The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation

M R S Kulenović,E Pilav,Connor O’Loughlin,Abdul Qadeer Khan

doi:10.1155/2021/2092709

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.

Highlights

Introduction and PreliminariesIn this paper, we consider the difference equation xn+1 ax2n + x2n x2n−, f n 0, 1, . . . , (1)where the parameters a and f are positive numbers and the initial conditions x− 1 and x0 are nonnegative numbers
E rest of the paper is organized as follows; Section 2 gives local and global stability analysis of the zero equilibrium and positive equilibrium solutions in some regions of parameters; Section 3 presents the computation of Neimark–Sacker bifurcation; Section 4 presents the approximations of stable, unstable, and center manifolds of the equilibrium solutions of equation (1); Section 5 establishes that the rate of convergence of the solutions that converge to the zero equilibrium is quadratic while the rate of convergence of the solutions that converge to any positive equilibrium solution is linear
We give the global dynamics of the difference equation xn+1 x2n/(ax2n + x2n− 1 + f) where a and f are positive numbers and the initial conditions x− 1 and x0 are nonnegative numbers in a part of parametric space

Summary

Introduction and Preliminaries

Where the parameters a and f are positive numbers and the initial conditions x− 1 and x0 are nonnegative numbers. E Neimark–Sacker bifurcation is the discrete counterpart to the Hopf bifurcation for a system of ordinary differential equations in two or more dimensions, see [16,17,18] It occurs for such a discrete system depending on a parameter, λ for instance, along with a fixed point, the Jacobian matrix of which has a pair of complex conjugate eigenvalues. E rest of the paper is organized as follows; Section 2 gives local and global stability analysis of the zero equilibrium and positive equilibrium solutions in some regions of parameters; Section 3 presents the computation of Neimark–Sacker bifurcation; Section 4 presents the approximations of stable, unstable, and center manifolds of the equilibrium solutions of equation (1); Section 5 establishes that the rate of convergence of the solutions that converge to the zero equilibrium is quadratic while the rate of convergence of the solutions that converge to any positive equilibrium solution is linear

Local and Global Stability

The Neimark–Sacker Bifurcation

The Invariant Manifolds

Rate of Convergence

Conclusions