by employing the theory of an integral Markov semigroup. Finally, the numerical examples are presented to illustrate the analytical findings.

scholarly journals Common Weak Linear Copositive Lyapunov Functions for Positive Switched Linear Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Yuangong Sun ◽  
Zhaorong Wu ◽  
Fanwei Meng
Keyword(s):  
Stability Analysis ◽  
Complex Systems ◽  
Linear Systems ◽  
Algebraic Structure ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Switched Linear Systems ◽  
Numerical Examples ◽  
The Stability ◽  
Necessary And Sufficient

Lyapunov functions play a key role in the stability analysis of complex systems. In this paper, we study the existence of a class of common weak linear copositive Lyapunov functions (CWCLFs) for positive switched linear systems (PSLSs) which generalize the conventional common linear copositive Lyapunov functions (CLCLFs) and can be used as handy tool to deal with the stability of PSLSs not covered by CLCLFs. We not only establish necessary and sufficient conditions for the existence of CWCLFs but also clearly describe the algebraic structure of all CWCLFs. Numerical examples are also given to demonstrate the effectiveness of the obtained results.

scholarly journals A Mathematical Model of Cancer Treatment by Radiotherapy

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Zijian Liu ◽  
Chenxue Yang
Keyword(s):  
Mathematical Model ◽  
Periodic Solution ◽  
Asymptotic Stability ◽  
Cancer Treatment ◽  
Cancer Cells ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Numerical Examples ◽  
Globally Asymptotic Stability

A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shown to verify the validity of the results. A discussion is presented for further study.

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