scholarly journals An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Zhen Wang ◽  
Yuanfu Shao ◽  
Xianjia Fang ◽  
Xiangmin Ma
Keyword(s):  
Comparison Theorem ◽  
Lyapunov Functions ◽  
Floquet Theory ◽  
Control Strategies ◽  
Sufficient Conditions ◽  
Mutual Interference ◽  
Square Root ◽  
Functional Responses ◽  
Multiple Lyapunov Functions ◽  
Predator Model

An impulsive two-prey and one-predator model with square root functional responses, mutual interference, and integrated pest management is constructed. By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated. We use some methods and sufficient conditions to prove the permanence of the system which involve multiple Lyapunov functions and differential comparison theorem. Numerical simulations are given to portray the complex behaviors of this system. Finally, we analyze the biological meanings of these results and give some suggestions for feasible control strategies.

Stability of a Class of Piezoelectric State-Switching Methods

2000 ◽  
Author(s):  
Andrew J. Kurdila ◽  
William W. Clark ◽  
Weijian Wang ◽  
Dwayne E. McDaniel
Keyword(s):  
Lyapunov Functions ◽  
Closed Loop ◽  
Control Strategies ◽  
Short Circuit ◽  
Open Circuit ◽  
Hybrid Dynamical Systems ◽  
Multiple Lyapunov Functions ◽  
Control Laws ◽  
State Switching ◽  
The Stability

Abstract Experimental and anecdotal evidences have shown that state-switched control strategies for piezoelectric actuators can be advantageous. However, most discussions of the stability of these systems has relied on heuristic, or physically motivated, arguments. In this paper, we show that recent open-circuit/short-circuit state-switching control laws can be viewed as hybrid dynamical systems of Witsenhausen type. Within this framework, the closed-loop stability of OC/SC switching is rigorously established using the method of multiple Lyapunov functions.

scholarly journals Permanence of a Discrete Periodic Volterra Model with Mutual Interference and Beddington-DeAngelis Functional Response

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Runxin Wu
Keyword(s):  
Difference Equation ◽  
Functional Response ◽  
Comparison Theorem ◽  
Sufficient Conditions ◽  
Mutual Interference ◽  
Volterra Model

This paper discuss a discrete periodic Volterra model with mutual interference and Beddington-DeAngelis functional response. By using the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system. After that,we give an example to show the feasibility of our main result.

scholarly journals Impulsive Control on Seasonally Perturbed General Holling Type Two-Prey One-Predator Model

2016 ◽  
Vol 2016 ◽  
pp. 1-20
Author(s):  
Chandrima Banerjee ◽  
Pritha Das
Keyword(s):  
Floquet Theory ◽  
Impulsive Control ◽  
Impulsive System ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Functional Responses ◽  
Dynamical Behaviors ◽  
Predator Model ◽  
Conditions Of Stability ◽  
Fixed Times

We investigate the dynamical behaviors of two-prey one-predator model with general Holling type functional responses. The effect of seasonal perturbation on the model has been discussed analytically as well as numerically. The periodic fluctuation is considered in prey growth rate and the predator mortality rate of the model. The impulsive effects involving biological and chemical control strategy, periodic releasing of natural enemies, and spraying pesticide at different fixed times are introduced in the model with seasonal perturbation. We derive the conditions of stability for impulsive system using Floquet theory, small amplitude perturbation skills. A local asymptotically stable prey (pest) eradicated periodic solution is obtained when the impulsive period is less than some critical value. Numerical simulations of the model with and without seasonal disturbances exhibit different dynamics. Also we simulate numerically the model involving seasonal perturbations without impulse and with impulse. Finally, concluding remarks are given.

scholarly journals Analysis of an Impulsive One-Predator and Two-Prey System with Stage-Structure and Generalized Functional Response

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Xiangmin Ma ◽  
Yuanfu Shao ◽  
Zhen Wang ◽  
Xianjia Fang ◽  
Zhenguo Luo
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Global Attractivity ◽  
Control Strategies ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Reasonable Assumption ◽  
Functional Responses ◽  
Prey System ◽  
Dynamical Properties

An impulsive one-predator and two-prey system with stage-structure and generalized functional response is proposed and analyzed. By reasonable assumption and theoretical analysis, we obtain conditions for the existence and global attractivity of the predator-extinction periodic solution. Sufficient conditions for the permanence of this system are established via impulsive differential comparison theorem. Furthermore, abundant results of numerical simulations are given by choosing two different and concrete functional responses, which indicate that impulsive effects, stage-structure, and functional responses are vital to the dynamical properties of this system. Finally, the biological meanings of the main results and some control strategies are given.

scholarly journals Robust stability of impulsive switched systems with disturbance

2006 ◽  
Vol 48 (2) ◽  
pp. 259-270
Author(s):  
Xinzhi Liu ◽  
Hongtao Zhang
Keyword(s):  
Switched Systems ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Simple Approach ◽  
Disturbance Attenuation ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Multiple Lyapunov Functions ◽  
Impulsive Switched Systems ◽  
Bounded Disturbance

AbstractThis paper studies a class of impulsive switched systems with persistent bounded disturbance using robust attractor analysis and multiple Lyapunov functions. Some sufficient conditions for internal stability of the systems are obtained in terms of linear matrix inequalities (LMI). Based on the results, a simple approach for the design of a feedback controller is presented to achieve a desired level of disturbance attenuation. Numerical examples are also worked out to illustrate the obtained results.

scholarly journals Finite-Time Stability of Hybrid Systems With Unstable Modes

2021 ◽  
Vol 2 ◽  
Author(s):  
Kunal Garg ◽  
Dimitra Panagou
Keyword(s):  
Lyapunov Function ◽  
Hybrid Systems ◽  
Hybrid System ◽  
Finite Time ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Stability ◽  
Multiple Lyapunov Functions ◽  
Finite Time Stability ◽  
Continuous Flows

In this work, we study finite-time stability of hybrid systems with unstable modes. We present sufficient conditions in terms of multiple Lyapunov functions for the origin of a class of hybrid systems to be finite-time stable. More specifically, we show that even if the value of the Lyapunov function increases during continuous flow, i.e., if the unstable modes in the system are active for some time, finite-time stability can be guaranteed if the finite-time convergent mode is active for a sufficient amount of cumulative time. This is the first work on finite-time stability of hybrid systems using multiple Lyapunov functions. Prior work uses a common Lyapunov function approach, and requires the Lyapunov function to be decreasing during the continuous flows and non-increasing at the discrete jumps, thereby, restricting the hybrid system to only have stable modes, or to only evolve along the stable modes. In contrast, we allow Lyapunov functions to increase both during the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the earlier results in the related literature, and in effect allow the hybrid system to have unstable modes.

THE PERIODIC VOLTERRA MODEL WITH MUTUAL INTERFERENCE AND IMPULSIVE EFFECT

2012 ◽  
Vol 05 (03) ◽  
pp. 1260005 ◽  
Author(s):  
YUJUAN ZHANG ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Sufficient Conditions ◽  
Mutual Interference ◽  
Volterra Model ◽  
Impulsive Effect ◽  
Special Cases ◽  
Coexistence State ◽  
Theoretical Results

In this paper, a periodic Volterra model with mutual interference and impulsive effect is proposed and analyzed. By applying the Floquet theory of impulsive differential equation, some conditions are obtained for the linear stability of semi-trivial periodic solution. Some sufficient conditions are also given for the permanence of the system. Further, standard bifurcation theory is used to show the existence of coexistence state which arises near the semi-trivial periodic solution. Finally, theoretical results are confirmed by some special cases of the system.

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 Finite Time Stability of Stochastic Hybrid Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ying Yang ◽  
Guopei Chen
Keyword(s):  
Hybrid Systems ◽  
Finite Time ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Impulsive Effect ◽  
Stochastic Hybrid Systems ◽  
Time Stability ◽  
Multiple Lyapunov Functions ◽  
Common Lyapunov Function ◽  
Finite Time Stability

This paper considers the finite time stability of stochastic hybrid systems, which has both Markovian switching and impulsive effect. First, the concept of finite time stability is extended to stochastic hybrid systems. Then, by using common Lyapunov function and multiple Lyapunov functions theory, two sufficient conditions for finite time stability of stochastic hybrid systems are presented. Furthermore, a new notion called stochastic minimum dwell time is proposed and then, combining it with the method of multiple Lyapunov functions, a sufficient condition for finite time stability of stochastic hybrid systems is given. Finally, a numerical example is provided to illustrate the theoretical results.

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.

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