Dynamic Analysis of a Phytoplankton-Fish Model with the Impulsive Feedback Control Depending on the Fish Density and Its Changing Rate

Abstract This paper proposes a comprehensive fishing strategy that takes into consideration the population density of fish and its current growth rate, which provides new ideas for fishing strategies. Firstly, we establish a phytoplankton-fish model with the impulsive feedback control depending on the density and rate of change of the fish. Secondly, the complex phase and impulse sets of this model are divided into three cases, then the Poincar´e map for the model is defined, and analyzed the properties of Poincar´e map. In addition, the sufficient and necessary conditions for the global asymptotic stability of the order-1 periodic solution and existence condition of order- k ( k ≥ 2) periodic solution are discussed. The action threshold depends on the density and rate of change of the fish, which is reasonable than earlier studies. The analysis method proposed in this paper also plays an important role in the analysis of impulse models with complex phase sets or impulse sets.

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Dynamic Complexity of a Phytoplankton-Fish Model with the Impulsive Feedback Control by means of Poincaré Map

Complexity ◽

10.1155/2020/8974763 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Dezhao Li ◽

Yu Liu ◽

Huidong Cheng

Keyword(s):

Periodic Solution ◽

Feedback Control ◽

Poincaré Map ◽

Dynamic Change ◽

Sufficient Conditions ◽

Reference Value ◽

Poincare Map ◽

Dynamic Complexity ◽

Fish Model ◽

Necessary And Sufficient

The phytoplankton-fish model for catching fish with impulsive feedback control is established in this paper. Firstly, the Poincaré map for the phytoplankton-fish model is defined, and the properties of monotonicity, continuity, differentiability, and fixed point of Poincaré map are analyzed. In particular, the continuous and discontinuous properties of Poincaré map under different conditions are discussed. Secondly, we conduct the analysis of the necessary and sufficient conditions for the existence, uniqueness, and global stability of the order-1 periodic solution of the phytoplankton-fish model and obtain the sufficient conditions for the existence of the order-kk≥2 periodic solution of the system. Numerical simulation shows the correctness of our results which show that phytoplankton and fish with the impulsive feedback control can live stably under certain conditions, and the results have certain reference value for the dynamic change of phytoplankton in aquatic ecosystems.

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Dynamic Analysis of a Plankton-Herbivore State-Dependent Impulsive Model With Action Threshold Depending On The Density and Its Changing Rate

10.21203/rs.3.rs-710111/v1 ◽

2021 ◽

Author(s):

Wei Li ◽

Tonghua Zhang ◽

Yufei Wang ◽

Huidong Cheng

Keyword(s):

Periodic Solution ◽

Poincaré Map ◽

Complex Dynamics ◽

Economic Benefits ◽

Rate Of Change ◽

Poincare Map ◽

The Sustainable Development ◽

State Dependent ◽

Action Threshold ◽

Impulsive Model

Abstract A plankton-herbivore state-dependent impulsive model with nonlinear impulsive functions and action threshold including population density and rate of change is proposed. Since the use of action threshold makes the model have complex phase set and pulse set, we adopt the Poincaré map as a tool to study its complex dynamics. The Poincaré map is defined on the phase set and its properties in different situations are analyzed. Furthermore, the periodic solution of model are discussed, including the existence and stability conditions of the order-1 periodic solution and the existence of the order-k (k ≥ 2) periodic solutions. Compared with the fixed threshold in the existing literature, our results show that the use of action threshold is more practical, which is conducive to the sustainable development of population and makes people obtain more economic benefits. The analysis method used in this paper can study the complex dynamics of the model more comprehensively and deeply.

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

Advances in Difference Equations ◽

10.1186/s13662-021-03324-w ◽

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.

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The permanence and periodic solution of a competitive system with infinite delay, feedback control, and Allee effect

Advances in Difference Equations ◽

10.1186/s13662-018-1860-z ◽

2018 ◽

Vol 2018 (1) ◽

Cited By ~ 3

Author(s):

Lei Shi ◽

Hua Liu ◽

Yumei Wei ◽

Ming Ma ◽

Jianhua Ye

Keyword(s):

Periodic Solution ◽

Feedback Control ◽

Allee Effect ◽

Infinite Delay ◽

Competitive System ◽

Delay Feedback Control

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Poincaré Map Approach to Global Dynamics of the Integrated Pest Management Prey-Predator Model

Complexity ◽

10.1155/2020/2376374 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Zhenzhen Shi ◽

Qingjian Li ◽

Weiming Li ◽

Huidong Cheng

Keyword(s):

Periodic Solution ◽

Integrated Pest Management ◽

Pest Management ◽

Poincaré Map ◽

Global Dynamics ◽

Poincare Map ◽

Existence And Stability ◽

Sufficient And Necessary Conditions ◽

Predator Model ◽

Order One Periodic Solution

An integrated pest management prey-predator model with ratio-dependent and impulsive feedback control is investigated in this paper. Firstly, we determine the Poincaré map which is defined on the phase set and discuss its main properties including monotonicity, continuity, and discontinuity. Secondly, the existence and stability of the boundary order-one periodic solution are proved by the method of Poincaré map. According to the Poincaré map and related differential equation theory, the conditions of the existence and global stability of the order-one periodic solution are obtained when ΦyA<yA, and we prove the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution when ΦyA>yA. Furthermore, we prove the existence of the order-kk≥2 periodic solution under certain conditions. Finally, we verify the main results by numerical simulation.

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A Pest Management Model with Stage Structure and Impulsive State Feedback Control

Discrete Dynamics in Nature and Society ◽

10.1155/2015/617379 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12 ◽

Cited By ~ 4

Author(s):

Guoping Pang ◽

Zhiqing Liang ◽

Weijian Xu ◽

Lijie Li ◽

Gang Fu

Keyword(s):

Periodic Solution ◽

Feedback Control ◽

Pest Management ◽

State Feedback ◽

Stage Structure ◽

State Feedback Control ◽

Management Model ◽

Continuous Systems ◽

Impulsive State Feedback Control ◽

The Stability

A pest management model with stage structure and impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semicontinuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincarè criterion. Finally, we analyze numerically the theoretical results obtained.

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Impulsive state feedback control during the sulphitation reaction in process of manufacture of sugar

International Journal of Biomathematics ◽

10.1142/s179352452050076x ◽

2020 ◽

pp. 2050076

Author(s):

Guoping Pang ◽

Xianbo Sun ◽

Zhiqing Liang ◽

Silian He ◽

Xiaping Zeng

Keyword(s):

Periodic Solution ◽

Feedback Control ◽

Limit Cycles ◽

Switched Systems ◽

State Feedback ◽

State Feedback Control ◽

Continuous Systems ◽

Stability Criteria ◽

Impulsive State Feedback Control ◽

Existence And Stability

In this paper, the system with impulsive state feedback control corresponding to the sulphitation reaction in process of manufacture of sugar is considered. By means of square approximation and a series of switched systems, the periodic solution is approximated by a series of continuous hybrid limit cycles. Similar to the analysis of limit cycles of continuous systems, the existence and stability criteria of the order-1 periodic solution are obtained. Further, numerical analysis and discussion are given.

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Terrain-blind walking of planar underactuated bipeds via velocity decomposition-enhanced control

The International Journal of Robotics Research ◽

10.1177/0278364919870242 ◽

2019 ◽

Vol 38 (10-11) ◽

pp. 1307-1323 ◽

Cited By ~ 5

Author(s):

Martin Fevre ◽

Bill Goodwine ◽

James P Schmiedeler

Keyword(s):

Feedback Control ◽

Biped Robot ◽

Rate Of Change ◽

Energetic Cost ◽

Practical Implementation ◽

Leg Length ◽

Cost Of Transport ◽

Velocity Decomposition ◽

Novel Approach ◽

Hybrid Zero Dynamics

In this article, we develop and assess a novel approach for the control of underactuated planar bipeds that is based on velocity decomposition. The new controller employs heuristic rules that mimic the functionality of transverse linearization feedback control and that can be layered on top of a conventional hybrid zero dynamics (HZD)-based controller. The heuristics sought to retain HZD-based control’s simplicity and enhance disturbance rejection for practical implementation on realistic biped robots. The proposed control strategy implements a feedback on the time rate of change of the decomposed uncontrolled velocity and is compared with conventional HZD-based control and transverse linearization feedback control for both vanishing and non-vanishing disturbances. Simulation studies with a point-foot, three-link biped show that the proposed method has nearly identical performance to transverse linearization feedback control and outperforms conventional HZD-based control. For the non-vanishing case, the velocity decomposition-enhanced controller outperforms HZD-based control, but takes fewer steps on average before failure than transverse linearization feedback control when walking on uneven terrain without visual perception of the ground. The findings were validated experimentally on a planar, five-link biped robot for eight different uneven terrains. The velocity decomposition-enhanced controller outperformed HZD-based control while maintaining a relatively low specific energetic cost of transport (~0.45). The biped robot “blindly” traversed uneven terrains with changes in terrain height accumulating to 5% of its leg length using the stand-alone low-level controller.

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