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 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 Dynamic Analysis of General Integrated Pest Management Model with Double Impulsive Control

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Yingke Li ◽  
Zhidong Teng ◽  
Kai Wang ◽  
Ahmadjan Muhammadhaji
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Impulsive Control ◽  
Impulsive Differential Equations ◽  
Threshold Value ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Globally Attractive

A general predator-prey model with disease in the prey and double impulsive control is proposed and investigated for the purpose of integrated pest management. By using the Floquet theory, the comparison theorem of impulsive differential equations, and the persistence theory of dynamical systems, we obtain that if threshold valueR0<1, then the susceptible pest eradication periodic solution is globally asymptotically stable and ifR0>1, then the model is permanent. The numerical examples not only illustrate the theoretical results, but also show that when the model is permanent, then it may possess a unique globally attractiveT-periodic solution.

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.

scholarly journals A Holling Type II Pest and Natural Enemy Model with Density Dependent IPM Strategy

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Xia Wang ◽  
Yuan Tian ◽  
Sanyi Tang
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Pest Control ◽  
Natural Enemies ◽  
Natural Enemy ◽  
Impulsive Control ◽  
Control Measures ◽  
Threshold Condition ◽  
Density Dependent

Resource limitations and density dependent releasing of natural enemies during the pest control and integrated pest management will undoubtedly result in nonlinear impulsive control. In order to investigate the effects of those nonlinear control strategies on the successful pest control, we have proposed a pest-natural enemy system concerning integrated pest management with density dependent instant killing rate and releasing rate. In particular, the releasing rate depicts how the number of natural enemy populations released was guided by their current density at the fixed moment. The threshold condition which ensures the existence and global stability of pest-free periodic solution has been discussed first, and the effects of key parameters on the threshold condition reveal that reducing the pulse period does not always benefit pest control; that is, frequent releasing of natural enemies may not be beneficial to the eradication of pests when the density dependent releasing method has been implemented. Moreover, the forward and backward bifurcations could occur once the pest-free periodic solution becomes unstable, and the system could exist with very complex dynamics. All those results confirm that the control actions should be carefully designed once the nonlinear impulsive control measures have been taken for pest management.

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.

scholarly journals Dynamic Analysis of a Predator-Prey (Pest) Model with Disease in Prey and Involving an Impulsive Control Strategy

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Min Zhao ◽  
Yanzhen Wang ◽  
Lansun Chen
Keyword(s):  
Integrated Pest Management ◽  
Pest Management ◽  
Control Strategy ◽  
Impulsive Control ◽  
Critical Parameters ◽  
Critical Conditions ◽  
Dynamic Complexity ◽  
Predator Prey ◽  
Impulsive Control Strategy ◽  
Disease In Prey

The dynamic behaviors of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy to release infected prey at fixed times are investigated for the purpose of integrated pest management. Mathematical theoretical works have been pursuing the investigation of the local asymptotical stability and global attractivity for the semitrivial periodic solution and population persistent, which depicts the threshold expression of some critical parameters for carrying out integrated pest management. Numerical analysis indicates that the impulsive control strategy has a strong effect on the dynamical complexity and population persistent using bifurcation diagrams and power spectra diagrams. These results show that if the release amount of infective prey can satisfy some critical conditions, then all biological populations will coexist. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.

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