Dynamics of a density-dependent predator-prey biological system with nonlinear impulsive control

Yuan Tian;  ; Sanyi Tang;

doi:10.3934/mbe.2021362

Dynamics of a density-dependent predator-prey biological system with nonlinear impulsive control

Mathematical Biosciences and Engineering ◽

10.3934/mbe.2021362 ◽

2021 ◽

Vol 18 (6) ◽

pp. 7318-7343

Author(s):

Yuan Tian ◽

◽

Sanyi Tang ◽

Keyword(s):

Periodic Solution ◽

Integrated Pest Management ◽

Pest Management ◽

Natural Enemies ◽

Impulsive Control ◽

Control Strategies ◽

Integrated Management ◽

Rapid Development ◽

Dynamical Behavior ◽

Density Dependent

<abstract><p>Spraying insecticides and releasing natural enemies are two commonly used methods in the integrated pest management strategy. With the rapid development of biotechnology, more and more realistic factors have been considered in the establishment and implementation of the integrated pest management models, such as the limited resources, the mutual restriction between pests and natural enemies, and the monitoring data of agricultural insects. Given these realities, we have proposed a pest-natural enemy integrated management system, which is a nonlinear state-dependent feedback control model. Besides the anti-predator behavior of the pests to the natural enemies is considered, the density dependent killing rate of pests and releasing amount of natural enemies are also introduced into the system. We address the impulsive sets and phase sets of the system in different cases, and the analytic expression of the Poincaré map which is defined in the phase set was investigated. Further we analyze the existence, uniqueness, global stability of order-1 periodic solution. In addition, the existence of periodic solution of order-$ k $ ($ k\geq2 $) is discussed. The theoretical analyses developed here not only show the relationship between the economic threshold and the other key factors related to pest control, but also reveal the complex dynamical behavior induced by the nonlinear impulsive control strategies.</p></abstract>

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

Mathematical Problems in Engineering ◽

10.1155/2017/8683207 ◽

2017 ◽

Vol 2017 ◽

pp. 1-12 ◽

Cited By ~ 3

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.

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A NEW MATHEMATICAL MODEL FOR OPTIMAL CONTROL STRATEGIES OF INTEGRATED PEST MANAGEMENT

Journal of Biological System ◽

10.1142/s0218339007002143 ◽

2007 ◽

Vol 15 (02) ◽

pp. 219-234 ◽

Cited By ~ 14

Author(s):

XINZHU MENG ◽

ZHITAO SONG ◽

LANSUN CHEN

Keyword(s):

Periodic Solution ◽

Control Problem ◽

Integrated Pest Management ◽

Pest Management ◽

Control Strategy ◽

Impulsive Control ◽

Control Strategies ◽

Economic Damage ◽

Pest Population ◽

The Given

A state-dependent impulsive SI epidemic model for integrated pest management (IPM) is proposed and investigated. We shall examine an optimal impulsive control problem in the management of an epidemic to control a pest population. We introduce a small amount of pathogen into a pest population with the expectation that it will generate an epidemic and that it will subsequently be endemic such that the number of pests is no larger than the given economic threshold (ET), so that the pests cannot cause economic damage. This is the biological control strategy given in the present paper. The combination strategy of pulse capturing (susceptible individuals) and pulse releasing (infective individuals) is implemented in the model if the number of pests (susceptible) reaches the ET. Firstly, the impulsive control problem is to drive the pest population below a given pest level and to do so in a manner which minimizes a weighted sum of the cost of using the control. Hence, for a one time impulsive effect we obtain the optimal strategy in terms of total cost such that the number of pests is no larger than the given ET. Secondly, we show the existence of periodic solution with the number of pests no larger than ET, and by using the Analogue of the Poincaré Criterion we prove that it is asymptotically stable under a planned impulsive control strategy. Further, the period T of the periodic solution is calculated, which can be used to estimate how long the pest population will take to return back to its pre-control level. The main feature of the present paper is to apply an SI infectious disease model to IPM, and some pests control strategies are given.

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IMPULSIVE CONTROL STRATEGIES FOR PEST MANAGEMENT

Journal of Biological System ◽

10.1142/s0218339007002106 ◽

2007 ◽

Vol 15 (02) ◽

pp. 235-260 ◽

Cited By ~ 20

Author(s):

HONG ZHANG ◽

LANSUN CHEN ◽

PAUL GEORGESCU

Keyword(s):

Periodic Solution ◽

Pest Management ◽

Impulsive Control ◽

Control Strategies ◽

Integrated Management ◽

Management Strategies ◽

Critical Value ◽

Globally Asymptotically Stable ◽

Pest Eradication ◽

Low Levels

In this paper, we propose two impulsive differential systems concerning biological and, respectively, integrated pest management strategies. In each case, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution on condition that the amount of infective pests released periodically is larger than a certain critical value. When the amount of infective pests released is less than this critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its stability. Further, the existence of a non-trivial periodic solution is also studied by means of numerical simulations. In the case in which a single control is used, one can only use the amount of infective pests which are periodically released in order to control pests at desirable low levels, while in the case in which integrated management is used, one can use the proportion of pests removed by means of spraying chemical pesticides together with the amount of infective pests which are periodically released to control pests at desirable low levels.

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Two Generalized Predator-Prey Models for Integrated Pest Management with Stage Structure and Disease in the Prey Population

Abstract and Applied Analysis ◽

10.1155/2013/424723 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Ruiqing Shi ◽

Sanyi Tang ◽

Wenli Feng

Keyword(s):

Integrated Pest Management ◽

Pest Management ◽

Natural Enemies ◽

Impulsive Control ◽

Control Strategies ◽

Sufficient Conditions ◽

Stage Structure ◽

Predator Prey ◽

Predator Prey Models ◽

Impulsive Release

Stage-structured predator-prey models with disease in the prey are constructed. For the purpose of integrated pest management, two types of impulsive control strategies (impulsive release of infective prey and impulsive release of predator) are used. For Case 1, infective prey applications are more frequent than releases of predator (natural enemies). For Case 2, predator (natural enemies) releases are more frequent than infective prey applications. In both cases, we get the sufficient conditions for the global attractivity of the susceptible prey-eradication periodic solution. In addition, the persistence of the systems is also discussed. At last, the results are discussed and some possible future work is put forward.

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

Advances in Difference Equations ◽

10.1186/s13662-020-02971-9 ◽

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.

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

Complexity ◽

10.1155/2020/8829235 ◽

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.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2015/839097 ◽

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.

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Intercropping Rosemary (Rosmarinus officinalis) with Sweet Pepper (Capsicum annum) Reduces Major Pest Population Densities without Impacting Natural Enemy Populations

Insects ◽

10.3390/insects12010074 ◽

2021 ◽

Vol 12 (1) ◽

pp. 74

Author(s):

Xiao-wei Li ◽

Xin-xin Lu ◽

Zhi-jun Zhang ◽

Jun Huang ◽

Jin-ming Zhang ◽

...

Keyword(s):

Integrated Pest Management ◽

Pest Management ◽

Pest Control ◽

Natural Enemies ◽

Natural Enemy ◽

Sweet Pepper ◽

Population Densities ◽

Pest Population ◽

Major Pest ◽

Population Suppression

Intercropping of aromatic plants provides an environmentally benign route to reducing pest damage in agroecosystems. However, the effect of intercropping on natural enemies, another element which may be vital to the success of an integrated pest management approach, varies in different intercropping systems. Rosemary, Rosmarinus officinalis L. (Lamiaceae), has been reported to be repellent to many insect species. In this study, the impact of sweet pepper/rosemary intercropping on pest population suppression was evaluated under greenhouse conditions and the effect of rosemary intercropping on natural enemy population dynamics was investigated. The results showed that intercropping rosemary with sweet pepper significantly reduced the population densities of three major pest species on sweet pepper, Frankliniella intonsa, Myzus persicae, and Bemisia tabaci, but did not affect the population densities of their natural enemies, the predatory bug, Orius sauteri, or parasitoid, Encarsia formosa. Significant pest population suppression with no adverse effect on released natural enemy populations in the sweet pepper/rosemary intercropping system suggests this could be an approach for integrated pest management of greenhouse-cultivated sweet pepper. Our results highlight the potential of the integration of alternative pest control strategies to optimize sustainable pest control.

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Effects of Different Water Regimes and Spacing on Insect Pest Infestation and Efficacy of Control Measures in Cucumber (Cucumis sativus)

Journal of Experimental Agriculture International ◽

10.9734/jeai/2019/v41i530415 ◽

2019 ◽

pp. 1-8

Author(s):

A. A. Oso ◽

G. O. Awe

Keyword(s):

Integrated Pest Management ◽

Pest Management ◽

Pest Control ◽

Control Strategies ◽

Insect Pest ◽

Cultural Control ◽

Irrigated Agriculture ◽

Yield Attributes ◽

Pest Infestation ◽

And Control

Aim: Information on the influence of water availability during different seasons of rainfed or irrigated agriculture as it relates to insect pest population build-up in crops could assist in the development of integrated pest management. A study was therefore conducted to investigate effects of spacing, pest infestation and control on cucumber under rainfed and irrigated conditions. Place and Duration of Study: At the Teaching and Research Farm, Ekiti State University, Ado Ekiti, Nigeria during the 2016/2017 rainy and dry seasons. Methodology: The experiment was laid out using randomized complete block design (RCBD) in a split-plot arrangement in five replications, with spacing (60 x 60 cm, 60 x 90 cm and 60 x 120 cm) as the main plot treatments and the sub-plot treatments were different pest control strategies. The pest control strategies include synthetic insecticide (Lambda-cyhalothrin), botanical insecticide (Anogeissus leiocarpus) and control. Growth parameters and yield attributes were recorded. Insect pest occurrence, their build-up and percentage infestation on cucumber and the efficacy of the management strategies were monitored. Results: The results showed that yield was enhanced in irrigated system with the widest spacing of 60 x 120 cm botanical treatment interaction. Bemisia tabaci was the most prominent insect pest attacking cucumber under irrigated system. Conclusion: Other cultural control practices such as the use of trap crops with little or no financial implication should also be added to botanical pesticides as an integrated pest management tactic for effective management and control of the pest.

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