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. 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)

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Official URL: https://doi.org/10.1080/10236198.2023.2222855

Abstract / 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.

Item Type: Article
Uncontrolled Keywords: Planar Ricker model; non-invertible map; Lyapunov function; global stability; stable and unstable manifolds
Subjects: 500 Natural Sciences and Mathematics > 510 Mathematics
500 Natural Sciences and Mathematics > 570 Life sciences; biology
Department: School of Computing and Digital Media
Depositing User: Zhanyuan Hou
Date Deposited: 27 Jun 2023 12:47
Last Modified: 01 Aug 2023 14:13
URI: https://repository.londonmet.ac.uk/id/eprint/8616

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