Chaos control in a discrete-time predator–prey model with weak Allee effect

2018 ◽  
Vol 11 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Saheb Pal ◽  
Sourav Kumar Sasmal ◽  
Nikhil Pal
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

The stability of the predator–prey model subject to the Allee effect is an interesting topic in recent times. In this paper, we investigate the impact of weak Allee effect on the stability of a discrete-time predator–prey model with Holling type-IV functional response. The mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. We provide sufficient conditions for the flip bifurcation by considering Allee parameter as the bifurcation parameter. We observe that the model becomes stable from chaotic dynamics as the Allee parameter increases. Further, we observe bi-stability behavior of the model between only prey existence equilibrium and the coexistence equilibrium. Our analytical findings are illustrated through numerical simulations.

An Eco-Epidemic Predator–Prey Model with Allee Effect in Prey

2020 ◽  
Vol 30 (13) ◽  
pp. 2050194
Author(s):  
Absos Ali Shaikh ◽  
Harekrishna Das
Keyword(s):  
Allee Effect ◽  
Predator Population ◽  
Bifurcation And Chaos ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Disease In Predator ◽  
The Stability ◽  
The Impact ◽  
Existence Of Equilibria

This article describes the dynamics of a predator–prey model with disease in predator population and prey population subject to Allee effect. The positivity and boundedness of the solutions of the system have been determined. The existence of equilibria of the system and the stability of those equilibria are analyzed when Allee effect is present. The main objective of this study is to investigate the impact of Allee effect and it is observed that the system experiences Hopf bifurcation and chaos due to Allee effect. The results obtained from the model may be useful for analyzing the real-world ecological and eco-epidemiological systems.

BIFURCATION AND CHAOS IN A DISCRETE PREDATOR–PREY MODEL WITH HOLLING TYPE-III FUNCTIONAL RESPONSE AND HARVESTING EFFECT

2021 ◽  
pp. 1-28
Author(s):  
ANURAJ SINGH ◽  
PREETI DEOLIA
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Flip Bifurcation ◽  
Bifurcation Structure ◽  
Bifurcation And Chaos ◽  
Predator Prey ◽  
Type Iii ◽  
Predator Prey Model ◽  
Prey Model ◽  
Wide Range

In this paper, we study a discrete-time predator–prey model with Holling type-III functional response and harvesting in both species. A detailed bifurcation analysis, depending on some parameter, reveals a rich bifurcation structure, including transcritical bifurcation, flip bifurcation and Neimark–Sacker bifurcation. However, some sufficient conditions to guarantee the global asymptotic stability of the trivial fixed point and unique positive fixed points are also given. The existence of chaos in the sense of Li–Yorke has been established for the discrete system. The extensive numerical simulations are given to support the analytical findings. The system exhibits flip bifurcation and Neimark–Sacker bifurcation followed by wide range of dense chaos. Further, the chaos occurred in the system can be controlled by choosing suitable value of prey harvesting.

Qualitative analysis of a diffusive predator–prey model with Allee and fear effects

2021 ◽  
pp. 2150037
Author(s):  
Jia Liu
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Allee Effects ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Spatially Homogeneous ◽  
Fear Effects

In this study, we consider a diffusive predator–prey model with multiple Allee effects induced by fear factors. We investigate the existence, boundedness and permanence of the solution of the system. We also discuss the existence and non-existence of non-constant solutions. We derive sufficient conditions for spatially homogeneous (non-homogenous) Hopf bifurcation and steady state bifurcation. Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.

scholarly journals Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1280
Author(s):  
Liyun Lai ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

scholarly journals Discrete-Time Predator-Prey Model with Bifurcations and Chaos

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
K. S. Al-Basyouni ◽  
A. Q. Khan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

THE ROLE OF ADDITIONAL FOOD IN A PREDATOR–PREY MODEL WITH A PREY REFUGE

2016 ◽  
Vol 24 (02n03) ◽  
pp. 345-365 ◽  
Author(s):  
SUDIP SAMANTA ◽  
RIKHIYA DHAR ◽  
IBRAHIM M. ELMOJTABA ◽  
JOYDEV CHATTOPADHYAY
Keyword(s):  
Stable Equilibrium ◽  
Predator Population ◽  
Stable Limit ◽  
Prey Refuge ◽  
Additional Food ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

Hopf Bifurcation in a Delayed Diffusive Leslie–Gower Predator–Prey Model with Herd Behavior

2019 ◽  
Vol 29 (04) ◽  
pp. 1950055
Author(s):  
Fengrong Zhang ◽  
Yan Li ◽  
Changpin Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we consider a delayed diffusive predator–prey model with Leslie–Gower term and herd behavior subject to Neumann boundary conditions. We are mainly concerned with the impact of time delay on the stability of this model. First, for delayed differential equations and delayed-diffusive differential equations, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated respectively. It is observed that when time delay continues to increase and crosses through some critical values, a family of homogeneous and inhomogeneous periodic solutions emerge. Then, the explicit formula for determining the stability and direction of bifurcating periodic solutions are also derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are shown to support the analytical results.

Bifurcations and Control in a Discrete Predator–Prey Model with Strong Allee Effect

2018 ◽  
Vol 28 (05) ◽  
pp. 1850062 ◽  
Author(s):  
Limin Zhang ◽  
Lan Zou
Keyword(s):  
Allee Effect ◽  
Hybrid Control ◽  
Approximate Expression ◽  
Point Of View ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Strong Resonance ◽  
Prey Model ◽  
Strong Allee Effect ◽  
The Stability

In this paper, the bifurcations and the control of a discrete predator–prey model with strong Allee effect on the prey are investigated. This shows that the model undergoes a supercritical Neimark–Sacker bifurcation. Meanwhile, the explicit approximate expression of the stable closed invariant curve caused by the Neimark–Sacker bifurcation is given. 1:3 strong resonance is investigated through approximation by a flow, and the bifurcation curves around 1:3 resonance are obtained. Moreover, for the sake of regulating the stability of the biological system, we extend the hybrid control strategy to control the Neimark–Sacker bifurcation and 1:3 strong resonance. The theoretical analyses are validated by numerical simulations and are explained from the biological point of view.

Stability and Bifurcation Analysis in a Delayed Predator-Prey Model with Stage-Structure and Harvesting

2010 ◽  
Vol 143-144 ◽  
pp. 1358-1363
Author(s):  
Zhi Chao Jiang ◽  
Ming Wei Nie
Keyword(s):  
Bifurcation Analysis ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Stability And Bifurcation Analysis ◽  
Characteristic Values ◽  
The Stability

In this paper, we investigate a delayed stage-structured predator-prey model with continuous harvesting on prey. Positivity and boundness of solutions and sufficient conditions of the stability of equilibria are obtained. Using and as bifurcation parameters, the existence of Hopf bifurcations at equilibria is established by analyzing the distribution of the characteristic values.

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