On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator

Abstract

In this paper, a discrete predator-prey model with Allee effect which is obtained by the forward Euler method has been investigated. The local stability conditions of the model at the fixed point have been discussed. In addition, it is shown that the model undergoes Neimark-Sacker bifurcation by using bifurcation theory. Then, the direction of Neimark-Sacker bifurcation has been given. The OGY method is applied in order to control chaos in considered model due to emergence of Neimark-Sacker bifurcation. Some numerical simulations such as phase portraits and bifurcation figures have been presented to support the theoretical results. Also, the chaotic features are justified numerically by computing Lyapunov exponents. Because of consistency with the biological facts, the parameter values have been taken from literature [Controlling chaos and Neimark-Sacker bifurcation discrete-time predator-prey system, Hacet.

Qualitative Analysis of the Discretization of a Continuous Fractional Order Prey-Predator Model with the Effects of Harvesting and Immigration in the Population

“…Numerous investigations of the predator-prey relationship with Ivlev-type functional responses have been conducted. Te results suggested that Iviev-type relationships between the species have a number of models in ecological applications, including dynamics in host-parasite models [16], predator-prey models [17][18][19][20][21][22][23][24][25], animal coat patterns [26], and phytoplankton-zooplankton model [27].…”

“…Global dynamics, Neimark-Sacker bifurcation, and hybrid control in a Leslie's prey-predator model were all examined by Khan et al in [22]. Te discrete-time predator-prey model's stability, bifurcation, and chaos control are examined by the authors in [21], along with the Allee impact on the predator. Santra et al systematically examined the bifurcation analysis and chaos management of a discrete prey-predator model using a unique prey-refuge idea in [24].…”

Bifurcation and Chaos of a Nonlinear Discrete-Time Predator-Prey Model Involving the Nonlinear Allee Effect

“…In recent decades, the Allee efect in the discrete-time predator-prey model [22][23][24][25][26][27][28] has developed quickly. Celik and Duman [29] studied the stability of the positive equilibrium point for the discrete predator-prey model without the Allee efect:…”

Period-doubling Bifurcation and Stability in a Two Dimensional Discrete Prey-predator Model with Allee Effect and Immigration Parameter on Prey

“…In fact, nonlinear continuous models are discretized since nonlinear systems generally do not have analytic solutions expressible in terms of a finite representation of elementary functions. Authors in [5][6][7][8][9][10][11] analyzed dynamical analysis of different types of discrete-time predator-prey systems.…”