The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation. (26th November 2021)

• Record Type: Journal Article
• Authors: Kulenović, M. R. S.; O'Loughlin, Connor; Pilav, E.
• Other Names: Khan Abdul Qadeer Academic Editor.
• Is Part Of: Discrete dynamics in nature and society. Volume 2021(2021)
• Publication Date: 2021-11-26
• Subjects: System analysis -- Periodicals; Dynamics -- Periodicals; Chaotic behavior in systems -- Periodicals; Differentiable dynamical systems -- Periodicals; 003.05
• DOI: 10.1155/2021/2092709
• Languages: English
• ISSNs: 1026-0226

Abstract

Abstract : We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.