scholarly journals Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Xue Zhang ◽  
Qing-ling Zhang
Keyword(s):  
Time Delay ◽  
Allee Effect ◽  
Control Method ◽  
Handling Time ◽  
Delayed Feedback Control ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
Predator Model ◽  
And Control

This paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic revenue increases through zero, which causes impulsive phenomenon. It can be noted that the impulsive phenomenon can be much weaker by strengthening Allee effect in numerical simulation. On the other hand, at a critical value of time delay, the model undergoes a Hopf bifurcation; that is, the increase of time delay destabilizes the model and bifurcates into small amplitude periodic solution. Moreover, a state delayed feedback control method, which can be implemented by adjusting the harvesting effort for biological populations, is proposed to drive the differential-algebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.

scholarly journals Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
P. K. Santra ◽  
G. S. Mahapatra ◽  
G. R. Phaijoo
Keyword(s):  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Phase Portraits ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
And Control

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

scholarly journals The Dynamics of a Delayed Predator-Prey Model with Double Allee Effect

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Boli Xie ◽  
Zhijun Wang ◽  
Yakui Xue ◽  
Zhenmin Zhang
Keyword(s):  
Time Delay ◽  
Allee Effect ◽  
Spatial Dynamics ◽  
Threshold Value ◽  
Positive Equilibrium ◽  
Spatiotemporal Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Turing Structures

We study the dynamics of a delayed predator-prey model with double Allee effect. For the temporal model, we showed that there exists a threshold of time delay in predator-prey interactions; when time delay is below the threshold value, the positive equilibriumE∗is stable. However, when time delay is above the threshold value, the positive equilibriumE∗is unstable and period solution will emerge. For the spatiotemporal model, through numerical simulations, we show that the model dynamics exhibit rich parameter space Turing structures. The obtained results show that this system has rich dynamics; these patterns show that it is useful for a delayed predator-prey model with double Allee effect to reveal the spatial dynamics in the real model.

scholarly journals Hopf bifurcation analysis in a stage-structure predator–prey model with two time delay

2022 ◽  
Vol 355 ◽  
pp. 03048
Author(s):  
Bochen Han ◽  
Shengming Yang ◽  
Guangping Zeng
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Time Delays ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Model

In this paper, we consider a predator-prey system with two time delays, which describes a prey–predator model with parental care for predators. The local stability of the positive equilibrium is analysed. By choosing the two time delays as the bifurcation parameter, the existence of Hopf bifurcation is studied. Numerical simulations show the positive equilibrium loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold.

scholarly journals Dynamics of a Diffusive Predator-Prey Model with Allee Effect on Predator

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaoqin Wang ◽  
Yongli Cai ◽  
Huihai Ma
Keyword(s):  
Allee Effect ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Flux Boundary Conditions ◽  
The Rich ◽  
Predator Model

The reaction-diffusion Holling-Tanner prey-predator model considering the Allee effect on predator, under zero-flux boundary conditions, is discussed. Some properties of the solutions, such as dissipation and persistence, are obtained. Local and global stability of the positive equilibrium and Turing instability are studied. With the help of the numerical simulations, the rich Turing patterns, including holes, stripes, and spots patterns, are obtained.

Hopf analysis of a differential-algebraic predator–prey model with Allee effect and time delay

2015 ◽  
Vol 08 (03) ◽  
pp. 1550041 ◽  
Author(s):  
Xue Zhang ◽  
Qingling Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Allee Effect ◽  
Functional Differential Equations ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey Model ◽  
Predator Model

A differential-algebraic prey–predator model with time delay and Allee effect on the growth of the prey population is investigated. Using differential-algebraic system theory, we transform the prey–predator model into its normal form and study its dynamics in terms of local analysis and Hopf bifurcation. By analyzing the associated characteristic equation, it is observed that the model undergoes a Hopf bifurcation at some critical value of time delay. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, and an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.

EXPERIMENTAL EVIDENCE FOR SPATIOTEMPORAL STABILITY AND CONTROL OF A ONE-WAY OPEN CML

2002 ◽  
Vol 12 (12) ◽  
pp. 2937-2944 ◽  
Author(s):  
TAKUYA IMAI ◽  
KEIJI KONISHI ◽  
HIDEKI KOKAME ◽  
KENTARO HIRATA
Keyword(s):  
Feedback Control ◽  
Experimental Evidence ◽  
Control Method ◽  
Spatiotemporal Chaos ◽  
Coupled Map Lattice ◽  
Delayed Feedback Control ◽  
Electronic Circuits ◽  
Stability And Control ◽  
Feedback Control Method ◽  
And Control

We present an experimental evidence for spatiotemporal stability of a real one-way open coupled map lattice implemented by electronic circuits. Furthermore, it is shown that the decentralized delayed feedback control method can suppress the spatial instability and the spatiotemporal chaos in the coupled map lattice circuits.

scholarly journals Chaotic Dynamics and Control of Discrete Ratio-Dependent Predator-Prey System

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Sarker Md. Sohel Rana
Keyword(s):  
Chaotic Dynamics ◽  
Control Method ◽  
State Feedback Control ◽  
Phase Portraits ◽  
Predator Prey ◽  
Prey System ◽  
Dynamics And Control ◽  
Feedback Control Method ◽  
Ratio Dependent ◽  
And Control

This study examines the complexity of a discrete-time predator-prey system with ratio-dependent functional response. We establish algebraically the conditions for existence of fixed points and their stability. We show that under some parametric conditions the system passes through a bifurcation (flip or Neimark-Sacker). Numerical simulations are presented not only to justify theoretical results but also to exhibit new complex behaviors which include phase portraits, orbits of periods 9, 19, and 26, invariant closed circle, and attracting chaotic sets. Moreover, we measure numerically the Lyapunov exponents and fractal dimension to confirm the chaotic dynamics of the system. Finally, a state feedback control method is applied to control chaos which exists in the system.

scholarly journals Dynamic Study of a Predator-Prey Model with Weak Allee Effect and Delay

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Ye ◽  
Hua Liu ◽  
Yu-mei Wei ◽  
Ming Ma ◽  
Kai Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Theoretical Analysis ◽  
Bifurcation Analysis ◽  
Allee Effect ◽  
Stability Switches ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Predator Model

In this paper, a prey-predator model and weak Allee effect in prey growth and its dynamical behaviors are studied in detail. The existence, boundedness, and stability of the equilibria of the model are qualitatively discussed. Bifurcation analysis is also taken into account. After incorporating the searching delay and digestion delay, we establish a delayed predator-prey system with Allee effect. The results show that there exist stability switches and Hopf bifurcation occurs while the delay crosses a set of critical values. Finally, we present some numerical simulations to illustrate our theoretical analysis.

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