On the Analysis of Stability, Bifurcation, and Chaos Control of Discrete-Time Predator-Prey Model with Allee Effect on Predator

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.

Download Full-text

PD Bifurcation and Chaos Behavior in a Predator-Prey Model With Allee Effect and Seasonal Perturbation

Recent Advances in Chaotic Systems and Synchronization ◽

10.1016/b978-0-12-815838-8.00011-x ◽

2019 ◽

pp. 211-232

Author(s):

Afef Ben Saad ◽

Olfa Boubaker ◽

Zeraoulia Elhadj

Keyword(s):

Allee Effect ◽

Bifurcation And Chaos ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model

Download Full-text

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420501941 ◽

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.

Download Full-text

Qualitative properties and bifurcations of discrete-time Bazykin–Berezovskaya predator–prey model

International Journal of Biomathematics ◽

10.1142/s1793524520500400 ◽

2020 ◽

Vol 13 (06) ◽

pp. 2050040

Author(s):

A. A. Elsadany ◽

Qamar Din ◽

S. M. Salman

Keyword(s):

Discrete Time ◽

Allee Effect ◽

Initial Conditions ◽

Qualitative Properties ◽

Step Size ◽

Time Model ◽

Predator Prey ◽

Discrete Time Model ◽

Predator Prey Model ◽

Prey Model

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.

Download Full-text