A study of stability and bifurcation analysis in discrete-time predator–prey system involving the Allee effect

2019 ◽  
Vol 12 (01) ◽  
pp. 1950011 ◽  
Author(s):  
Seval Işık
Keyword(s):  
Discrete Time ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Small Neighborhood ◽  
Invariant Circle ◽  
Predator Prey ◽  
Discrete Dynamic System ◽  
Local Asymptotic Stability ◽  
Prey System ◽  
Theoretical Results

This paper deals with a discrete-time predator–prey system which is subject to an Allee effect on prey. We investigate the existence and uniqueness and find parametric conditions for local asymptotic stability of fixed points of the discrete dynamic system. Moreover, using bifurcation theory, it is shown that the system undergoes Neimark–Sacker bifurcation in a small neighborhood of the unique positive fixed point and an invariant circle will appear. Then the direction of bifurcation is given. Furthermore, numerical analysis is provided to illustrate the theoretical discussions with the help of Matlab packages. Thus, the main theoretical results are supported with numerical simulations.

scholarly journals Dynamical behaviors in a discrete fractional-order predator-prey system

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5857-5874 ◽  
Author(s):  
Yao Shi ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Fractional Order ◽  
Control Strategies ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Theoretical Results

This paper is related to the dynamical behaviors of a discrete-time fractional-order predatorprey model. We have investigated existence of positive fixed points and parametric conditions for local asymptotic stability of positive fixed points of this model. Moreover, it is also proved that the system undergoes Flip bifurcation and Neimark-Sacker bifurcation for positive fixed point. Various chaos control strategies are implemented for controlling the chaos due to Flip and Neimark-Sacker bifurcations. Finally, numerical simulations are provided to verify theoretical results. These results of numerical simulations demonstrate chaotic behaviors over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behaviors in the model.

Analysis of a discrete-time predator–prey system with Allee effect

2011 ◽  
Vol 8 (1) ◽  
pp. 81-85 ◽  
Author(s):  
Wan-Xiong Wang ◽  
Yan-Bo Zhang ◽  
Chang-zhong Liu
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Predator Prey ◽  
Prey System

Dynamics in a discrete-time predator-prey system with Allee effect

2013 ◽  
Vol 29 (1) ◽  
pp. 143-164 ◽  
Author(s):  
Xian-wei Chen ◽  
Xiang-ling Fu ◽  
Zhu-jun Jing
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Predator Prey ◽  
Prey System

Impact of Prey Refuge on a Discrete Time Predator–Prey System with Allee Effect

2014 ◽  
Vol 24 (09) ◽  
pp. 1450106 ◽  
Author(s):  
Sourav Rana ◽  
Amiya Ranjan Bhowmick ◽  
Sabyasachi Bhattacharya
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Chaotic Behavior ◽  
Large Fraction ◽  
Chaotic Oscillation ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
The Stability ◽  
The Impact

We study the impact of the Allee effect and prey refuge on the stability of a discrete time predator–prey system. We focus on the stability behavior of the system with the Allee effect in predator, prey and both populations. Based on the combination of analytical and numerical results, we observe that: (1) the Allee effect stabilizes the systems dynamics in a moderate value of prey refuge. (2) For a large fraction of prey refuge no significant improvement in stability is observed due to Allee effect. (3) Refuge may play an important role in managing the populations which are subject to the Allee effect. The population remains stable at an intermediate level of refuge parameter, whereas at relatively low and high refuge effect, prey exhibits chaotic oscillation. Such chaotic behavior is suppressed in the presence of Allee effect. The Allee mechanism and refuge are considered simultaneously on the populations and is shown to have a significant impact on the predator–prey dynamics that may be helpful in the conservation of endangered species.

Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

2021 ◽  
Author(s):  
Haixia Li ◽  
Wenbin Yang ◽  
Meihua Wei ◽  
Aili Wang
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
A Priori ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
A Priori Bound ◽  
Steady State Solutions

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

scholarly journals Asymmetrical Impact of Allee Effect on a Discrete-Time Predator-Prey System

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Wenting Wang ◽  
Yujuan Jiao ◽  
Xiuping Chen
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Stable State ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Stable Conditions ◽  
Periodic Dynamics ◽  
Asymmetrical Impact

A discrete-time predator-prey model is proposed with Leslie-type numerical response, and the asymmetrical influence of Allee effect on the proposed system is investigated. By mathematical analysis, locally stable conditions for the equilibrium points of the considered systems with or without Allee effect are obtained firstly. Furthermore, numerical simulation is used to verify the results and detect some new outcomes. The results show that Allee effect on predator leads the system to its stable state in much longer time. Conversely, the prey population with Allee effect makes it much faster. In particular, a large value of Allee effect on prey results in periodic dynamics of the system.

Allee effect in a discrete-time predator–prey system

2009 ◽  
Vol 40 (4) ◽  
pp. 1956-1962 ◽  
Author(s):  
Canan Çelik ◽  
Oktay Duman
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Predator Prey ◽  
Prey System
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