The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation
We present the bifurcation results for the difference equation xn+1=xn2/axn2+xn−12+f where a and f are positive numbers and the initial conditions x−1 and x0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a maj...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2021/2092709 |
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| author | M. R. S. Kulenović Connor O’Loughlin E. Pilav |
| author_facet | M. R. S. Kulenović Connor O’Loughlin E. Pilav |
| author_sort | M. R. S. Kulenović |
| collection | DOAJ |
| description | We present the bifurcation results for the difference equation xn+1=xn2/axn2+xn−12+f where a and f are positive numbers and the initial conditions x−1 and x0 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. |
| format | Article |
| id | doaj-art-7d3aed31dcdb45a4b6cc332f3e429d00 |
| institution | Kabale University |
| issn | 1607-887X |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-7d3aed31dcdb45a4b6cc332f3e429d002025-02-03T05:53:26ZengWileyDiscrete Dynamics in Nature and Society1607-887X2021-01-01202110.1155/2021/2092709The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference EquationM. R. S. Kulenović0Connor O’Loughlin1E. Pilav2Department of MathematicsDepartment of MathematicsDepartment of MathematicsWe present the bifurcation results for the difference equation xn+1=xn2/axn2+xn−12+f where a and f are positive numbers and the initial conditions x−1 and x0 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.http://dx.doi.org/10.1155/2021/2092709 |
| spellingShingle | M. R. S. Kulenović Connor O’Loughlin E. Pilav The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation Discrete Dynamics in Nature and Society |
| title | The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation |
| title_full | The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation |
| title_fullStr | The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation |
| title_full_unstemmed | The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation |
| title_short | The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation |
| title_sort | neimark sacker bifurcation and global stability of perturbation of sigmoid beverton holt difference equation |
| url | http://dx.doi.org/10.1155/2021/2092709 |
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