Stability, bifurcation, and chaos control in a discrete predator-prey model with strong Allee effect

Khabyah, Ali Al; Ahmed, Rizwan; Akram, Muhammad; Akhtar, Shahnaz

doi:10.3934/math.2023408

Cited by 14 publications

(10 citation statements)

References 33 publications

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“…Improper management of bifurcations can lead to disturbances in population dynamics and the consequent destruction of ecosystems. For a detailed bifurcation analysis, we recommend that readers refer to [31,[33][34][35][36][37][38][39].…”

Section: Local Bifurcation Analysismentioning

confidence: 99%

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

Ahmed,

Almatrafi

2023

Int. J. Anal. Appl.

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The complex dynamics of a predator-prey system in discrete time are studied. In this system, we consider the prey’s Gompertz growth and the square-root functional response. The existence of fixed points and stability are examined. Using the center manifold and bifurcation theory, we found that the system undergoes transcritical bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation. In addition, numerical examples are presented to illustrate the consistency of the analytical findings. The bifurcation diagrams show that the positive fixed point is stable if the death rate of the predator is greater than a threshold value. Biologically, it means that to prevent the predator population from growing uncontrollably and stability of the positive fixed point, the predator’s death rate should be greater than the threshold value.

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Section: Local Bifurcation Analysismentioning

confidence: 99%

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

Ahmed,

Almatrafi

2023

Int. J. Anal. Appl.

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“…Other recent studies on the dynamics of such games have been recently reported in the literature [28][29][30][31][32]. Recent studies on the complicated dynamics of such games have been reported in literature [33][34][35][36][37][38][39][40]. In addition, the case of Cournot-Bertrand on where decision variables are quantity and price has been recently studied in [41].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Analysis of a Simple Cournot Duopoly Model Based on a Computed Cost

Askar,

M. Alshamrani

2024

Discrete Dynamics in Nature and Society

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The paper is organized to study some mathematical properties and dynamics of a simple Cournot duopoly game based on a computed quadratic cost. The time evolution of this game is described by a two-dimensional noninvertible discrete time map using the bounded rationality mechanism. For this map, some dynamic characteristics such as multistability and synchronization are investigated. Its equilibrium points are obtained for the asymmetric case, and their conditions of stability are obtained. Our results investigate that the Nash equilibrium point may be unstable due to flip bifurcation and under certain parameter values, and Neimark–Sacker bifurcation is born after the period-4 cycle. Through some restrictions, the coordinate axes of the map construct an invariant manifold, and therefore, their dynamics can be analyzed by using a one-dimensional map. In the symmetric case, both firms behave identically, and this implies that the diagonal set forms an invariant manifold, and hence the synchronization phenomena take place. Furthermore, the global bifurcation of the map is confirmed through contact between critical curves and the boundaries of infeasible domains.

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“…Discrete-time models provide the most precise depiction of the dynamics shown by animals that participate in seasonal reproduction and have nonoverlapping generations. In addition, discrete models exhibit much more complicated dynamical patterns as compared to continuous-time models [1,2,5,14,19,29,30,44]. Thus, discrete-time models are more appealing than continuous ones.…”

Section: Introductionmentioning

confidence: 99%

Stability and bifurcation analysis of a discrete Leslie predator-prey model with fear effect

Abbas,

Ahmed

2024

VFAST trans. math.

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This study examines a predator-prey model that includes the impact of fear and a square-root functional responseto represent herd behavior in the prey population. Our investigation aims to investigate the existence and stabilityof fixed points in this model. Through conducting an extensive analysis, we have uncovered valuable observations onthe model's behavior, namely recognizing the occurrence of period-doubling and Neimark-Sacker bifurcations.These findings provide an understanding of the intricate dynamics that govern predator-prey interactions in the presence of fear and herd behavior. We provide numerical examples to support our conclusions.

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Stability, bifurcation, and chaos control in a discrete predator-prey model with strong Allee effect

Cited by 14 publications

References 33 publications

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

Dynamic Analysis of a Simple Cournot Duopoly Model Based on a Computed Cost

Stability and bifurcation analysis of a discrete Leslie predator-prey model with fear effect

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