The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems

Emma D'Aniello; Saber Elaydi

doi:10.3934/dcdsb.2019195

Article Contents

2020, Volume 25, Issue 3: 903-915. Doi: 10.3934/dcdsb.2019195

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The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems

Emma D'Aniello^1, and
Saber Elaydi^2, ,

1.
Dipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli", Viale Lincoln n.5, 81100 Caserta, Italia
2.
Department of Mathematics, Trinity University, San Antonio, TX 78212-7200, USA

^* Corresponding author: Saber Elaydi
^* Corresponding author: Saber Elaydi

Revised: November 30, 2018

Early access: September 2019

Published: March 2020

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Abstract

We consider a discrete non-autonomous semi-dynamical system generated by a family of continuous maps defined on a locally compact metric space. It is assumed that this family of maps uniformly converges to a continuous map. Such a non-autonomous system is called an asymptotically autonomous system. We extend the dynamical system to the metric one-point compactification of the phase space. This is done via the construction of an associated skew-product dynamical system. We prove, among other things, that the omega limit sets are invariant and invariantly connected. We apply our results to two populations models, the Ricker model with no Allee effect and Elaydi-Sacker model with the Allee effect, where it is assumed that the reproduction rate changes with time due to habitat fluctuation.

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Mathematics Subject Classification: Primary: 54H20, 37C70; Secondary: 39A05.

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Figure 1. The space $ \widehat {\cal F} = \left\{ {{f_n}:n = 0,1,2, \ldots } \right\} \cup \left\{ f \right\} $, where $ f_{n} \rightarrow f $, uniformly, as $ n \rightarrow \infty $. If $ x_{0} $ is on the fiber $ {\mathcal X}_{0} $, then $ f_{0}(x_{0}) = x_{1} $ is in the fiber $ {\mathcal X}_{1} $, and $ f_{1}(x_{1}) = x_{2} $ is on the fiber $ {\mathcal X}_{2} $, etc

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Figure 2. This commuting diagram illustrates the notion of a skew product discrete semidynamical system. Here $ p $ is the projection map, $ p: X \times {\mathcal F} \rightarrow {\mathcal F} $, that is $ p(x,g) = g $, for each $ (x, g) \in X \times {\mathcal F} $, $ i_{d} $ is the identity map and $ \sigma $ is the shift map, where $ \sigma(f_{i}, n) = f_{i+n} $

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Figure 3. Beverton-Holt maps with $ {B}_{0} $, $ {B}_{1} $, $ {B}_{2} $, $ {B}_{3} $, $ {B}_{4} $, $ \dots $, with time-dependent $ {r}_{n} $ converging to the map $ B $

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Figure 4. The phase space diagram of the 2-species hierarchical model with four interior fixed points and five fixed points on the axes

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