Bifurcation of nontrivial periodic solutions for an impulsively controlled pest management model

Georgescu, Paul; Zhang, Hong; Chen, Lansun

doi:10.1016/j.amc.2008.03.012

Cited by 47 publications

(30 citation statements)

References 13 publications

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“…It is also to be noted that in a related prey-dependent model (Georgescu et al, 2008;Zhang et al, 2007a) the condition which ensures the local stability of the trivial periodic solution is actually enough to imply its global stability as well, and this condition displays a significant dependence on the functional response of the predator. In our model, featuring a predator-dependent Beddington-DeAngelis functional response (and probably in any similar predator-dependent model which is not ratio-dependent), the local and global stability conditions, depending on the (per-pest) minimal and, respectively, maximal additional loss of pests in a period T due to predator consumption, are not the same.…”

Section: Resultsmentioning

confidence: 99%

“…See, for instance Liu et al (2004); Zhang et al (2004); Tang and Chen (2004); Liu et al (2005); Zhang et al (2007a, b); Georgescu et al (2008). All these prey-predator (target pest-its natural enemy) models do, however, assume that predators do not interfere with one another and consequently their functional and numerical responses depend upon the size of the prey population only.…”

Section: Introductionmentioning

confidence: 99%

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On the impulsive controllability and bifurcation of a predator–pest model of IPM

Zhang

Georgescu

Chen³

2008

Biosystems

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From a practical point of view, the most efficient strategy for pest control is to combine an array of techniques to control the wide variety of potential pests that may threaten crops in an approach known as integrated pest management (IPM). In this paper, we propose a predator-prey (pest) model of IPM in which pests are impulsively controlled by means of spraying pesticides (the chemical control) and releasing natural predators (the biological control). It is assumed that the biological and chemical control are used with the same periodicity, but not simultaneously. The functional response of the predator is allowed to be predator-dependent, in the form of a Beddington-DeAngelis functional response, rather than to have a perhaps more classical prey-only dependence. The local and global stability of the pest-eradication periodic solution, as well as the permanence of the system, are obtained under integral conditions which are shown to have biological significance. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the impulsive controllability and bifurcation of a predator–pest model of IPM

Zhang

Georgescu

Chen³

2008

Biosystems

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“…[12] (see also Ref. [16]) and denote by Φ(t; X 0 ) = Φ 1 (t, X 0 ), Φ 2 (t, X 0 ) the solution of the pulse-free system corresponding to (2) with the initial data X 0 = (x 1 0 , x 2 0 ). We also define two operators I 1 , I 2 : R 2 → R 2 by…”

Section: Bifurcationmentioning

confidence: 99%

Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield

Zhang

Georgescu

Nieto

et al. 2009

Appl. Math. Mech.-Engl. Ed.

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In this paper, we consider a variable yield model of a single-species growth in a well-stirred tank containing fresh medium, assuming the instances of time as triggering factors in which the nutrient refilling process and the removal of microorganisms by the uptake of lethal external antibiotic are initiated. It is also assumed that the periodic nutrient refilling and the periodic antibiotic injection occur with the same periodicity, but not simultaneously. The model is then formulated in terms of autonomous differential equations subject to impulsive perturbations. It is observed that either the population of microorganisms essentially washes out, or more favorably, the system is permanent. To describe this dichotomy, some biologically significant integral conditions are introduced. Further, it is shown that in a certain critical situation, a nontrivial periodic solution emerges via a bifurcation phenomenon. Finally, the dynamics of the model is illustrated with numerical experiments and computer simulations.

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“…From then on, the predator-prey model considering both pulse spraying pesticides and releasing natural enemies have been extensively studied by many researchers [3,4,13,[15][16][17][18][19][20][21]. Note that for all those studies the pulse functions introduced to the models could be summarized as the following generalized linear functions:…”

Section: Introductionmentioning

confidence: 99%

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

Wang

Tian

Tang

2017

Mathematical Problems in Engineering

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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.

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Bifurcation of nontrivial periodic solutions for an impulsively controlled pest management model

Cited by 47 publications

References 13 publications

On the impulsive controllability and bifurcation of a predator–pest model of IPM

On the impulsive controllability and bifurcation of a predator–pest model of IPM

Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield

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

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