scholarly journals Dynamics of a Guanaco-sheep Competitive System with Unilateral and Bilateral Control

2021 ◽  
Author(s):  
Jing Xu ◽  
Mingzhan Huang ◽  
Xinyu Song
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Control System ◽  
Control Strategy ◽  
Impulsive Control ◽  
Bilateral Control ◽  
Competitive System ◽  
Existence And Stability ◽  
Uncontrolled System ◽  
Unilateral Control

Abstract In this paper, based on a guanaco-sheep competitive system, we develop and analyze mathematical models with unilateral and bilateral control for the management of overgrazing. We first analyze the dynamics of the uncontrolled system. Then, we investigate the system with impulsive control by differential equation geometry theory. And we mainly prove the existence and stability of order-1 periodic solution for unilateral control system and order-2 periodic solution for bilateral control system. Comparing the unilateral and bilateral control strategy, we encourage bilateral control rather than unilateral control for the management of sheep species.

scholarly journals Dynamics of Unilateral and Bilateral Control Systems with State Feedback for Renewable Resource Management

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Mingzhan Huang ◽  
Shouzong Liu ◽  
Xinyu Song ◽  
Lansun Chen
Keyword(s):  
Periodic Solution ◽  
Control System ◽  
Periodic Solutions ◽  
State Feedback ◽  
Control Strategies ◽  
Management Strategies ◽  
Sufficient Conditions ◽  
Renewable Resource ◽  
Bilateral Control ◽  
Unilateral Control

In this paper, mathematical models for the management of biological resources based on a given predator-prey relationship are proposed, and two types of control strategies, unilateral and bilateral control with impulsive state feedback, are studied. The existence of the order-1 homoclinic orbit, order-1 periodic solution, and bifurcation of homoclinic of the unilateral control system are obtained, and the attraction region of this system is also discussed. Besides, sufficient conditions for the existence and stability of the order-1 and order-2 periodic solutions of the bilateral control system are also gained. A series of numerical simulations including bifurcation diagrams of periodic solution are performed, which not only verify the theoretical results we get but also reveal some peculiar dynamical phenomena, such as the appearance of high-order periodic solutions and existence of parameter intervals with drastic order change of periodic solution. By comparing the two management strategies, our study encourages bilateral control rather than unilateral control for the risk of predator extinction.

Dynamical analysis of a two-species competitive system with state feedback impulsive control

2020 ◽  
Vol 13 (05) ◽  
pp. 2050007
Author(s):  
Jing Xu ◽  
Mingzhan Huang ◽  
Xinyu Song
Keyword(s):  
Periodic Solution ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Competitive System ◽  
Competitive Systems ◽  
State Dependent ◽  
Existence And Stability ◽  
The Stability ◽  
Theoretical Results

In this paper, three competitive systems with different kinds of state-dependent control are presented and investigated. The existence of the order-1 homoclinic orbit and order-1 periodic solution of the two systems that incorporate just one kind of state-dependent control is obtained by applying differential equation geometry theory, and the stability of the order-1 periodic solution of each system is also given. Besides, sufficient conditions for the existence and stability of the order-2 periodic solution of the system that incorporate two kinds of state-dependent control are gained by successor function method and analogue of Poincaré criterion, respectively. Finally, numerical simulations are carried out to verify the theoretical results.

scholarly journals Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Nattawut Khansai ◽  
Akapak Charoenloedmongkhon
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Impulsive Control ◽  
Positive Periodic Solution ◽  
Existence Condition ◽  
Predator Prey ◽  
Type Iv ◽  
Prey System ◽  
Existence And Stability ◽  
Predator Behavior

AbstractIn the present article, we propose and analyze a new mathematical model for a predator–prey system including the following terms: a Monod–Haldane functional response (a generalized Holling type IV), a term describing the anti-predator behavior of prey populations and one for an impulsive control strategy. In particular, we establish the existence condition under which the system has a locally asymptotically stable prey-eradication periodic solution. Violating such a condition, the system turns out to be permanent. Employing bifurcation theory, some conditions, under which the existence and stability of a positive periodic solution of the system occur but its prey-eradication periodic solution becomes unstable, are provided. Furthermore, numerical simulations for the proposed model are given to confirm the obtained theoretical results.

scholarly journals Multi-State Dependent Impulsive Control for Holling I Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Huidong Cheng ◽  
Fang Wang ◽  
Tongqian Zhang
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Chemical Control ◽  
Impulsive Control ◽  
Control Methods ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
State Dependent ◽  
Order One Periodic Solution

According to the different effects of biological and chemical control, we propose a model for Holling I functional response predator-prey system concerning pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove that the existence of order one periodic solution of such system and the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results which show that our method used in this paper is more efficient and easier than the existing ones for proving the existence of order one periodic solution.

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

2007 ◽  
Vol 15 (02) ◽  
pp. 219-234 ◽  
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.

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 Existence and Attractiveness of Order One Periodic Solution of a Holling I Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Huidong Cheng ◽  
Tongqian Zhang ◽  
Fang Wang
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Impulsive Control ◽  
Management Strategies ◽  
Type I ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
State Dependent ◽  
Order One Periodic Solution

According to the integrated pest management strategies, a Holling type I functional response predator-prey system concerning state-dependent impulsive control is investigated. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution, and the attractivity of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results which show that our method used in this paper is more efficient than the existing ones for proving the existence and attractiveness of order one periodic solution.

scholarly journals An Impulse Dynamic Model for Computer Worms

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chunming Zhang ◽  
Yun Zhao ◽  
Yingjiang Wu
Keyword(s):  
Periodic Solution ◽  
Numerical Simulations ◽  
Dynamic Model ◽  
Control Strategy ◽  
Impulsive Control ◽  
Vaccination Rate ◽  
Computer Worms ◽  
Spread Model ◽  
Impulsive Control Strategy ◽  
Globally Attractive

A worm spread model concerning impulsive control strategy is proposed and analyzed. We prove that there exists a globally attractive virus-free periodic solution when the vaccination rate is larger thanθ1. Moreover, we show that the system is uniformly persistent if the vaccination rate is less thanθ1. Some numerical simulations are also given to illustrate our main results.

scholarly journals Multi-State Dependent Impulsive Control for Pest Management

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Huidong Cheng ◽  
Fang Wang ◽  
Tongqian Zhang
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Qualitative Analysis ◽  
Pest Management ◽  
Pest Control ◽  
Impulsive Control ◽  
Management Strategies ◽  
Control Methods ◽  
State Dependent ◽  
Order One Periodic Solution

According to the integrated pest management strategies, we propose a model for pest control which adopts different control methods at different thresholds. By using differential equation geometry theory and the method of successor functions, we prove the existence of order one periodic solution of such system, and further, the attractiveness of the order one periodic solution by sequence convergence rules and qualitative analysis. Numerical simulations are carried out to illustrate the feasibility of our main results. Our results show that our method used in this paper is more efficient and easier than the existing ones for proving the existence of order one periodic solution.

scholarly journals Complex Dynamical Behaviors in a Predator-Prey System with Generalized Group Defense and Impulsive Control Strategy

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Shunyi Li
Keyword(s):  
Periodic Solution ◽  
Control Strategy ◽  
Impulsive Control ◽  
Period Doubling ◽  
Periodic Oscillation ◽  
Critical Value ◽  
Predator Prey ◽  
Prey System ◽  
Group Defense ◽  
Impulsive Control Strategy

A predator-prey system with generalized group defense and impulsive control strategy is investigated. By using Floquet theorem and small amplitude perturbation skills, a local asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, the system is permanent if the impulsive period is larger than the critical value. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest eradication lost its stability. Numerical examples show that the system considered has more complicated dynamics, including (1) high-order quasiperiodic and periodic oscillation, (2) period-doubling and halving bifurcation, (3) nonunique dynamics (meaning that several attractors coexist), and (4) chaos and attractor crisis. Further, the importance of the impulsive period, the released amount of mature predators and the degree of group defense effect are discussed. Finally, the biological implications of the results and the impulsive control strategy are discussed.

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