scholarly journals Dynamics of a Beddington-DeAngelis Type Predator-Prey Model with Impulsive Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Sun Shulin ◽  
Guo Cuihua
Keyword(s):  
Small Amplitude ◽  
Floquet Theory ◽  
Impulsive Control ◽  
Control Strategies ◽  
Impulsive Effect ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Impulsive Equation ◽  
Predator System ◽  
Comparison Technique

In view of the logical consistence, the model of a two-prey one-predator system with Beddington-DeAngelis functional response and impulsive control strategies is formulated and studied systematically. By using the Floquet theory of impulsive equation, small amplitude perturbation method, and comparison technique, we obtain the conditions which guarantee the global asymptotic stability of the two-prey eradication periodic solution. We also proved that the system is permanent under some conditions. Numerical simulations find that the system appears the phenomenon of competition exclusion.

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.

scholarly journals Extinction and Permanence of a Three-Species Lotka-Volterra System with Impulsive Control Strategies

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
Hunki Baek
Keyword(s):  
Chemical Control ◽  
Small Amplitude ◽  
Floquet Theory ◽  
Impulsive Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Perturbation Techniques ◽  
Top Predator ◽  
Volterra System ◽  
Comparison Results

A three-species Lotka-Volterra system with impulsive control strategies containing the biological control (the constant impulse) and the chemical control (the proportional impulse) with the same period, but not simultaneously, is investigated. By applying the Floquet theory of impulsive differential equation and small amplitude perturbation techniques to the system, we find conditions for local and global stabilities of a lower-level prey and top-predator free periodic solution of the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. We also give a numerical example that seems to indicate the existence of chaotic behavior.

scholarly journals Complex Dynamics of Beddington–DeAngelis-Type Predator-Prey Model with Nonlinear Impulsive Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Changtong Li ◽  
Xiaozhou Feng ◽  
Yuzhen Wang ◽  
Xiaomin Wang
Keyword(s):  
Periodic Solution ◽  
Pest Management ◽  
Pest Control ◽  
Natural Enemy ◽  
Complex Dynamics ◽  
Impulsive Control ◽  
Control Strategies ◽  
Resource Limitation ◽  
Predator Prey ◽  
Predator Prey Model

According to resource limitation, a more realistic pest management is that the impulsive control actions should be adjusted according to the densities of both pest and natural enemy in the field, which result in nonlinear impulsive control. Therefore, we have proposed a Beddington–DeAngelis interference predator-prey model concerning integrated pest management with both density-dependent pest and natural enemy population. We find that the pest-eradication periodic solution is globally stable if the impulsive period is less than the critical value by Floquet theorem. The condition of permanent is established, and a stable positive periodic solution appears via a supercritical bifurcation by bifurcation theorem. Finally, in order to investigate the effects of those nonlinear control strategies on the successful pest control, the bifurcation diagrams showed that the model exists with very complex dynamics. Consequently, the resource limitation may result in pest outbreak in complex ways, which means that the pest control strategies should be carefully designed.

scholarly journals Seasonal Effects on a Beddington-DeAngelis Type Predator-Prey System with Impulsive Perturbations

2009 ◽  
Vol 2009 ◽  
pp. 1-19 ◽  
Author(s):  
Hunki Baek ◽  
Younghae Do
Keyword(s):  
Numerical Simulations ◽  
Periodic Solutions ◽  
Small Amplitude ◽  
Floquet Theory ◽  
Dynamical Behavior ◽  
Seasonal Effects ◽  
Predator Prey ◽  
Impulsive Equation ◽  
Prey System ◽  
Impulsive Perturbations

We study a Beddington-DeAngelis type predator-prey system with impulsive perturbation and seasonal effects. First, we numerically observe the influence of seasonal effects on the system without impulsive perturbations. Next, we find the conditions for the local and global stabilities of prey-free periodic solutions by using Floquet theory for the impulsive equation and small amplitude perturbation skills, and for the permanence of the system via comparison theorem. Finally, we show that seasonal effects and impulsive perturbation can give birth to various kinds of dynamical behavior of the system including chaotic phenomena by numerical simulations.

scholarly journals Dynamic Analysis of an Impulsive Predator-Prey Model with Disease in Prey and Ivlev-Type Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yuanfu Shao ◽  
Peiluan Li ◽  
Guoqiang Tang
Keyword(s):  
Functional Response ◽  
Small Amplitude ◽  
Floquet Theory ◽  
Complex Dynamics ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Pest Eradication ◽  
Disease In Prey

A predator-prey model with disease in prey, Ivlev-type functional response, and impulsive effects is proposed. By using Floquet theory and small amplitude perturbation skill, sufficient conditions of the existence and global stability of susceptible pest-eradication periodic solution are obtained. By impulsive comparison theorem, conditions ensuring the permanence of the system are established. Examples and simulation are given to show the complex dynamics for the key parameters.

scholarly journals Dynamic Analysis of an Impulsively Controlled Predator-Prey Model with Holling Type IV Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Yanzhen Wang ◽  
Min Zhao
Keyword(s):  
Functional Response ◽  
Impulsive Control ◽  
Control Strategies ◽  
Positive Periodic Solution ◽  
Pitchfork Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
Bifurcation Chaos

The dynamic behavior of a predator-prey model with Holling type IV functional response is investigated with respect to impulsive control strategies. The model is analyzed to obtain the conditions under which the system is locally asymptotically stable and permanent. Existence of a positive periodic solution of the system and the boundedness of the system is also confirmed. Furthermore, numerical analysis is used to discover the influence of impulsive perturbations. The system is found to exhibit rich dynamics such as symmetry-breaking pitchfork bifurcation, chaos, and nonunique dynamics.

scholarly journals Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Changtong Li ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Pest Control ◽  
Impulsive Control ◽  
Period Doubling ◽  
Dynamical Behaviour ◽  
Period Doubling Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model

Abstract An expectation for optimal integrated pest management is that the instantaneous numbers of natural enemies released should depend on the densities of both pest and natural enemy in the field. For this, a generalised predator–prey model with nonlinear impulsive control tactics is proposed and its dynamics is investigated. The threshold conditions for the global stability of the pest-free periodic solution are obtained based on the Floquet theorem and analytic methods. Also, the sufficient conditions for permanence are given. Additionally, the problem of finding a nontrivial periodic solution is confirmed by showing the existence of a nontrivial fixed point of the model’s stroboscopic map determined by a time snapshot equal to the common impulsive period. In order to address the effects of nonlinear pulse control on the dynamics and success of pest control, a predator–prey model incorporating the Holling type II functional response function as an example is investigated. Finally, numerical simulations show that the proposed model has very complex dynamical behaviour, including period-doubling bifurcation, chaotic solutions, chaos crisis, period-halving bifurcations and periodic windows. Moreover, there exists an interesting phenomenon whereby period-doubling bifurcation and period-halving bifurcation always coexist when nonlinear impulsive controls are adopted, which makes the dynamical behaviour of the model more complicated, resulting in difficulties when designing successful pest control strategies.

scholarly journals Periodicity and Permanence of a Discrete Impulsive Lotka-Volterra Predator-Prey Model Concerning Integrated Pest Management

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Chang Tan ◽  
Jun Cao
Keyword(s):  
Discrete System ◽  
Numerical Experiments ◽  
Critical Value ◽  
Euler Method ◽  
Impulsive Effect ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.

DYNAMICS ON A HOLLING II PREDATOR–PREY MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL

2012 ◽  
Vol 05 (03) ◽  
pp. 1260006 ◽  
Author(s):  
BING LIU ◽  
YE TIAN ◽  
BAOLIN KANG
Keyword(s):  
Periodic Solution ◽  
Chemical Control ◽  
Impulsive Control ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
State Dependent ◽  
Continuous Dynamic System ◽  
Continuous Dynamic ◽  
Holling Ii Functional Response

According to biological and chemical control strategy for pest control, a Holling II functional response predator–prey system concerning state-dependent impulsive control is investigated. We define the successor functions of semi-continuous dynamic system and give an existence theorem of order 1 periodic solution of such a system. By means of sequence convergence rules and qualitative analysis, we successfully get the conditions of existence and attractiveness of order 1 periodic solution. Our results show that our method used in this paper is more efficient and easier than the existing methods to prove the existence and attractiveness of order 1 periodic solution.

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