ON A GENERALIZED YORKE CONDITION FOR SCALAR DELAYED POPULATION MODELS

For a scalar delayed differential equation ẋ(t) = f(t, xt), we give sufficient conditions for the global attractivity of its zero solution. Some technical assumptions are imposed to insure boundedness of solutions and attractivity of nonoscillatory solutions. For controlling the behaviour of oscillatory solutions, we require a very general condition of Yorke type, together with a 3/2-condition. The results are particularly interesting when applied to scalar differential equations with delays which have served as models in populations dynamics, and can be written in the general form ẋ(t) = (1+ x(t))F (t, xt). Applications to several models are presented, improving known results in the literature.

Our research is mainly motivated by the applications of the so-called 3/2 stability results (see e.g. [6,Section 4.5]) to scalar population models which can be written in the formẋ (t) = (1 + x(t))F (t, x t ), t ≥ 0. (1.2) Recently, the global stability of the zero solution of (1.2) was investigated in [1] assuming that F satisfies the following generalization of the well-known Yorke condition ( [15], [6, p. 141] [15]. We remark that condition (1.3) with F = f and λ(t) ≡ a > 0 was first introduced in [15], not in the setting of equation (1.2), but to study the stability of the zero solution of (1.1); later, Yoneyama [13] generalized Yorke's hypothesis by replacing the constant a with a continuous function λ(t) ≥ 0.
Connecting and unifying the approaches in [1] and [7] (in which another generalization of the Yorke condition was introduced, see Theorem 1.1 below), in the present paper we improve the results in the referred works: we establish a global stability result for (1.1), from which the global stability of (1.2) is obtained under a Yorke condition (see assumption (A3) in Section 3) more general than the ones considered in both [1] for (1.1) and [7] for (1.2). Our result is easy to apply, and allows us to improve some results in the literature for a number of concrete examples.
In the following, the next hypotheses will be considered for f as in ( where the first inequality holds for all ϕ ∈ C and the second one for ϕ ∈ C such that ϕ > −1/b ∈ [−∞, 0), and M(ϕ) is the Yorke functional (H4) for λ(t) as in (H3), there is T ≥ h such that, for Without loss of generality, by a time scaling, we may assume h = 1. Also, if b > 0, for b as in (H3), the scaling x → bx allows us to consider b = 1. By the change of variables x → y = −x, we may as well consider a function f (t, ϕ) such that g(t, ϕ) = −f (t, −ϕ) satisfies (H1)-(H4).
In [7] the following result was proven: Our first purpose is to prove Theorem 1.1 under hypotheses (H1)-(H4) with general β(t), λ(t). Actually, by a change of variables introduced in [7] (see (2.2)-(2.3) below), it turns out that the framework in (H3) can be reduced to the situation of (H3) with λ(t) ≡ a > 0, if the additional condition λ(t) > 0 for large t is imposed (cf. Lemma 2.3). As we shall show in Section 3, the application of this result to general delayed scalar population models (1.2) provides a generalization of the criterion for global stability established in [1] (see Theorem 3.1 below). In Section 4, some particular models that have been considered in the literature are addressed within the present framework, and weaker sufficient conditions for the global asymptotic stability of equilibria or periodic solutions of such models are obtained. Also some open problems and counter-examples will be presented.
2. Global stability for (1.1). We start this section with some preliminary lemmas.
We are now in position to state the following improvement of Theorem 1.1.
Then the zero solution of (1.1) is globally asymptotically stable.
Proof. The result follows immediately from Lemmas 2.3 and 2.4.
Remark 2.6. For the case b = 0, it was proven in [8] that, for the particular case of for all t ≥ 0 and ϕ ∈ C with ϕ(θ) = 0, θ ∈ [−h, 0]. For the general case of distributed delays, some additional conditions on the behaviour of f (t, ϕ) might be required.
Another interesting question is whether it is possible to replace (H3) in Theorem 2.5 by the following weaker condition: (H3') there is a piecewise continuous function λ : [0, ∞) → [0, ∞) and there are where the first inequality holds for all ϕ ∈ C and the second one The following result shows that, under additional restrictions on (H4) and on the size of b 1 /b 2 , we can replace (H3) by (H3') in Theorem 2.5.

Theorem 2.7. Assume (H1), (H2), (H3') and that, for λ(t) as in (H3'), there is
Assume also that Then all solutions x(t) of (1.1) satisfy x(t) → 0 as t → ∞. Proof. As already mentioned, we may consider h = 1. Arguing as in the proof of Lemma 2.1, one deduces that all solutions of (1.1) are defined and bounded on [0, ∞), and that non-oscillatory solutions of (1.1) go to zero as t → ∞. Hence, only oscillatory solutions x(t) will be considered. Let x(t) be an oscillatory solution of (1.1), and define Fix ε > 0, and for T as in (2.6) choose T 0 ≥ T such that Thus, for t ∈ [η n , s n ], we have Hence, By letting n → ∞ and ε → 0 + , we obtain By letting n → ∞ and ε → 0 + , we obtain (2.9) From (2.8) and (2.9), we get we obtain a contradiction. Thus, u = 0. From (2.8), it follows that also v = 0.
From the above proof, it is easy to see that the result holds if, instead of (2.6) and (2.7), we have Remark 2.8. As the next example shows, it is impossible to extend the result in Theorem 2.7 up to α = 3/2 even if b 1 /b 2 > 1 − δ, for δ > 0 arbitrarily small.
Let us consider the following T (M )-periodic delay differential equation and then extended over all R in a periodic way. Here, and T (M ) will be defined in the proof of the next theorem, which, for convenience of the reader, will be presented in Appendix A at the end of the paper. 3. Scalar population models. We apply now Theorem 2.5 to scalar population models that can be written in the forṁ where F : [0, ∞) × C → R is a continuous function. Due to biological reasons (cf. Section 4), we only consider admissible initial conditions  (A4) for λ(t) as in (A3), there is T ≥ h such that, for α ≤ 3/2 if b = 1/2, and α < 3/2 if b = 1/2. As in Section 2, we again remark that, for case b = 0, (A3) and (A4) imply (A1). Clearly, hypothesis (A2) is imposed to force non-oscillatory solutions to go to zero, as t goes to infinity. The case b = 0 was studied in [1], where a hypothesis (conditions (H1)-(H2) in [1]) slightly stronger than (A2) was assumed. However, it was noticed that conditions (H1)-(H2) in [1] could be replaced by the above assumption (A2), similarly to what was considered in [12] for Eq. (1.1), hence the following result follows from [1]. Under (A1)-(A4), we now prove that the statement in Theorem 3.1 is still true for any b > 0, where an additional condition similar to the one in Theorem 2.5 may be required. Proof. The case b = 0 was addressed in Theorem 3.1, so we assume b > 0. From [1], it follows that all solutions x(t) of (3.1) with admissible initial conditions (3.2) are defined, bounded and bounded below away from −1 on [0, ∞).
The change of variables y(t) = log(1 + x(t)), t ≥ 0, transforms (3.1) intȯ On the other hand, clearly We first consider the case b ≥ 1/2. Define We deduce that x + e x 1 2 x − 1 ≥ 0, x ≥ 0, which can be proven easily by studying the signs of w (x) and w (x). Analogously, we prove (3.6). From (3.4), (3.5) and (3.6), we obtain for t ≥ 0 where the first inequality holds for every ϕ ∈ C and the second one for ϕ ∈ C such that ϕ > −1/(b − 1/2). Thus, the function f in where the first inequality holds for all ϕ ∈ C −1 and the second one for ϕ ∈ C such that ϕ > max{−1/b 2 , −1};

Then all solutions x(t) of (3.1) satisfy x(t) → 0 as t → ∞.
Remark 3.4. By the translation x → 1 + x, the scalar FDE (7) in [7] can be seen as a particular case of Eq. (3.1). Thus, Theorem 3 in [7] is a consequence of Theorem 3.2 above. Condition λ(t) > 0 for all t ≥ 0 in [7] can be weakened according to the statement in Theorem 3.2. On the other hand, it seems that, in the statement of [7,Theorem 3], the authors have forgotten to include a hypothesis similar to (A1) (or (H1)), to assure that all solutions are bounded on [0, ∞). That is, for model (7) in [7], if b > 1, it seems necessary to further impose that, for all q ∈ (−1, 0), there is η(q) ∈ R such that f (t, x + 1) ≤ η(q) for all t ≥ 0, x ∈ [q, 0).
Remark 3.6. As remarked in [1,Remark 3.11], the present setting can be applied to Eq. (3.1) with time-dependent bounded discrete delays of the forṁ

Applications.
In this section, we consider some scalar FDEs which have served as models for the growth of a single population, and improve some results in the literature.
Since the functions s, c, K have positive lower and upper bounds, it is easy to check that (A1) and (A2) hold for (4.6). On the other hand, for and define For x ≥ 0, we have Analogously, for −1 ≤ x < 0, This implies that (4.6) satisfies (A3), for r(x) defined as above, i.e., with b = v0 1+v0 , and Applying Theorem 3.2, we obtain the following criterion for the global stability of the periodic solution N * (t) of (4.4), which improves the result in [1,Theorem 4.8].
Example 4.7. Consider the delay differential equatioṅ . . , n, and ρ(t) > 0 for large t. Eq. (4.16) was studied in [1,9]. In [9], possible unbounded delays were allowed. In [1], Theorem 3.1 (i.e., Theorem 3.2 with b = 0) was applied to the study of the global attractivity of the positive equilibrium of (4.16). However, our method with b > 0 is not easy to apply to certain models with more than one discrete delay. Therefore, here we only consider the case n = 1. Let n = 1, τ 1 (t) = τ (t), a = a 1 and S(t) = s 1 (t), so that (4.16) reads aṡ Eq. (4.17) has been considered by many authors, since it has been proposed as an alternative model to the delayed logistic equation for a food limited single population model (see [1,2,4,7] and references therein).
As usual, because of its biological context, we only consider positive solutions of (4.17). Following the approach in [1], we effect the change of variables is the unique positive equilibrium of (4.17). Thus, Eq. (4.17) is written aṡ , t ≥ 0. (4.18) The following result was proven in [7,Corollary 3]. However, here we present a simpler proof.
Then all positive solutions of (4.17) tend to the positive equilibrium N * as t → ∞. If S 0 = a, the same result holds if we assume (4.19) and Proof. Note that (4.18) has the form (3.1), for F defined by where .
Finally, using the step by step method, we easily find that Π α (φ) converges to Π ρ (φ), uniformly on φ from bounded subsets of C + , when α → ρ in the space of all 5periodic piecewise continuous functions equipped with the norm |α| 1 = 5 0 |α(s)|ds. This implies that we can perturb the coefficient ρ slightly (in the stated norm) to transform it into a positive continuous function ρ * which satisfies Π ρ * Ū (z 2 , ε 2 ) ⊂ U(z 2 , ε 2 ). Since Π is compact, an application of the Schauder fixed point theorem assures the existence of a non-trivial 5-periodic solution of the perturbed equation.