scholarly journals Complex Behavior in a Fish Algae Consumption Model with Impulsive Control Strategy

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Jin Yang ◽  
Min Zhao
Keyword(s):  
Control Strategy ◽  
Small Amplitude ◽  
Impulsive Control ◽  
Asymptotically Stable ◽  
Perturbation Techniques ◽  
Globally Asymptotically Stable ◽  
Impulsive Equations ◽  
Consumption Model ◽  
The Rich ◽  
Impulsive Control Strategy

This paper investigates a dynamic mathematical model of fish algae consumption with an impulsive control strategy analytically. It is proved that the system has a globally asymptotically stable algae-eradication periodic solution and is permanent by using the theory of impulsive equations and small-amplitude perturbation techniques. Numerical results for impulsive perturbations demonstrate the rich dynamic behavior of the system. Further, we have also compared biological control with chemical control. All these results may be useful in controlling eutrophication.

scholarly journals A Mathematical Model for the Dynamics of a Fish Algae Consumption Model with Impulsive Control Strategy

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Jin Yang ◽  
Min Zhao
Keyword(s):  
Mathematical Model ◽  
Control Strategy ◽  
Floquet Theory ◽  
Impulsive Control ◽  
Impulsive Differential Equations ◽  
Asymptotically Stable ◽  
Perturbation Techniques ◽  
Globally Asymptotically Stable ◽  
The Rich ◽  
Impulsive Control Strategy

A dynamic mathematical model of fish algae consumption with an impulsive control strategy is proposed and analyzed in detail. It is shown that the system has a globally asymptotically stable algae-eradication periodic solution which can be obtained using the Floquet theory of impulsive differential equations and small-amplitude perturbation techniques. The conditions for the permanence of the system can also be determined. Numerical results for impulsive perturbations show the rich dynamic behavior of the system. All these results may be useful in controlling eutrophication.

DYNAMIC COMPLEXITIES IN A LOTKA–VOLTERRA PREDATOR–PREY MODEL CONCERNING IMPULSIVE CONTROL STRATEGY

2005 ◽  
Vol 15 (02) ◽  
pp. 517-531 ◽  
Author(s):  
BING LIU ◽  
YUJUAN ZHANG ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Control Strategy ◽  
Impulsive Control ◽  
Period Doubling ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pest Eradication ◽  
Impulsive Control Strategy

Based on the classical Lotka–Volterra predator–prey system, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the system can be permanent. We observe that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Numerical results show that the system we considered has more complex dynamics including period-doubling bifurcation, symmetry-breaking bifurcation, period-halving bifurcation, quasi-periodic oscillation, chaos and nonunique dynamics, meaning that several attractors coexist. Finally, a pest–predator stage-structured model for the pest concerning this kind of impulsive control strategy is proposed, and we also show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some threshold.

COMPLEX DYNAMICS OF ONE-PREY MULTI-PREDATOR SYSTEM WITH DEFENSIVE ABILITY OF PREY AND IMPULSIVE BIOLOGICAL CONTROL ON PREDATORS

2005 ◽  
Vol 08 (04) ◽  
pp. 483-495 ◽  
Author(s):  
YONGZHEN PEI ◽  
CHANGGUO LI ◽  
LANSUN CHEN ◽  
CHUNHUA WANG
Keyword(s):  
Biological Control ◽  
Control Strategy ◽  
Complex Dynamics ◽  
Impulsive Control ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Behavior Dynamics ◽  
Impulsive Control Strategy ◽  
Predator Model ◽  
Predator System

This work investigates the dynamic behaviors of one-prey multi-predator model with defensive ability of the prey by introducing impulsive biological control strategy. By using the Floquent theorem and the small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value, and a permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that the multi-predator impulsive control strategy is more effective than the classical one and makes the behavior dynamics of the system more complex.

scholarly journals An Impulsively Controlled Three-Species Prey-Predator Model with Stage Structure and Birth Pulse for Predator

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Yanyan Hu ◽  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Small Amplitude ◽  
Floquet Theory ◽  
Stage Structure ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Birth Pulse ◽  
Impulsive Equation ◽  
Predator Model ◽  
Predator System

We investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator. By using the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we study the permanence of the investigated model. Our results provide valuable strategy for biological economics management. Numerical analysis is also inserted to illustrate the results.

Dynamical Behaviors of a Pest Epidemic Model with Impulsive Control Over a Patchy Environment

2018 ◽  
Vol 28 (14) ◽  
pp. 1850173 ◽  
Author(s):  
Zhichun Yang ◽  
Cheng Chen ◽  
Lanzhu Zhang ◽  
Tingwen Huang
Keyword(s):  
Epidemic Model ◽  
Impulsive Control ◽  
Threshold Value ◽  
Impulsive System ◽  
Asymptotically Stable ◽  
Patchy Environment ◽  
Globally Asymptotically Stable ◽  
Dynamical Behaviors ◽  
Persistence Theory ◽  
Chaotic Characteristic

An epidemic model for pest management with impulsive control over a patchy environment is proposed in this paper. We investigate the dynamical behaviors on extinction and permanence and obtain the threshold value [Formula: see text] of dynamics for the impulsive system by utilizing a small amplitude perturbation method, matrix spectral analysis and persistence theory. We prove that the periodic pest-eradication solution of the system is globally asymptotically stable if [Formula: see text], while the system is persistent if [Formula: see text]. Furthermore, by discussion on the two-patch case, we analyze the effects of the dispersal and impulsive control on dynamical behaviors of the system. Some numerical examples are given to illustrate the effectiveness of the obtained results and to demonstrate the complexity such as chaotic characteristic of the system.

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