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

Complexity ◽  
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.

scholarly journals The Dynamics of a Predator-Prey System with State-Dependent Feedback Control

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hunki Baek
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Poincare Map ◽  
Predator Prey ◽  
Prey System ◽  
State Dependent ◽  
Fold Bifurcation

A Lotka-Volterra-type predator-prey system with state-dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period-one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period-doubling bifurcations.

scholarly journals The Impulsive Model with Pest Density and Its Change Rate Dependent Feedback Control

2020 ◽  
Vol 2020 ◽  
pp. 1-20 ◽  
Author(s):  
Ihsan Ullah Khan ◽  
Sanyi Tang
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Control Measures ◽  
Poincare Map ◽  
Change Rate ◽  
Action Threshold ◽  
Pest Density

The idea of action threshold depends on the pest density and its change rate is more general and furthermore can produce new modelling techniques related to integrated pest management (IPM) as compared with those that appeared in earlier studies, which definitely bring challenges to analytical analysis and generate new ideas to the state control measures. Keeping this in mind, using the strategies of IPM, we develop a prey-predator system with action threshold depending on the pest density and its change rate, and study its dynamical behavior. We develop new criteria guaranteeing the existence, uniqueness, local and global stability of order-1 periodic solutions. Applying the properties of Lambert W function, the Poincaré map is portrayed for the exact phase set, which is helpful to provide the sufficient conditions for the existence and stability of the interior order-1 periodic solutions and boundary order-1 periodic solution, also confirmed by numerical simulations. It is studied in detail that how and under what conditions the fixed point of Poincaré map and its stability are affected by the newly introduced action threshold. The analytical methods developed in this paper will be very beneficial to study other generalized models with state-dependent feedback control.

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

Complexity ◽  
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.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

scholarly journals Constrained control and rate or increment for linear systems with additive disturbances

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
Fouad Mesquine ◽  
Fernando Tadeo ◽  
Abdellah Benzaouia
Keyword(s):  
Feedback Control ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Ball Screw ◽  
Control Law ◽  
Constrained Control ◽  
Stabilizing Feedback Control ◽  
Bounded Disturbances ◽  
Control Of Linear Systems ◽  
Necessary And Sufficient

This paper is devoted to the control of linear systems with constrained control and rate or increment with additive bounded disturbances. Necessary and sufficient conditions such that the system evolution respects rate or increment constraints are used to derive stabilizing feedback control. The control law respects both constraints on control and its rate or increment and is robust against additive bounded disturbances. An application to a surface mount robot, where theY-axis of the machine uses a typical ball screw transmission driven by a DC motor to position circuits boards, is achieved.

scholarly journals On a Periodic Solution of the 4-Body Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-3 ◽  
Author(s):  
Jian Chen ◽  
Bingyu Li
Keyword(s):  
Periodic Solution ◽  
Angular Velocity ◽  
Periodic Motion ◽  
Equilateral Triangle ◽  
Sufficient Conditions ◽  
Constant Angular Velocity ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
The Masses ◽  
Necessary And Sufficient

We study the necessary and sufficient conditions on the masses for the periodic solution of planar 4-body problems, where three particles locate at the vertices of an equilateral triangle and rotate with constant angular velocity about a resting particle. We prove that the above periodic motion is a solution of Newtonian 4-body problems if and only if the resting particle is at the origin and the masses of the other three particles are equal and their angular velocity satisfies a special condition.

scholarly journals On periodic solutions to autonomous retarded functional differential equations

1990 ◽  
Vol 41 (3) ◽  
pp. 347-354
Author(s):  
Zhanyuan Hou
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Retarded Functional Differential Equations ◽  
Retarded Functional Differential Equation ◽  
Functional Differential ◽  
Necessary And Sufficient

Under the assumption that Ca = C([−r, 0], Sn−1(a)) is positively invariant for a > 0, two necessary and sufficient conditions are obtained for an autonomous retarded functional differential equation to have a non-trivial periodic solution in Ca. Moreover, a feasible sufficient condition is given, which is better for n = 2 than that given by Dos Reis and Baroni.

scholarly journals On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response

2021 ◽  
Vol 73 (4) ◽  
pp. 523-543
Author(s):  
N. N. Pelen
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Analytic Structure ◽  
Predator Prey ◽  
Periodic Environment ◽  
Globally Attractive ◽  
Necessary And Sufficient

UDC 517.9 In this study, the two-dimensional predator-prey system with Beddington–DeAngelis type functional response with impulses is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the considered system when it has at least one -periodic solution. This result is mainly based on the continuation theorem in the coincidence degree theory and to get the globally attractive -periodic solution of the given system, an inequality is given as the necessary and sufficient condition by using the analytic structure of the system.  

scholarly journals Robust set stabilization of Boolean control networks with impulsive effects

2018 ◽  
Vol 23 (4) ◽  
pp. 553-567 ◽  
Author(s):  
Xiaojing Xu ◽  
Yansheng Liu ◽  
Haitao Li ◽  
Fuad E. Alsaadi
Keyword(s):  
Feedback Control ◽  
State Feedback ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
State Feedback Control ◽  
Impulsive Effects ◽  
Algebraic Form ◽  
Control Networks ◽  
Set Stabilization ◽  
Necessary And Sufficient

This paper addresses the robust set stabilization problem of Boolean control networks (BCNs) with impulsive effects via the semi-tensor product method. Firstly, the closed-loop system consisting of a BCN with impulsive effects and a given state feedback control is converted into an algebraic form. Secondly, based on the algebraic form, some necessary and sufficient conditions are presented for the robust set stabilization of BCNs with impulsive effects under a given state feedback control and a free-form control sequence, respectively. Thirdly, as applications, some necessary and sufficient conditions are presented for robust partial stabilization and robust output tracking of BCNs with impulsive effects, respectively. The study of two illustrative examples shows that the obtained new results are effective.

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