The Structure of Ω-Limit Sets of Asymptotically Non-Autonomous Discrete Dynamical Systems

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DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019195 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 3, March 2020 pp. 903–915

THE STRUCTURE OF ω-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 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.

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 autonomous 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 ∈ 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) = {π(x, t): t ∈ Z+}, and the ω-limit set of π at x ∈ X is + ω(x, π) = {z : ∃{nk} ⊆ Z with π(x, nk) → z, nk → ∞ as k → ∞}. For brevity, in the sequel, we use O(x) instead of O(x, π) and ω(x) instead of ω(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 ∈ X and A and E are subsets of X, then d(x, A) = inf{d(x, y): y ∈ A}, d(A, E) = inf{d(x, y): x ∈ A, y ∈ E}, and, for > 0, B(E) = {y ∈ X : ∃z ∈ E with d(y, z) < }. 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 → +∞. If the orbit closure of x, that is the closure of the orbit of x, O(x), is compact then ω(x) attracts x, that is lim d(ω(x), f n(x)) = 0 n→+∞ (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 ω-LIMIT SETS 905

Theorem 1.1. Let (X, Z+, π) be a dynamical system, and x ∈ X such that its orbit closure O(x) is compact. Then, ω(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 ∈ X, the set ω(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) = {f n(x): n ∈ Z+}. 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 ω-limit set of π at x ∈ X, ω(x, π), or simply ω(x), is the set

+ nk ω(x, f) = {z : ∃{nk} ⊆ Z with f (x) → z, nk → ∞ as k → ∞} n = ∩m≥0∪n≥mf (x).

The ω-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 = {f0, f1, . . . , fn,... } with fi : X → X, i ∈ 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 ∈ X and i ∈ 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 ∈ X, x0 = x, x1 = f0(x), x2 = (f1 ◦f0)(x0), ..., and for each n ∈ 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 = {f0, f1, . . . , fn,...} ⊆ 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 ∪ {f} is compact in the compact open topology. 906 EMMA D’ANIELLO AND SABER ELAYDI

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 Fˆ = {fn : n = 0, 1, 2,...} ∪ {f}, where fn → f, uniformly, as n → ∞. 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]). Recall that, in metric spaces, the convergence in this topology can be characterized by the uniform convergence on every compact subset of X. Hence, by Ascoli-Arzela’s theorem [15], it follows that F is compact in the space of continuous functions C(X,X).

Deﬁnition 2.2. 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. We can extend π toπ ˜ : X × F × Z+ → X × F in the following way π˜((x, f), j) = (f j(x), f), for each x ∈ X and j ∈ Z+.

Theorem 2.3. The map π˜ : X × F × Z+ → X × F is a dynamical system, that is (i) π˜(x, g, 0) = (x, g) for all x ∈ X and g ∈ F = F ∪ {f}, (ii) π˜(˜π((x, g), s), t) =π ˜((x, g), s + t), for all x ∈ X and g ∈ F = F ∪ {f} and s, t ∈ Z+, and (iii) π˜ is continuous, that is, for each j ∈ Z+,

lim(xn , fn ) = (x, f) ⇒ lim π˜((xn , fn ), j) =π ˜((x, f), j). k k k k k k THE STRUCTURE OF ω-LIMIT SETS 907

π X × F × Z+ X × F

p×id p σ F × Z+ F

Figure 2. This commuting diagram illustrates the notion of a skew product discrete semidynamical system. Here p is the projection map, p : X × F → F, that is p(x, g) = g, for each (x, g) ∈ X × F, id is the identity map and σ is the shift map, where σ(fi, n) = fi+n.

Proof. (i) is straightforward. (ii) The proof is direct if g = f and, for fi ∈ F,

π(π((x, fi), s), t) = π((Φs,i(x), fs+i), t)

= (fi+s+t−1 ◦ · · · ◦ fs+i(Φs,i(x)), fs+i)

= π((x, fi), s + t). (iii) What remains to show is the continuity. For this, we argue by induction on j. The basis of induction is j = 1. We have

lim π˜((xn , fn ), 1) k k k = lim(Φ1,n (xn ), fn +1) = lim(fn (xn ), fn +1) k k k k k k k k = (f(x), f) =π ˜((x, f), 1). Now assume that

lim π˜((xn , fn ), j) =π ˜((x, f), j), k k k that is j lim((fn +j−1◦fn +j−2◦· · ·◦fn )(xn ), fn +j) = lim(Φj,n (xn ), fn +j) = (f (x), f). k k k k k k k k k k Then

lim π˜((xn , fn ), j + 1) k k k

= lim((fn +j ◦ fn +j−1 ◦ fn +j−2 ◦ · · · ◦ fn )(xn ), fn +j+1) k k k k k k k = lim(Φj+1,n (xn ), fn +1) k k k k

= (lim fn +j(fn +j−1 ◦ fn +j−2 ◦ · · · ◦ fn )(xn )), f) k k k k k k = (f(f j(x)), f) = (f j+1(x), f) =π ˜((x, f), j + 1).

Remark 2.4. Let (X ×F, π˜) be the dynamical system discussed above. Note that, if (x, fi) ∈ X × F, then the omega-limit set ω(x, fi) is + ω(x, fi) = {(y, f): ∃(nk) ⊆ Z with π((x, fi), nk) → (y, f), nk → ∞ as k → ∞}. 908 EMMA D’ANIELLO AND SABER ELAYDI

In other words,

+ (y, f) ∈ ω(x, fi) ⇔ ∃(nk) ⊆ Z such that lim(Φnk,i(x)) = y. k

Note that ω(x, f0) is commonly ([23]) called the omega limit set of x in the non-autonomous system, and the deﬁnition used in the present paper includes the special case when fi = f0. Let (x, f) ∈ X × F, then, clearly,

nk n ω(x, f) = {(z, f): f (x) → z, nk → ∞} = (∩m≥0∪n≥mf (x)) × {f}.

3. The extension of a discrete semidynamical system on X to the one- point compactification Xˆ = X ∪ {∞}. In this section, we shall assume that the metric space X is not compact. Definition 3.1. [29] Let X = (X, d) be a metric space and F be a set in X. For any r > 0, let Br(F ) = {x ∈ X : d(x, y) < r for some y ∈ F }. If G ⊆ X is such that Br(F ) ⊂ G for some r > 0, we say that G is a uniform neighborhood of F . Definition 3.2. ([29]) A metric space X = (X, d) is Mandelkern locally compact ∞ (M-locally compact) if it is a countable union ∪k=1Hk of compact sets, where each set Hk+1 is a uniform neighborhood of Hk, that is Hk ⊂ int(Hk+1) (where int(Hk+1 denotes the interior of Hk+1). Theorem 3.3. ([29]) The following metric spaces are locally compact: 1. Any compact space. 2. The Euclidean space Rn. 3. Open spheres in Rn. 4. Any nonvoid space in which any bounded subset is contained in a compact subset. Under the traditional definition of local compactness, which merely requires each point have a compact neighborhood, there exist locally compact spaces whose one- point compactifications are not metrizable. Metrizability of the one-point compactification requires the point at infinity be the intersection of infinitely many neighborhoods, therefore the space must be σ-compact. This is the reason why Mandelkern in [29] introduces a new definition of local compactness which uses σ- compactness with the addition of a strong monotonicity condition on the sequence of compact subsets. Let (X, d) be a locally compact metric space in the sense of Mandelkern. Then, by Theorem 2 in [29] (Theorem 3.4 below), the metric space (X, d) has a metric one-point compactification (X,ˆ dˆ). The construction is as follows. We adjoint a point x1 to X to obtain the space X1 = X ∪ {x1}. Then, we define a new metric space (X1, d1) as follows.

For each k ∈ N, let ck > 0 be such that Bck (Hk) ⊂ Hk+1 and ck ↓ 0. Deﬁne, for each x ∈ X,

h(x) = sup{ck − d(x, Hk): k ∈ N}. Mandelkern has illustrated the deﬁnition of h in the case when X = [0, ∞[, 1 n Hk = [0, k], and ck = k . In the more general case, when X = [0, +∞[ , n ≥ 1, THE STRUCTURE OF ω-LIMIT SETS 909

n n n take Hk = [0, k] , and in the case when X = R , n ≥ 1, take Hk = [−k, k] . Take √1 n ck = k n in both cases. Then, in both cases, for each x ∈ R , 1 h(x) = sup{ √ − d(x, Hk)}. k∈N k n

√1 Clearly, 0 < h ≤ c1 = n . For each given point, the supremum defining h is the maximum of finitely many terms, some of them corresponding to the value cm if x ∈ Hm. Hence, h is well- defined, with 0 < h < c1. We now define d1 as follows:  min{d(x, y), h(x) + h(y)} for x, y ∈ X   h(x) if y = x1 and x ∈ X d1(x, y) = h(y) if x = x1 and y ∈ X   0 if x = y = x1

Then d1 is a metric on X1 under which X1 is totally bounded. The completion ˆ (X,ˆ d) of (X1, d1) will be called the one-point compactiﬁcation of (X, d)[15]. The mapping i : X → Xˆ is the restriction to X of the usual inclusion map i : X1 → Xˆ of a space into its completion (for the topological aspects relative to completion see [15]. We denote the image of x1 in Xˆ by∞ ˆ .

In the sequel, when we write “locally compact”, we mean “M-locally compact”. Theorem 3.4. ([29]: Theorem 2) Every locally compact metric space (X, d) has a metric one-point compactiﬁcation (X,ˆ dˆ). The mapping i : X → Xˆ is a uniformly continuous injection, and a homeomorphism between X and Xˆ \ ∞ˆ . Theorem 3.5. Let (X, d) be a locally compact space. If f :(X, d) → (X, d) is uniformly continuous, then it can be extended to a uniformly continuous map fˆ : (X,ˆ dˆ) → (X,ˆ dˆ).

Proof. Since the inclusion map i is a homeomorphism between X and Xˆ \ ∞ˆ , we ˆ shall write Xˆ as X ∪ {∞}ˆ . Let {xn} be a sequence in (X,ˆ d) convergent to ∞ with ˆ respect to the metric d. Then, given δ0, there exists a positive integer N such that ˆ d(xn, xm) < δ0, for all n, m > N. By uniform continuity of f restricted to X, given > 0, there exists δ ≤ δ0, such that dˆ(x, x0) < δ ⇒ dˆ(f(x), f(x0)) < . Then ˆ d(f(xn), f(xm)) < , ∀n, m > N. ˆ ˆ Hence, the sequence {xn} is Cauchy in (X,ˆ d). Since (X,ˆ d) is complete, the sequence ˆ {f(xn)} must converge to a point z ∈ Xˆ in (X,ˆ d). We deﬁne

z if z ∈ X fˆ(∞ ˆ ) = ∞ˆ if z =∞ ˆ Clearly, fˆ is the continuous extension of f to Xˆ we are looking for. The map fˆ is well deﬁned because if there exist {xn} and {yn} both having limit∞ ˆ , then, by the uniform continuity, it follows that limn(f(xn)) = limn(f(yn)). 910 EMMA D’ANIELLO AND SABER ELAYDI

Theorem 3.6. Let (X, d) be a locally compact metric space, let f : X → X be uniformly continuous, and let fˆ :(X,ˆ dˆ) → (X,ˆ dˆ) be the uniformly continuous extension of f to the one-point compactification (X,ˆ dˆ). For any x ∈ Xˆ, the omega limit set ω(x, fˆ) is non-empty, closed, invariant and invariantly connected. More- over, if ∞ˆ 6∈ ω(x, fˆ), then ω(x, fˆ) = ω(x, f), and hence ω(x, f) is non-empty, closed, invariant and invariantly connected. Proof. The first part follows from Theorem 1.2. We use this completion of (X, d), (X,ˆ dˆ), in order to obtain a compact space. Next, we assume∞ ˆ 6∈ ω(x, fˆ) and show that ω(x, f) = ω(x, fˆ). + “ ⊆ ” Let z ∈ ω(x, f). Then ∃{nk} ⊆ Z such that lim d(z, f nk (x)) = 0. k→∞ As dˆ≤ d, it is also lim dˆ(z, f nk (x)) = 0. k→∞ Hence z ∈ ω(x, fˆ). ˆ + “ ⊇ ” Let y ∈ ω(x, f). Then y ∈ X. Let {nk} ⊆ Z such that lim dˆ(y, f nk (x)) = 0. k→∞ ∞ Since X is locally compact, it follows that X = ∪k=1Hk of compact sets, where each set Hk+1 is a uniform neighborhood of Hk, that is Hk ⊂ int(Hk+1). Hence, there exists a positive integer k0 such that y ∈ Hk0 so that h(y) ≥ ck0 . Now, by the ˆ definition of d, there exists an index s0 such that

ˆ nk nk d(z, f (x)) = d(z, f (x)), ∀k ≥ s0. Therefore, y ∈ ω(x, f). This implies that ω(x, f) is nonempty, closed, invariant and invariantly connected.

4. Non-autonomous Dynamical Systems generated by convergent sequences. Based on the previous section, we shall assume in the sequel that our space X is a compact metric space. In this section we shall assume that there is a sequence of continuous maps {fn} and the associated non-autonomous diﬀerence equation is

xn+1 = fn(xn), n = 0, 1, 2,.... + Let π : X ×F ×Z → X ×F, where F = {fi : i = 0, 1, 2,... } the space of functions equipped with the compact open topology such that

π((x, fi), n) = (Φn,i(x), fi+n), where

Φn,i = fi+n−1 ◦ fi+n−2 ◦ · · · ◦ fi. As seen before (X × F, Z+, π) = (X × F, π) is a discrete dynamical system on X × F. Now, let us assume that the sequence {fn} converges uniformly to a function f : X → X. Clearly, f is continuous as well. Let F = {fn} ∪ {f}. We extend the skew-product to the space X × F by lettingπ ˆ((x, f), n) = (f n(x), f). By Theorem 2.3 the extended system (X × F, πˆ) is also a discrete dynamical system. THE STRUCTURE OF ω-LIMIT SETS 911

Figure 3. Beverton-Holt maps with B0, B1, B2, B3, B4, ... , with time-dependent rn converging to the map B.

Theorem 4.1. Let (X × Fˆ, πˆ) be as above. Then, for any x ∈ X and fi ∈ F, the omega limit set ω(x, fi, πˆ) = ω(x, fi) is closed, invariant and invariantly connected. Moreover, for each i ∈ {0, 1, 2,... }, πˆ(ω(x, fi), 1) = (f(ω(x, fi)), f).

Proof. Fix (x, fi) ∈ X × Fˆ. The ﬁrst part follows from Theorem 3.6 applied to the skew product dynamical system (X × Fˆ, πˆ). For the second part, observe that, as ˆ + ω(x, fi) = {(y, f) ∈ X × F : ∃{nk} ⊆ Z

withπ ˆ((x, fi), nk) → (y, f), nk → ∞ as k → ∞}, then πˆ((y, f), 1) = (f(y), f) and, consequently,π ˆ(ω(x, fi), 1) = (f(ω(x, fi)), f).

Remark 4.2. Note that ω(x, f0) is commonly ([23]) called the omega limit set of x in the non-autonomous system. 912 EMMA D’ANIELLO AND SABER ELAYDI

Example 4.3. Consider the Ricker population model with no Allee eﬀect, in which the reproduction rate rn is time-dependent due to habitat ﬂuctuation. This results in a sequence of maps

Rn(x) = x exp(rn − x), which generates the non-autonomous diﬀerence equation

xn+1 = Rn(xn) + + for all n ∈ Z , defined on R . We assume that the sequence {rn} converges to r, where 0 < r ≤ 2. Hence the family {Rn} converges to the autonomous Ricker map R(x) = x exp(r − x). By Theorem 4.1, every orbit in the non-autonomous system converges to the fixed point x∗ = r. Example 4.4. Consider the following population model with the Allee effect [20], in which the reproduction rate rn is time-dependent due to habitat fluctuation. This results in a sequence of maps

2 2 Bn(x) = rnxn/(1 + bxn), for all n ∈ Z+, and B(x) = rx2/(1 + bx2). 2 We assume that the sequence {rn} converges to r, where rn > 4b. Hence the family {Bn} converges to the autonomous Beverton-Holt type map (see Figure 3) 2 2 B(x) = rxn/(1 + bxn). ∗ The map B has three fixed points, the extinction fixed point x1 = 0, the Allee fixed ∗ ∗ point x2 = A, and the carrying capacity x3 = K. If the population size or density falls below the Allee threshold A, then the population goes extinct. By Theorem 4.1, every orbit in the non-autonomous system converges to either the fixed point ∗ ∗ x1 = 0 or to the fixed point x3 = K. Remark 4.5. If X is a closed and bounded interval of the real line, it is known that an ω-limit set of a non-autonomous dynamical system on X is non-empty, compact and invariant ([23] and [10]). In [10], the author investigated the case when the uniform limit f is a 2n function for some n ≥ 1; and in [16] the authors analyzed the case when f is a 2∞ function answering positively a question posed in [10].

5. Triangular maps. A special class of maps deﬁned on the Euclidean space Rk, for which the Jacobian at every point in Rk is a triangular matrix, are called triangular maps. They are of the form

f(x1, x2, . . . , xk) = (f1(x1), f2(x1, x2), . . . , fk(x1, x2, . . . , xk)). As noted in [26], it is interesting to mention that, unlike the interval case, a triangular map can possess only fixed points and have, simultaneously, an infinite ω-limit set. In this paper, we assume that our triangular maps are smooth and hyperbolic, in which case the latter statement is not possible. In fact it was shown in [7][3], [2], [17] that triangular maps have only fixed points and the ω-limit sets are fixed points. THE STRUCTURE OF ω-LIMIT SETS 913

These maps are important, not only for theoretical considerations, but also for representing hierarchical competition biological models. For instance, a discrete- time multi-species hierarchical model for species x1, x2, . . . , xk, may be written as

F (x) = x1g1(x1), x2g2(x1, x2), . . . , xkgk(x1, x2, . . . , xk), which generates the diﬀerence equation

xn+1 = F (xn)

In this general model, species x1 is the dominant species which is not affected by the competition of the other species on resources in its habitat, species x2 is less dominant and is affected by the competition of species x1 only on resources, and lastly there is the “wimpy” species xk which is affected by all the competition on resources of the species in its habitat. Definition 5.1. Let I = [a, b] ⊂ R.A C1 function f : Ik → Ik is said to be hyperbolic if all the eigenvalues of its Jacobian matrix at every point in Ik are off the unit circle in the complex plane. Theorem 5.2. [7] Let f : Ik → Ik be a C1 triangular map which is hyperbolic. Then every orbit converges to a fixed point if and only if there are no periodic orbits of prime (minimal) period 2. Theorem 5.3. ([8]; [18]) Let f : I → I be a continuous map. Then every orbit converges to a fixed point if and only if f has no periodic orbit of prime (minimal) period 2. 1 Corollary 5.4. Let F = {fn}n≥0 be a sequence of hyperbolic, triangular, C maps, from Ik to Ik uniformly convergent to a hyperbolic map f. Then every omega limit set ω((x, fi), πˆ) is a fixed point of f if and only if the map f has no periodic orbits of prime period 2.

Proof. First, since each fi is a triangular map, it follows that the uniform limit f is also a C1 triangular map. Hence, by Theorem 5.2, every orbit converges to a ﬁxed point, under the map f. Now the omega limit set ω(x, fi) is closed and invariant k under the map f (meaning that the projection of ω(x, fi) on X = I is invariant under f, as ω(x, fi) ⊆ X × {f}). But the only closed invariant sets under the map f are the ﬁxed points of f. The result of the corollary now follows.

Corollary 5.5. Let F = {fn}n≥0 be a sequence of continuous real valued maps on [a, b] ⊂ R uniformly convergent to a map f. Then every omega limit set ω(x, fi) of x ∈ X is a fixed point of f if and only if f has no periodic orbits of prime period 2. Proof. This follows from Theorem 5.3. Example 5.6. Consider the following population model with the Allee effect [17], [3], [2], in which the reproduction rates rn and sn are time-dependent due to habitat fluctuation. This results in a sequence of maps

m1 rn−xn− xn+1 = fn(xn) = xne 1+s1xn m2 (1) sn−yn−cxn− yn+1 = gn(xn, yn) = yne 1+s2yn + for all n ∈ Z , where rn and sn are the intrinsic growth rates of species x and y, respectively, and c is a scaled interspeciﬁc competition parameter. This model may be written in the compact form (xn+1, yn+1) = Fn(xn, yn), where Fn(x, y) = 914 EMMA D’ANIELLO AND SABER ELAYDI

(fn(x), gn(x, y)) , where x is the dominant species whose growth is limited only by its own intraspecific competition, while y is the “wimpy” species whose growth rate is affected by its own intraspecific competition and by the interspecific competition caused by the presence of species x. An example is an invasive species x that − m1 − m2 strongly competes with a native species y. The terms e 1+s1x and e 1+s2y represent the positive density dependence of x and y, respectively, which is the hallmark of the Allee effect caused by predator saturation [1], [20]. The parameter mi represents the predation intensity, and si is a parameter proportional to the “handling time”, that is, the duration of the capture and eating phase, i = 1, 2. We assume that the sequences {rn} and sn converge to r, and s, respectively. Hence the family {Rn = (fn, gn)} converges uniformly to the 2-species Ricker competition model with the Allee effect R = (f, g). The map R may have up to four interior fixed points and five fixed points on the axes. Under certain conditions on the parameters ([17], [3], [2]), by Theorem 5.2, every orbit in the non-autonomous system converges to one of the fixed points.

Four interior fixed points

(A1, yA12) K2

(K1, yK12)

(K1, yK11)

(A1, yA11) A2

A1 K1

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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