Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)

Global asymptotic and exponential stability of equilibria for the following class of functional differential equations with distributed delay is investigated \begin{document}$ x'(t)=-f(x(t))+\int_{0}^{\tau}h(a)g(x(t-a))da.$\end{document} We make our analysis by introducing a new approach, combining a Lyapunov functional and monotone semiflow theory. The relevance of our results is illustrated by studying the well-known integro-differential Nicholson's blowflies and Mackey-Glass equations, where some delay independent stability conditions are provided. Furthermore, new results related to exponential stability region of the positive equilibrium for these both models are established.


(Communicated by Chongchun Zeng)
Abstract. Global asymptotic and exponential stability of equilibria for the following class of functional differential equations with distributed delay is investigated x (t) = −f (x(t)) + τ 0 h(a)g(x(t − a))da.
We make our analysis by introducing a new approach, combining a Lyapunov functional and monotone semiflow theory. The relevance of our results is illustrated by studying the well-known integro-differential Nicholson's blowflies and Mackey-Glass equations, where some delay independent stability conditions are provided. Furthermore, new results related to exponential stability region of the positive equilibrium for these both models are established.
1. Introduction. In this paper, we study the following general class of functional differential equations with distributed delay x (t) = −f (x(t)) + τ 0 h(a)g(x(t − a))da, t > 0, (1) Various forms of system (1) can be found in the literature, we invite the reader to see [3], [6], [7], [8], [12], [15], [20], [21], [38], [39] and references therein. As known, the dynamics of the subfamily of quasi-monotone functional differential equations of (1) is well understood, see [26], [28]; it admits a comparison principle and therefore, the chaotic behavior of this family is not possible [26]. However, much is unknown about the global dynamics of non-monotone functional differential equations (1) see [11], [13], [28], [30]. These models need more attention to elucidate the complex dynamics of its solutions. In fact, despite the existence of various techniques for the global study of functional differential equations, the determination of the asymptotic behavior of its solutions even for the simplestlooking equations is difficult. For example, in the case where g is an unimodal feedback (when g has exactly one maximum and changes monotonicity at only one point) and f (x) = x, the trajectory may lead to complicated dynamics, as chaotic behavior [22]. In this context, based on the idea stated in the works of Mallet-Paret 4392 TARIK MOHAMMED TOUAOULA and Nussbaum [24], [25], many authors emphasize that the proofs of the convergence to the positive equilibrium is strongly related to the linearity of f and by the sequel to the convergence of the sequence defined by x n+1 = g(x n ). For the most related works in this direction see [4], [9], [14], [16], [18], [17], [19], [27] and further references therein. We also mention the work of Kuang in his book [15], where he investigated the following general functional differential equation, and under some conditions on f and g, he found a set of sufficient conditions which lead to the global attractivity of the positive equilibrium.
Recently, Yuan et al [39] analyzed the following model and obtained sufficient conditions under which the positive equilibrium is globally attractive, the approach used there is related to the fluctuation method. When a spatial diffusion is considered; many models are studied in the literature, see for example [34], [35], [36], [37]; some of the techniques used there can be adapted for our problem to obtain the stability region.
The purpose of this paper is to provide a new approach on analytic study of asymptotic behavior of solutions to problem (1), which can be applied to the nonmonotone feedback case. More precisely, we will present some results which will enable us to give general conditions on f and g that ensure global asymptotic and exponential stability of equilibria regardless of the length of the delay. To this end, fluctuation and monotone semi-flow methods combined with a Lyapunov functional are used.
The main idea of this work is to derive necessary and sufficient conditions on f and g which guarantee the existence of a nondecreasing function satisfying certain properties. From then, the monotone semi-flow theory is applied. For this, the first part of this paper is devoted to dealing with the monotone feedback i.e.(g is a nondecreasing function), where the global asymptotic stability of equilibria to (1) is proved. Even though this first case is simple to prevent, it will play an important role in the more interesting case, when g is a non-monotone feedback. This is the goal of the second part of the paper. In this framework, the first objective is to obtain some estimates of solutions. In order to do this, we will split our study into two cases : x * ≤ M and x * > M, (M being the value for which g in (1) reaches its maximum and x * being the positive equilibrium). When dealing with the most interesting situation, namely x * > M, and as mentioned above, we will build nondecreasing functions (in most cases) less or equal than our principal feedback function g in (1) and satisfying a suitable condition stated later, therefore, we can apply the obtained results to the monotone semi-flow. Using a comparison principle result will lead us to the required estimates of solutions. From then on, we will use these established results to get a series of criteria for global asymptotic stability of positive equilibrium in the case where the functions g are nondecreasing over (0, M ) and non-increasing over (M, B) (unimodal functions).
We point out here that our resulting criteria improve, among other things, some existing results by relaxing the conditions on f in system (1). Indeed, in [15] (respectively [39]), the condition that f in (2) (αx−f 1 (x) in (3)) is strictly increasing over (0, B) is necessary in both works. In addition, as part of our discussion, we will present some extended results of global stability concerning the nonlinear equation mentioned in [1] and studied in [5], Theorem 3.1. Furthermore, at our knowledge there is no result on exponential stability of equilibria of system (1); we will give a condition for which the exponential stability of equilibria is reached.
Our main results will be applied to two well-known models : integro-differential Nicholson's blowflies and Mackey-Glass equations. Our theorems allow us to prove the absolute global asymptotic stability of the positive equilibrium whenever it is absolutely locally stable. We will also give some new results concerning the exponential stability of the positive equilibrium for these two cited models.
The paper is organized as follows : In the next section, we establish existence and uniqueness of positive solution and present a suitable Lyapunov functional. In section 3, we prove the asymptotic stability of equilibria when the monotonic case is considered. In Section 4, we show the strong persistence and determine some useful estimates of solutions in the case of non-monotone feedback. Section 5 is devoted to presenting some theorems related to the global asymptotic stability of equilibria. The global exponential stability of equilibria is investigated in Section 6. We illustrate our results by studying the integro-differential Nicholson's blowflies and Mackey-Glass equations in Section 7. A discussion is included in the last section.
Throughout this paper, we will make the following assumptions: we suppose that the function h is positive and  We will also use the notation |φ(θ)|, and C + = {φ ∈ C; φ(θ) ≥ 0, −τ ≤ θ ≤ 0} is the positive cone of C. Then (C, C + ) is a strongly ordered Banach space, that is: for We define the ordered intervals and for any χ ∈ R, we write χ * for the element of C satisfying χ * (θ) = χ for all θ ∈ [−τ, 0]. The segment x t ∈ C of a solution is defined by the relation x t (θ) = x(t + θ) where θ ∈ [−τ, 0] and t ≥ 0, in particular x 0 = φ. The family of maps defines a continuous semiflow on C + , [31]. The map Φ(t, .) is defined from C + to C + which is denoted by Φ t , Φ t (φ) = Φ(t, φ).

TARIK MOHAMMED TOUAOULA
The set of equilibria of the semiflow generated by (1) is given by

2.
Preliminaries. In this section, we first provide existence, uniqueness and boundedness of solution to problem (1), then a Lyapunov functional candidate is presented, some estimates of solutions are also given. We begin by recalling this useful theorem related to a comparison principle, see ( [28], page 78, Theorem 1.1) and which is adapted, for the reader convenience, to the context of our discussion.
We consider the following system where F : Ω → R is continuous on Ω, an open subset of C. We denote x(t, φ, F ) the maximal defined solution of (4).
, holds for all t ≥ 0 for which both are defined.
The following lemma, states existence, uniqueness and boundedness of the positive solution to (1). See also [33], [39] for similar results. Moreover, the semi-flow Φ t admits a compact global attractor which attracts every bounded set in C + .
Proof. We refer to [11] for the proof of existence and uniqueness of solution. Now we focus on the positivity of solution. We first claim that x(t) ≥ 0, for all t > 0. Otherwise, suppose (without loss of generality) that there exists T ≥ τ such that x(T ) = 0 and x (T ) ≤ 0, and using the fact that g(s) > g(0) for all s > 0 we get, which is a contradiction. The claim is proved. Concerning the boundedness of solutions. We setḡ(s) := max v∈[0,s] g(v) and we consider the following system According to Theorem 2.1 observe that if φ ∈ [0 * , l * ] then 0 ≤ y(t, φ) ≤ l for all t ≥ −τ with l is any constant greater than B. We also have x(t) ≤ y(t) for all t ≥ −τ. Hence, 0 ≤ x(t) ≤ l for all t ≥ 0 and l ≥ B. Therefore, since l is arbitrarily large, then the solution is global and the orbits of bounded sets for the semi-flow Φ are bounded. Next we claim that lim sup On the other hand, from ( [32], Proposition A.22) there exists t n → ∞ such that y(t n ) → l and y (t n ) → 0. Thus, substituting y(t n ) in (1), by the definition of lim sup y(t) and the monotonicity of the continuous functionḡ we haveḡ ( lim n→∞ y(t n − a)) ≤ḡ( lim n→∞ y(t n )) :=ḡ(l), ∀a ∈ [0, τ ].
Passing to the limit in (7) and combining with (8) we get, and this provides a contradiction with (6). Therefore the solution semiflow is point dissipative on C + , see [31]. Further, let F : C → R be defined as in view of (T1), F is continuous and satisfies a Lipschitz condition on each bounded subset of C. So we can easily show that F is completely continuous. Moreover, since the orbits of bounded sets are bounded, then the Proposition 5.5, [29] shows that the semiflow Φ t is completely continuous for any t ≥ 0. Finally according to [10], Theorem 3.4.8, the existence of a compact global attractor is established and the result is reached. Next we will show that this solution is strictly positive if φ(0) > 0. In fact, due to (T1) and (T2), according to (T1) we obtain with L 1 is f -Lipschitz constant; this gives The following result concerns a useful equality which will play an important role when dealing with the global asymptotic stability of equilibria.
Let H : R → R be a convex and differentiable function, we define the functions W and J as with γ ∈ R + and J(φ) = We introduce the functional V γ : C → R by The derivative of V γ along the solution of (1) is given by the following theorem.
Theorem 2.3. All solutions of problem (1) satisfy the following estimate where Proof. We first compute the derivative of W γ along the solutions of (1), we have Adding and subtracting the term W γ (x(t))g(x(t)) in the above equation, we get On the other hand, a direct computation of the derivative of J along solutions of (1) leads to, Summing these two above equalities, we obtain In view of (10) and (11), it yields where 2.1. Global asymptotic stability of the trivial equilibrium. Now we focus on global asymptotic stability of the trivial equilibrium. We begin by stating the following hypothesis : We have the extinction result as follows Theorem 2.4. Assume that (15) holds. Then the trivial equilibrium of problem (1) is globally asymptotically stable.

2.2.
Global stability of the positive equilibrium when g is nondecreasing.
We first present some assumptions, that will be used in this subsection. Assume that there exists a positive constant x * such that, Suppose also that f (0) and g (0) exist and verify The hypothesis (18) is stated in order to rule out the oscillation of f in the neighborhood of x * . However and as it will be established in the next proof, we will not need it, if g is assumed to be a strictly increasing function.
The following result concerns the strong persistence of solutions of (1), for more details on this notion see [31].
Proof. Notice that x(t) > 0 for any t > 0 for φ ∈ C + and φ(0) > 0. We now suppose that there exists an increasing sequence {t n } n , t n → ∞, t 0 ≥ τ and a non-increasing sequence {x(t n )} n such that x(t n ) < x * , x(t n ) → 0 as t n → ∞ and x(t n ) = min x(t), by substituting x(t n ) in the equation (1), we have since g is a nondecreasing function, so g(x(t n )) ≤ g(x(s)) for all 0 ≤ s ≤ t n , and hence we get we reach a contradiction with 0 < x(t n ) < x * and (16).
The second case. we assume that g is a nondecreasing function. Then from (16)- (18) and for θ ∈ (0,θ) we can build two strictly increasing functions g θ and g θ (see Appendix) satisfying the following assertions Next, we introduce the following auxiliary problems then, Theorem 2.1, gives that Hence, letting t tends to infinity and using the first part of the proof, we reach that Now, since θ is arbitrary small, we prove that all solutions of (1) converge to the positive equilibrium. Further, in order to reach the result, it is sufficient to prove the local stability of the positive equilibrium, that is, |φ(t)|. Indeed, for θ = ε we have, In particular choosing ε 1 = ε and using the fact that On the other hand, we have As above, choosing ε 2 = ε, by the fact that x(t) ≤ y ε (t) for all t > 0 we get finally, combining the above estimates, for Consequently the positive equilibrium is globally asymptotically stable.
3. The case when g is non-monotone.
3.1. Persistence and estimates of solutions. Our main objective in this section is to exhibit some fundamental results, including the strong persistence and some estimates of solutions to problem (1) in the case where g is non-monotone. We make the following assumption, that will be used from now on. There exists a positive constant x * such that, It is easy to show that, x * is the unique value that satisfies g(x * ) = f (x * ).
Remark 1. The positive equilibrium of (1), which is the constant function x * (t) = x * for all t ∈ [−τ, 0] satisfying the equation of (1) exists and is unique.
With the aim to prove the strong persistence and to obtain some estimates of solutions of (1), we need to build a nondecreasing function having some properties in order to apply the results of the previous section; this will be done by the help of the next lemmas.
in addition f and g are strictly increasing over [0, m].
Proof. First, from (17) there exists σ 0 > 0 such that f and g are strictly increasing over [0, σ 0 ]. Let γ > 0 be defined as Next Indeed, since f is strictly increasing over [0, from (23), we have We define α := min withm is the constant satisfyingm < m and g(m) = f (m) where m is defined in (22). The following result is easily checked.
Lemma 3.2. Assume that (17) and (21) hold. Then g B m defined in (24) is a nondecreasing function over (0, B) satisfying Now we are in position to prove the strong persistence of solutions of (1). Lemma 3.3. Assume that (17), (21) hold, then the solution of problem (1) is strongly persistent provided the corresponding initial data satisfies φ(0) > 0.
Proof. We consider the following problem with g B m is defined in (24). In view of Theorem 2.1, we have y(t) ≤ x(t) for all t > 0. Consequently, since g B m is a nondecreasing function, then the result is reached by applying Lemma 2.5.
In the following, we focus on functions g having a maximum. More precisely, assume that the function g satisfies: there exists a positive constant M such that, We will investigate two cases, namely, x * ≤ M and x * > M.

3.2.
The Case x * ≤ M . In order to state our next result we need the following lemma, Lemma 3.4. Under the hypotheses (21), (25). Then for all solutions x of problem (1) there exists T > 0 such that Proof. First, suppose that x * < M it yields from (21), We begin by assuming that all solutions of (1) satisfy x(t) > M > x * for all t > 0 then, due to the second assertion of (21) we have, Moreover, by integrating the equation (1) over (0, r), r > 0 we obtain thus, Rearranging these terms, we find passing to the limit as r tends to infinity we get, K(x(s))ds ≤ x(0) + C.

TARIK MOHAMMED TOUAOULA
Further, notice that K(x(.)) is uniformly continuous; indeed, as all solutions x of (1) are uniformly bounded and from (T1) we have, with L 1 and L 2 are Lipschitz constants associated to f and g respectively. Since the solutions of (1) are of uniformly bounded first derivative, then with C is the upper bound of the derivative of x, accordingly, From this, all solutions x of (1) converge either to zero or to the positive equilibrium x * ; in both cases we reach a contradiction with the assumption. Therefore there exists T > 0 such that x(T ) ≤ M. Next, we claim that x(t) ≤ M for all t ≥ T. Indeed, again by contradiction, we suppose that there exists a positive constant t > T such that x(t) = M and x (t) ≥ 0, then consequently, we arrive at this is a contradiction with (26).
Observe that, from the second assertion of (21) we get g(x * ) < f (s) for all x * < s ≤ B, thus combining this with (27) we conclude that According to Lemma 2.2 (substituting the hypothesis (T2) by (28)) we show that lim sup t→∞ x(t) ≤ x * and the result is reached since the semiflow admits a global compact attractor. The lemma is proved.
Next let us turn our attention towards the more interesting case.

3.3.
The case x * > M . This situation is more delicate to study and we need to impose additional hypotheses on f and g.
Now, to avoid any possibility of infinitely oscillations of g around f (M ), we will assume that g satisfies the next hypotheses, setting The rest of this subsection is devoted to estimating the solutions of (1) into two different situations namely, either D = ∅ or min D exists. The following lemma deals with the first case.
we claim that the function g B M is nondecreasing and satisfies, Indeed, from (31), (32) and (33) it is easily checked that g B M is nondecreasing and g B M (s) ≤ g(s) for all s ∈ [0, B]. We first take 0 < s <m, then using the fact that M < x * , we have Next, we consider the following problem, and M is defined in (25). The following lemma gives some estimates of solutions of (1).
Then for all solutions x of problem (1) there exists T > 0 such that Proof. First, we claim that x * < A. Conversely, suppose x * ≥ A, then if x * > A, due to the first assertion of (21) and M < A, we obtain On the other hand, we prove that for all solutions x of (1) it is impossible to have T > 0 such that x(t) > A for all t ≥ T. Indeed, if the contrary is true, so we take t ≥ T + τ and let us consider the following problem the contradiction comes from x * < A < x(t) ≤ y(t) and y(t) converges to x * , sincē g is a nondecreasing function and consequently Theorem 2.6 can be applied. Then there exists t 0 > 0 such that x(t 0 ) ≤ A, we claim that x(t) ≤ A for all t ≥ t 0 .
Otherwise, there exists t 1 > 0 such that x(t 1 ) = A and x (t 1 ) ≥ 0 thus, from (1), this is a contradiction with (37). The upper bound of solutions x to problem (1) is proved.
Next we focus on the lower bound of x. From (29), (36) the function g A M defined in (33) (by substituting B by A) is nondecreasing and satisfies (34). So by introducing the following problem Indeed, according to (T2) we have since f is strictly increasing, we find The following corollary deals with the case when x * ≤ M and its proof is a direct consequence of Theorem 2.6.
We now establish the main theorem of this section related to the case x * > M. Proof. First of all, according to Lemma 3.6 we know that for every solution x of (1) there exists T > 0 such that M ≤ x(t) ≤ A for all t > T. Now we split the proof into two possible cases, non-oscillatory, that is the trajectory does not intersect the positive equilibrium infinitely many times, and oscillatory, that is the trajectories oscillate infinitely around the positive equilibrium : 1/ Non-oscillatory case. We first suppose that for all solutions x of (1) there exists T > 0 such that x(t) ≤ x * for all t > T. Then, we introduce the following function and the associated problem (y(t − a))da, t > 0, thus since g(s) ≤ g(s) for all s ≤ x * then y(t) ≤ x(t) for all t > 0 and y(t) converges to x * as t tends to infinity. Now if x(t) ≥ x * for all t ≥ T, as above, letḡ be defined asḡ the solution of the following problem, verify x(t) ≤ y(t) for all t > 0 and y(t) converges to x * as t goes to infinity.
Concerning the local stability we use the same arguments as in the proof of Theorem 2.6.
2/ Oscillatory case. We claim that this situation is not possible. Assume by contradiction that the solution x oscillates infinitely around the positive equilibrium x * , we set Using the fluctuations method, see [31], [32], there exist two sequences t n → ∞ and s n → ∞ such that lim n→∞ x(t n ) = x ∞ , x (t n ) = 0, ∀n ≥ 1, and lim n→∞ x(s n ) = x ∞ , x (s n ) = 0, ∀n ≥ 1, then substituting x(t n ) in problem (1) it follows that, now, using the fact that passing to the limit in (47) and since g is nonincreasing over [M, A], then Similarly by taking in consideration that we get, from g(0) = f (0) we have, Multiplying the expression (50) by g(x ∞ ) − g(0) > 0 and combining with (51) we obtain this fact together with the hypothesis (H1) give x ∞ ≤ x ∞ , so we reach a contradiction. Arguing as before we may conclude the results for (H2) and (H3). Now suppose that (H4) holds. First notice that G defined just before the present theorem make sense, that is, for all s ∈ In view of (48), (49) and the monotonicity off we arrive at, and with G(s) :=f −1 (g(s)). Now suppose that x ∞ < x * ≤ x ∞ , then, applying the function G, the inequalities (52)-(53) become due to (H4) it ensures that x * ≤ x ∞ , which is a contradiction. Moreover, if x ∞ ≤ x * < x ∞ then from (54) we get In addition, according to (48) we have the contradiction is also reached. Using the same arguments as in (H4) we establish the result for (H5). The Theorem is proved.
Proof. In order to prove this corollary, it suffices to show that x * ≤M and x(t) ≤M for all t ≥ T. Then, the result follows from (H1) in Theorem 3.7. First, from (56) and the fact that thus, K(M ) < 0 and therefore x * <M . Now we first suppose that there exists T > 0 such that x(t) ≥M for all t > T. Notice that M <M , then, as done in the first part of the proof of Theorem 3.7, we defineḡ and by the sequel y as in (45)-(46). So we have x * <M ≤ x(t) ≤ y(t) and y(t) converges to x * , which is a contradiction. As a conclusion, there exists t 0 > 0 such that x(t 0 ) <M . Now we claim that x(t) ≤M for all t > t 0 , otherwise, there exists t 1 > 0 such that On the other hand, by multiplying the expression (56) by f (M ) − f (0) we have, since the function H reaches its maximum inM it follows that, , which contradicts (58).

4.
Global exponential stability of equilibria. Now let us investigate the global exponential stability of equilibria. For this, we first establish the following lemma.
If A 1 > A 2 , then there exist two positive constants C and A 3 with Proof. First, observe that (59) can be reformulated as with z(t) = e A1t w(t). Now, we can choose C > 0 and A 2 < A 3 < A 1 such that Ce A3t is a supersolution of (61), that is Indeed, since A 1 > A 2 there exists ε so small such that Next, we set A 3 = A 1 − ε, it follows that  (62) Proof. First of all, in view of the proof of Theorem 3.7, notice that the oscillatory case is not possible. Thus, we set y(t) = x(t) − x * , and we suppose that, there exists T > 0 such that y(t) ≥ 0 for all t ≥ T (the proof will be the same if we assume that y(t) ≤ 0 for all t ≥ T ). Then y satisfies the following problem where θ(t) (θ 1 (t − a) respectively) is a value between x(t) and x * (x(t − a) and x * respectively). Now since x(t) and x * belong to [M, A] we obtain, The result is established from (62) and by applying Lemma 4. |g (s)|.

5.
Application to Nicholson's blowflies and Mackey-Glass models. The goal of this section is to apply our results to two well-known models, namely Blowflies and Mackey-Glass distributed delay equation. For a good survey in this direction see [1] and references therein. First, observe that the condition (21) is verified whenever the positive equilibrium exists. We set τ 0 h(a)da = 1.

5.1.
Nicholson's blowflies model. The blowflies model with distributed delay is defined as follows where the equation of equilibria is given by Thus either x * = 0 or x * = ln( 1 δ ) whenever 1 δ > 1. For more details and results concerning this equation see [1] and references therein.
Proof. The condition (15) is equivalent to δ ≥ 1, so the result is obtained by applying Theorem 2.4.
Theorem 5.2. The positive equilibrium of (63) is globally asymptotically stable provided that Moreover this equilibrium is globally exponentially stable if, Proof. First, for g(s) = se −s , notice that sup s∈R + g(s) = g(1) = e −1 . Now, assume that 1 < 1 δ ≤ e, then we have x * ≤ M := 1, with M is defined in (25). Therefore all assumptions of Corollary 2 are satisfied, hence we conclude that x * is globally asymptotically stable. Next, suppose that this case corresponds to x * > M := 1, thus we can define the constant A as in (36) On the other hand, the condition (62) holds if as a consequence, the exponential stability follows from Theorem 4.2.

Mackey-Glass model of hematopoiesis.
To model a blood cell production and haematological diseases, different authors studied the following Mackey-Glass model with distributed delay, For more results of this type of equations, see [2] and the references therein.
We set g(s) = s 1 + s n and K(s) = g(s) − δs, notice that sup By the application of our theorems stated in the previous sections, we obtain Theorem 5.3. Suppose that δ ≥ 1, then the trivial equilibrium of (68) is globally asymptotically stable.
Before stating the global stability of the positive equilibrium, we need to verify the requirement (37) of Lemma 3.6. The other conditions are clearly verified. This is the objective of the following lemma. with Proof. In the context of Mackey-Glass model, the condition (37) reads, and From Corollary 1 and Remark 2, the condition (71) is equivalent to we observe that the expression (71) is equivalent to, The lemma is proved.
We will now present our version of Theorem 4.5 in [35], that provides sufficient conditions for the global asymptotic stability of the positive equilibrium. Although the equation considered in [35] only has a discrete delay, but the dynamical system setting and method used there should also work for our problem.
Theorem 5.5. Suppose that δ < 1, then the positive equilibrium of (68) is globally asymptotically stable if one of the following conditions is satisfied n > 2 and 1 δ < n n − 2 . (74) Proof. First case. 0 < n ≤ 2. We will divide this case into three subcases Subcase 1. 0 < n ≤ 1. Observe that this corresponds to g is increasing, therefore the result follows from Theorem 2.6.
. This corresponds to x * ≤ M, so the Corollary 2 is applied to reach the global asymptotic stability of the positive equilibrium. and so, F ( n n − 1 ) < 0, hence from Lemma 5.4 the assertion (37) is verified. Moreover, by a straightforward computation we show that the function G(G(s)) s is decreasing for all s > 0 and all 0 < n ≤ 2, with G(s) = 1 δ s 1 + s n , thus the condition (H4) of Theorem 3.7 is satisfied. Thus the assertion (73) is proved.
Second case. n > 2 and 1 δ ≤ n n − 1 . As Corollary 2 can be applied for n > 2, by the same reasoning of Subcase 2 we get the result.
Third case. n > 2 and n n − 1 < 1 δ < n n − 2 . By a simple calculation we remark that G(G(s)) s is a decreasing function in this case. We conclude that (H4) of observe that the function F is concave for s < s * and convex for s > s * . In addition, in view of n > 2 and n n − 1 < 1 δ < n n − 2 , it is easy to show that as a consequence it is clear that F ( n n − 1 ) < 0. The result follows from Theorem 3.7.
The case leading to the exponential stability of the positive equilibrium is proved by applying Theorem 4.2.
6. Discussion. In this paper we have investigated the global behavior of solutions for a class of functional differential equations (1). We have presented some conditions to ensure global asymptotic and exponential stability of the unique positive equilibrium. We illustrated the obtained results in some well-known biologic models namely, the Nicholson and Mackey-Glass equations.
In addition, when dealing with the following equation, which is mentioned in [1], in the particular case of non autonomous equations, our methods give more explicit conditions than those presented in [5]. In fact the stability of equilibria of equation (75) is carried out by applying some of the previous results of this paper. More precisely we use the Theorem 2.4 for the trivial equilibrium and the Corollary 2, Theorem 3.7 (H1) together with the Remark 3, in order to prove global asymptotic stability of the positive equilibrium. We set f (s) = αs b + s and g(s) = qse −s , then we have Lemma 6.1. The trivial equilibrium of (75) is globally asymptotically stable if one of the following conditions holds: b) e b−1 < α q and b < 1.
Lemma 6.2. Assume that α q < b, then the positive equilibrium of (75) exists and is unique. Moreover this one is globally asymptotically stable if one of the following conditions holds: a) (b + 1)e −1 ≤ α q .
In view of (76), the assertion (T2) is verified for B = 1, thus due to Lemma 2.2 we have, for all solutions x of (75) there exists T > 0 such that x(t) ≤ 1 for all t ≥ T, we use Corollary 2 to reach the result. Next suppose that e −1 < α q < (b + 1)e −1 .