DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019195 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 3, March 2020 pp. 903{915 THE STRUCTURE OF !-LIMIT SETS OF ASYMPTOTICALLY NON-AUTONOMOUS DISCRETE DYNAMICAL SYSTEMS Emma D'Aniello Dipartimento di Matematica e Fisica Universit`adegli Studi della Campania \Luigi Vanvitelli" Viale Lincoln n.5, 81100 Caserta, Italia Saber Elaydi∗ Department of Mathematics Trinity University San Antonio, TX 78212-7200, USA (Communicated by Christian P¨otzsche) Abstract. We consider a discrete non-autonomous semi-dynamical system generated by a family of continuous maps defined on a locally compact met- ric 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. 1. Introduction. The theory of non-autonomous discrete dynamical systems has been developed by many authors including the main contributors Bernd Aulbach and his students [6], [5], [30], [4] and Peter Kloeden and his collaborators [25], [24]. Kempf [23] and Canovas [10] investigated the limit sets of non-autonomous discrete systems that are asymptotically autonomous. The special case of non- autonomous periodic discrete systems was, thoroughly, investigated by Elaydi and Sacker [18], and the references there in, by Kloeden [24], Silva [32] and Franco, Silva, and Sim~oes[21]. D'Aniello and Steele ([12], [13]) investigated the limit sets of 2- periodic (alternating) systems, and D'Aniello and Oliveira [11] investigated attracting periodic orbits of these systems. This paper focuses on the topological and the dynamical properties of the omega limit sets in non-autonomous discrete dynamical systems that are asymptotic to autonomous systems. It has been shown by La Salle [27] and by Dowker and Fried- lander [14] that an omega limit set in an autonomous discrete dynamical system 2010 Mathematics Subject Classification. Primary: 54H20, 37C70; Secondary: 39A05. Key words and phrases. Dynamical system, skew product, attractor. Saber Elaydi acknowledges the hospitality of the Department of Mathematics and Physics of the Universit`adegli Studi della Campania \Luigi Vanvitelli". ∗ Corresponding author: Saber Elaydi. 903 904 EMMA D'ANIELLO AND SABER ELAYDI is invariantly connected (see the definition below). This is in contrast with au- tonomous differential systems, in which omega limit sets are connected [29], [31]. Our main objective here is to extend this result to non-autonomous discrete systems that are asymptotic to autonomous systems in which the phase space is a locally compact metric space. It should be noted that all the studies in the cited literature above assume that the phase space is either a closed bounded interval or a compact subset of an Euclidean space. In this paper, the phase space is assumed to be a locally compact metric space. Using the method in [29], we extend the metric to its one point compactification of the phase space. Then we extend the dynamical system to the compactified space. The final step is to embed the non- autonomous system into an autonomous skew-product discrete system [19]. The theoretical results are then applied to population models, namely the Ricker model with no Allee effect and the Elaydi-Sacker model with the Allee effect [20]. The Allee effect is a phenomenon in biology characterized by a positive correlation between population density (size) and its per capita growth rate [1]. So as population size approaches a threshold, favorable influences stimulate its growth, and when it goes below it, unfavorable influences inhibit its growth. Finally, in section 5, we apply our results to non-autonomous triangular maps. Discrete dynamical systems Let (X; d) be a metric space, T a topological group, and let π : X × T ! X. Then the triple (X; T; π) is called a dynamical system ([31]; [28]) if (i) (identity axiom) π(x; 0) = x for all x 2 X, where 0 is the identity of T . (ii) (homomorphism axiom) π(π(x; s); t) = π(x; s + t). (iii) (continuity axiom) π is continuous. If T is a topological semigroup, then (X; T; π) is called a semi-dynamical system. If T = Z+, the set of non-negative integers, then the (forward) orbit of a point x with respect to π, is defined as O(x; π) = fπ(x; t): t 2 Z+g, and the !-limit set of π at x 2 X is + !(x; π) = fz : 9fnkg ⊆ Z with π(x; nk) ! z; nk ! 1 as k ! 1g: For brevity, in the sequel, we use O(x) instead of O(x; π) and !(x) instead of !(x; π) when no confusion arises. A subset A of X is invariant if π(A; n) = A for each n ≥ 0. Recall that, if x 2 X and A and E are subsets of X, then d(x; A) = inffd(x; y): y 2 Ag, d(A; E) = inffd(x; y): x 2 A; y 2 Eg, and, for > 0, B(E) = fy 2 X : 9z 2 E with d(y; z) < g. Let A, E be two (nonempty) subsets of X. The set A is said to attract E ([31]: page 895) if d(π(E; n);A) ! 0 as n ! +1: If the orbit closure of x, that is the closure of the orbit of x, O(x), is compact then !(x) attracts x, that is lim d(!(x); f n(x)) = 0 n!+1 (see, for instance, Proposition 2.11 in [31], page 896). A closed invariant subset A of X is said to be invariantly connected if it cannot be represented as the union of two nonempty, disjoint, closed, invariant sets ([31]: page 891). THE STRUCTURE OF !-LIMIT SETS 905 Theorem 1.1. Let (X; Z+; π) be a dynamical system, and x 2 X such that its orbit closure O(x) is compact. Then, !(x; π) is non-empty, closed, invariant, and invariantly connected. Proof. This follows from ([31]: Lemma 2.11 page 896, and Lemma 2.9, page 895). The following is a corollary of Theorem 1.1 since, clearly, when X is compact, then each non-empty closed subset of X is also compact. Theorem 1.2. Let (X; Z+; π) be a dynamical system, with X compact. Then, for each x 2 X, the set !(x; π) is non-empty, closed, invariant, and invariantly connected. If T = Z+, a semi-dynamical system may be generated by a continuous map f, where π(x; n) = f n(x). Then the (forward) orbit of a point x with respect to π, is the set O(x; f) = ff n(x): n 2 Z+g. Then a subset A of X is invariant if f(A) = A. Definition 1.3. ([8], [9], [31]) Let (X; Z+; π) be a dynamical system generated by a continuous map f. Then the !-limit set of π at x 2 X, !(x; π), or simply !(x), is the set + nk !(x; f) = fz : 9fnkg ⊆ Z with f (x) ! z; nk ! 1 as k ! 1g n = \m≥0[n≥mf (x): The !-limit sets of a dynamical system (X; Z+; π) are closed and invariant but not necessarily connected. But they do satisfy a connectedness property. 2. Non-autonomous difference equations. Let X = (X; d) be a compact met- + ric space and let F = ff0; f1; : : : ; fn;::: g with fi : X ! X, i 2 Z , continuous maps. We examine the semi-dynamical system + π :(X × F) × Z ! X × F + with π((x; fi); 0) = (x; fi), for any x 2 X and i 2 Z and, for each n ≥ 1, π((x; fi); n) = (Φn;i(x); fn+i), where Φn;i = fi+n−1 ◦ fi+n−2 ◦ ::: ◦ fi+1 ◦ fi. Note that Φ1;i = fi. Define, for x 2 X, x0 = x, x1 = f0(x), x2 = (f1 ◦f0)(x0), :::, and for each n 2 N, xn+1 = (fn ◦ fn−1 ◦ ::: ◦ f1 ◦ f0)(x0), so to obtain the difference equation xn+1 = fn(xn) which we treat in the setting of skew-product dynamical systems by considering the mappings fi : Xi !X(i+1); where Xi, the fiber over fi, is just a copy of X residing over fi and consisting of those x on which fi acts. The product space we consider is X × F where F = ff0; f1; : : : ; fn;:::g ⊆ C(X; X). Theorem 2.1. Let X = (X; d) be a compact metric space and let fn : X ! X, n = 0; 1;::: be a sequence of continuous maps uniformly convergent to a function f. Then F = F [ ffg is compact in the compact open topology. 906 EMMA D'ANIELLO AND SABER ELAYDI X x 3 x 4 x 1 x x 5 2 x 0 م َ n f f f f f f f 0 1 2 3 4 5 Figure 1. The space F^ = ffn : n = 0; 1; 2;:::g [ ffg, where fn ! f, uniformly, as n ! 1. If x0 is on the fiber X0, then f0(x0) = x1 is in the fiber X1, and f1(x1) = x2 is on the fiber X2, etc. Proof. It is straightforward to show that the set F is closed, bounded and equicon- tinuous in the space of compact open topology (for the compact open topology, see [15]).