Influence of the Fear Effect on a Holling Type II Prey–Predator System with a Michaelis–Menten Type Harvesting

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Binfeng Xie ◽  
Zhengce Zhang ◽  
Na Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Oscillation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Bifurcation Parameter ◽  
Fear Effect ◽  
Existence And Stability ◽  
Positive Equilibrium Point ◽  
The Stability ◽  
Predator System

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results.

scholarly journals Impact of the fear and Allee effect on a Holling type II prey–predator model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Binfeng Xie
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Jacobian Matrix ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Bifurcation Parameter ◽  
Fear Effect ◽  
Predator Model ◽  
The Stability ◽  
Theoretical Results

AbstractIn this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

Stability and Hopf Bifurcation in a Predator–Prey Model with the Cost of Anti-Predator Behaviors

2019 ◽  
Vol 29 (13) ◽  
pp. 1950185 ◽  
Author(s):  
Ting Qiao ◽  
Yongli Cai ◽  
Shengmao Fu ◽  
Weiming wang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle Oscillation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Existence And Stability ◽  
Cycle Oscillation ◽  
Predator Model ◽  
The Stability ◽  
The Cost

In this paper, we investigate the influence of anti-predator behavior in prey due to the fear of predators with a Beddington–DeAngelis prey–predator model analytically and numerically. We give the existence and stability of equilibria of the model, and provide the existence of Hopf bifurcation. In addition, we investigate the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also prevent the occurrence of limit cycle oscillation and increase the stability of the system.

scholarly journals Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lv-Zhou Zheng
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
The Stability

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

scholarly journals Dynamics Analysis of a Delayed Rumor Propagation Model in an Emergency-Affected Environment

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Chunru Li ◽  
Zujun Ma
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Propagation Model ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Rumor Propagation ◽  
Boundary Equilibrium ◽  
Normal Form Method ◽  
Positive Equilibrium Point ◽  
The Stability

Rumors influence people’s decisions in an emergency-affected environment. How to describe the spreading mechanism is significant. In this paper, we propose a delayed rumor propagation model in emergencies. By taking the delay as the bifurcation parameter, the local stability of the boundary equilibrium point and the positive equilibrium point is investigated and the conditions of Hopf bifurcation are obtained. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, some numerical simulations are also given to illustrate our theoretical results.

Oscillatory behavior of p53-Mdm2 system driven by transcriptional and translational time delays

2020 ◽  
Vol 13 (05) ◽  
pp. 2050034
Author(s):  
Chunyan Gao ◽  
Haihong Liu ◽  
Zengrong Liu ◽  
Yuan Zhang ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Cell Fate ◽  
Time Delays ◽  
Oscillatory Behavior ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Positive Equilibrium Point ◽  
The Stability

Biological experiments clarify that p53-Mdm2 module is the core of tumor network and p53 oscillation plays an important role in determining the tumor cell fate. In this paper, we investigate the effect of time delay on the oscillatory behavior induced by Hopf bifurcation in p53-Mdm2 system. First, the stability of the unique positive equilibrium point and the existence of Hopf bifurcation are investigated by using the time delay as the bifurcation parameter and by applying the bifurcation theory. Second, the explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. In addition, the combination of numerical simulation results and theoretical calculation results indicates that time delays in p53-Mdm2 system are critical for p53 oscillations. The results may help us to better understand the biological functions of p53 pathway and provide clues for treatment of cancer.

scholarly journals Fractional-Order Delayed Ross-Macdonald Model for Malaria Transmission

2021 ◽  
Author(s):  
Xinshu Cui ◽  
Dingyu Xue ◽  
Tingxue Li
Keyword(s):  
Hopf Bifurcation ◽  
Malaria Transmission ◽  
Dynamic Behavior ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Major Effect ◽  
Positive Equilibrium ◽  
Incubation Periods ◽  
Positive Equilibrium Point ◽  
The Stability

Abstract This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem and bifurcation theory, several sufficient conditions for the existence, uniqueness, local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of the system. The system becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.

Global Hopf Bifurcation for a Predator–Prey System with Three Delays

2017 ◽  
Vol 27 (07) ◽  
pp. 1750108 ◽  
Author(s):  
Zhichao Jiang ◽  
Lin Wang
Keyword(s):  
Hopf Bifurcation ◽  
Degree Theory ◽  
Positive Equilibrium ◽  
General Function ◽  
Normal Form Theory ◽  
Global Hopf Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Existence And Stability ◽  
The Stability

In this paper, a delayed predator–prey model is considered. The existence and stability of the positive equilibrium are investigated by choosing the delay [Formula: see text] as a bifurcation parameter. We see that Hopf bifurcation can occur as [Formula: see text] crosses some critical values. The direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are also determined by using the center manifold and normal form theory. Furthermore, based on the global Hopf bifurcation theorem for general function differential equations, which was established by J. Wu using fixed point theorem and degree theory methods, the existence of global Hopf bifurcation is investigated. Finally, numerical simulations to support the analytical conclusions are carried out.

scholarly journals Stability and Hopf Bifurcation in a Prey-Predator System with Disease in the Prey and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Predator System

This paper is concerned with a prey-predator system with disease in the prey and two delays. Local stability of the positive equilibrium of the system and existence of local Hopf bifurcation are investigated by choosing different combinations of the two delays as bifurcation parameters. For further investigation, the direction and the stability of the Hopf bifurcation are determined by using the normal form method and center manifold theorem. Finally, some numerical simulations are given to support the theoretical analysis.

scholarly journals Hopf Bifurcation of a Differential-Algebraic Bioeconomic Model with Time Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaojian Zhou ◽  
Xin Chen ◽  
Yongzhong Song
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Time Delays ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Bioeconomic Model ◽  
Numerical Tests ◽  
The Stability

We investigate the dynamics of a differential-algebraic bioeconomic model with two time delays. Regarding time delay as a bifurcation parameter, we show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Using the theories of normal form and center manifold, we also give the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical tests are provided to verify our theoretical analysis.

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