A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds
A quadratic Lyapunov function is demonstrated for the noninvertible planar Ricker map (x, y) → (xe^{r−x−αy}, ye^{s−y−βx}) which shows that for α,β > 0, and 0 < r, s ≤ 2 all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equi...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Taylor & Francis
2023
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| Online Access: | https://repository.londonmet.ac.uk/8616/1/A%20global%20picture%20for%20the%20planar%20Ricker%20map%20convergence%20to%20fixed%20points%20and%20identification%20of%20the%20stable%20unstable%20manifolds.pdf |
| _version_ | 1804072924914647040 |
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| author | Baigent, Stephen Hou, Zhanyuan Elaydi, Saber Balreira, E. C. Luís, Rafael |
| author_facet | Baigent, Stephen Hou, Zhanyuan Elaydi, Saber Balreira, E. C. Luís, Rafael |
| author_sort | Baigent, Stephen |
| collection | LMU |
| description | A quadratic Lyapunov function is demonstrated for the noninvertible planar Ricker map (x, y) → (xe^{r−x−αy}, ye^{s−y−βx}) which shows that for α,β > 0, and 0 < r, s ≤ 2 all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (0,∞)^2). Our approach bypasses and improves on methods that rely on monotonicity, which require 0 < r, s ≤ 1. We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle. |
| first_indexed | 2024-07-09T04:06:53Z |
| format | Article |
| id | oai:repository.londonmet.ac.uk:8616 |
| institution | London Metropolitan University |
| language | English |
| last_indexed | 2024-07-09T04:06:53Z |
| publishDate | 2023 |
| publisher | Taylor & Francis |
| record_format | eprints |
| spelling | oai:repository.londonmet.ac.uk:86162023-08-01T14:13:24Z http://repository.londonmet.ac.uk/8616/ A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds Baigent, Stephen Hou, Zhanyuan Elaydi, Saber Balreira, E. C. Luís, Rafael 510 Mathematics 570 Life sciences; biology A quadratic Lyapunov function is demonstrated for the noninvertible planar Ricker map (x, y) → (xe^{r−x−αy}, ye^{s−y−βx}) which shows that for α,β > 0, and 0 < r, s ≤ 2 all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (0,∞)^2). Our approach bypasses and improves on methods that rely on monotonicity, which require 0 < r, s ≤ 1. We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle. Taylor & Francis 2023-06-26 Article PeerReviewed text en cc_by_nc_nd_4 https://repository.londonmet.ac.uk/8616/1/A%20global%20picture%20for%20the%20planar%20Ricker%20map%20convergence%20to%20fixed%20points%20and%20identification%20of%20the%20stable%20unstable%20manifolds.pdf Baigent, Stephen, Hou, Zhanyuan, Elaydi, Saber, Balreira, E. C. and Luís, Rafael (2023) A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds. Journal of Difference Equations with Applications, 29 (5). pp. 575-591. ISSN 1023-6198 (print) 1563-5120 (online) https://doi.org/10.1080/10236198.2023.2222855 10.1080/10236198.2023.2222855 |
| spellingShingle | 510 Mathematics 570 Life sciences; biology Baigent, Stephen Hou, Zhanyuan Elaydi, Saber Balreira, E. C. Luís, Rafael A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds |
| title | A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds |
| title_full | A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds |
| title_fullStr | A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds |
| title_full_unstemmed | A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds |
| title_short | A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds |
| title_sort | global picture for the planar ricker map convergence to fixed points and identification of the stable unstable manifolds |
| topic | 510 Mathematics 570 Life sciences; biology |
| url | https://repository.londonmet.ac.uk/8616/1/A%20global%20picture%20for%20the%20planar%20Ricker%20map%20convergence%20to%20fixed%20points%20and%20identification%20of%20the%20stable%20unstable%20manifolds.pdf |
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