Stability and Bifurcation in a Logistic Model with Allee Effect and Feedback Control

2020 ◽  
Vol 30 (15) ◽  
pp. 2050231
Author(s):  
Zhenliang Zhu ◽  
Mengxin He ◽  
Zhong Li ◽  
Fengde Chen
Keyword(s):  
Feedback Control ◽  
Logistic Model ◽  
Allee Effect ◽  
Control Variable ◽  
Positive Equilibrium ◽  
Threshold Condition ◽  
Local Asymptotic Stability ◽  
Linearized System ◽  
Positive Equilibrium Point ◽  
Two Parameters

This paper aims to study the dynamic behavior of a logistic model with feedback control and Allee effect. We prove the origin of the system is always an attractor. Further, if the feedback control variable and Allee effect are big enough, the species goes extinct. According to the analysis of the Jacobian matrix of the corresponding linearized system, we obtain the threshold condition for the local asymptotic stability of the positive equilibrium point. Also, we study the occurrence of saddle-node bifurcation, supercritical and subcritical Hopf bifurcations with the change of parameter. By calculating a universal unfolding near the cusp and choosing two parameters of the system, we can draw a conclusion that the system undergoes Bogdanov–Takens bifurcation of codimension-2. Numerical simulations are carried out to confirm the feasibility of the theoretical results. Our research can be regarded as a supplement to the existing literature on the dynamics of feedback control system, since there are few results on the bifurcation in the system so far.

scholarly journals Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Jinlei Liu ◽  
Wencai Zhao
Keyword(s):  
Feedback Control ◽  
Deterministic Model ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Theoretical Derivation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Delays ◽  
Positive Equilibrium Point

In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.

CONSEQUENCES OF MULTIPLE ALLEE EFFECT IN AN OPEN ACCESS FISHERY MODEL

2015 ◽  
Vol 23 (supp01) ◽  
pp. S101-S121 ◽  
Author(s):  
EDUARDO GONZÁLEZ-OLIVARES ◽  
JOSÉ D. FLORES
Keyword(s):  
Open Access ◽  
Allee Effect ◽  
Fishing Effort ◽  
Positive Equilibrium ◽  
Allee Effects ◽  
Continuous Time Model ◽  
Time Model ◽  
Parsimony Principle ◽  
Positive Equilibrium Point ◽  
Fishery Model

This work deals with the dynamics of a bioeconomic continuous time model, where the combined action of the fishing effort exerted by men (as a predator) and multiple Allee effect or depensation on the growth rate of a self-regenerating resource (the prey) are considered. It has been recently established that a depensation phenomenon appears by diverse causes and new functions have been proposed to describe multiple Allee effects. One of these formalizations is here incorporated in the well-known Smith's model, one of the simplest models to open access fisheries. We prove that this new and complex expression is topologically equivalent to a simpler form. Then, we postulate that the parsimony principle must be used to describe this phenomenon. It is also shown that in the phase plane of biomass-effort on the proposed model, the origin is an attractor equilibrium for all parameters values as a consequence of the Allee effect. Moreover, there is a subset of the parameter values, for which two limit cycles exist surrounding the unique positive equilibrium point of the system, one of them being asymptotically stable (the non damped oscillatory tragedy of the commons); hence, multiestability exists, particularly three-stability.

scholarly journals Global Stability for a Semidiscrete Logistic System with Feedback Control

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Li Xu ◽  
Shanshan Lou ◽  
Ruiwen Han
Keyword(s):  
Feedback Control ◽  
Lyapunov Function ◽  
Boundary Conditions ◽  
Global Stability ◽  
Logistic Model ◽  
Solution Method ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Positive Equilibrium ◽  
Feedback Controls

In this paper, a semidiscrete logistic model with the Dirichlet boundary conditions and feedback controls is proposed. By means of the sub- and supper-solution method and eigenvalue theory, the unique positive equilibrium is proved. By constructing a suitable Lyapunov function, the global asymptomatic stability of the unique positive equilibrium is investigated. Finally, numerical simulations are presented to verify the effectiveness of the main results.

scholarly journals Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Qamar Din ◽  
A. A. Elsadany ◽  
Hammad Khalil
Keyword(s):  
Numerical Simulations ◽  
Chaos Control ◽  
Hybrid Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Pole Placement ◽  
Positive Equilibrium ◽  
Local Asymptotic Stability ◽  
Positive Equilibrium Point ◽  
Plant Herbivore

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.

scholarly journals A New Global Stable Conclusion on a Diffusive Leslie-Gower Predator-Prey System with Additive Allee Effect

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Liu Yang
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Comparison Method ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
Lyapunov Function Method ◽  
Positive Equilibrium Point

In this work, a diffusive Leslie-Gower predator-prey model with additive Allee effect on prey under a homogeneous Neumann boundary condition is reconsidered. We establish new sufficient conditions for the global stability of the unique positive equilibrium point of the system by using the comparison method rather than the Lyapunov function method. It is shown that our result supplements and complements one of the main results of Yang and Zhong, 2015. Furthermore, numerical simulations are performed to consolidate the analytic finding.

scholarly journals Stability and Bifurcation Analysis of a Nonlinear Discrete Logistic Model with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Daiyong Wu ◽  
Hai Zhang ◽  
Jinde Cao ◽  
Tasawar Hayat
Keyword(s):  
Asymptotic Stability ◽  
Logistic Model ◽  
Bifurcation Theory ◽  
Polynomial Equation ◽  
Positive Equilibrium ◽  
Stability And Bifurcation Analysis ◽  
Linearized System ◽  
Bifurcation Direction ◽  
The Stability ◽  
Theoretical Results

We consider a nonlinear discrete logistic model with delay. The characteristic equation of the linearized system at the positive equilibrium is a polynomial equation involving high order terms. We obtain the conditions ensuring the asymptotic stability of the positive equilibrium and the existence of Neimark-Sacker bifurcation, with respect to the parameter of the model. Based on the bifurcation theory, we discuss Neimark-Sacker bifurcation direction and the stability of bifurcated solutions. Finally, some numerical simulations are performed to illustrate the theoretical results.

Use of vent stack temperature as a feedforward variable for dissolver total titratable alkali (TTA) control

TAPPI Journal ◽  
2018 ◽  
Vol 17 (05) ◽  
pp. 261-269
Author(s):  
Wei Ren ◽  
Brennan Dubord ◽  
Jason Johnson ◽  
Bruce Allison
Keyword(s):  
Feedback Control ◽  
Current Practice ◽  
Feedforward Control ◽  
Control Variable ◽  
Indirect Measurement ◽  
Tight Control ◽  
Transfer Lines ◽  
Green Liquor ◽  
Process Dynamics ◽  
Downstream Processes

Tight control of raw green liquor total titratable alkali (TTA) may be considered an important first step towards improving the overall economic performance of the causticizing process. Dissolving tank control is made difficult by the fact that the unknown smelt flow is highly variable and subject to runoff. High TTA variability negatively impacts operational costs through increased scaling in the dissolver and transfer lines, increased deadload in the liquor cycle, under- and over-liming, increased energy consumption, and increased maintenance. Current practice is to use feedback control to regulate the TTA to a target value through manipulation of weak wash flow while simultaneously keeping dissolver density within acceptable limits. Unfortunately, the amount of variability reduction that can be achieved by feedback control alone is fundamentally limited by the process dynamics. One way to improve upon the situation would be to measure the smelt flow and use it as a feedforward control variable. Direct measurement of smelt flow is not yet possible. The use of an indirect measurement, the dissolver vent stack temperature, is investigated in this paper as a surrogate feedforward variable for dissolving tank TTA control. Mill trials indicate that significant variability reduction in the raw green liquor TTA is possible and that the control improvements carry through to the downstream processes.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

2020 ◽  
pp. 471-488
Author(s):  
M. N. Srinivas ◽  
G. Basava Kumar ◽  
V. Madhusudanan
Keyword(s):  
Equilibrium Point ◽  
Disease Model ◽  
Stochastic Perturbation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Unique Positive Solution ◽  
Stochastic Nature ◽  
Research Article ◽  
Positive Equilibrium Point

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

Sign in / Sign up

Export Citation Format

Share Document