Global stability for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity

: The present work is devoted to the global stability analysis for a class of functional di ﬀ erential equations with distributed delay and non-monotone bistable nonlinearity. First, we characterize some subsets of attraction basins of equilibria. Next, by Lyapunov functional and ﬂuctuation method, we obtain a series of criteria for the global stability of equilibria. Finally, we illustrate our results by applying them to a problem with Allee e ﬀ ect.

When g(x) allows Eq (1.1) to have two positive equilibria x * 1 and x * 2 in addition to the trivial equilibrium, Eq (1.1) is said to be a problem with bistable nonlinearity. In this case, Huang et al. [5] investigated the following general equation: x (t) = − f (x(t)) + g(x(t − τ)).
The authors described the basins of attraction of equilibria and obtained a series of invariant intervals using the decomposition domain. Their results were applied to models with Allee effect, that is, f (x) = µx and g(x) = βx 2 e −αx . We point out that the authors in [5] only proved the global stability for x * 2 ≤ M, where g(M) = max s∈R + g(s). In this paper, we are interested in the dynamics of the bistable nonlinearity problem (1.1). More precisely, we will present some attracting intervals which will enable us to give general conditions on f and g that ensure global asymptotic stability of equilibria x * 1 and x * 2 in the both cases x * 2 < M and x * 2 ≥ M. The paper is organized as follows: in Section 2, we give some preliminary results including existence, uniqueness and boundedness of the solution as well as a comparison result. We finish this section by proving the global asymptotic stability of the trivial equilibrium. Section 3 is devoted to establish the attractive intervals of solutions and to prove the global asymptotic stability of the positive equilibrium x * 2 . In Section 4, we investigate an application of our results to a model with Allee effect. In Section 5, we perform numerical simulation that supports our theoretical results. Finally, Section 6 is devoted to the conclusion.
In the whole paper, we suppose that the function h is positive and τ 0 h(a)da = 1.
(T4) There exist two positive constants x * 1 and x * 2 such that g( . We define the following ordered interval: 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, defines a continuous semiflow on C + [28]. For each t ≥ 0, the map Φ(t, .) is defined from The set of equilibria of the semiflow, which is generated by (1.1), is given by

Preliminary results
In this section, we first provide existence, uniqueness and boundedness of solution to problem (1.1). We then present a Lyapunov functional and show the global asymptotic stability of the trivial equilibrium. We begin by recalling a useful theorem related to a comparison principle (see Theorem 1.1 in page 78 in [29]).
We consider the following problem: where F : Ω → R is continuous on Ω, which is an open subset of C. We write x(t, φ, F) for the maximal defined solution of problem (2.1). When we need to emphasize the dependence of a solution on initial data, we write x(t, φ) or x(φ).
Theorem 2.1. Let f 1 , f 2 : Ω → R be continuous, Lipschitz on each compact subset of Ω, and assume that either f 1 or f 2 is a nondecreasing function, with f 1 (φ) ≤ f 2 (φ) for all φ ∈ Ω. Then holds for all t ≥ 0, for which both are defined.
The following lemma states existence, uniqueness and boundedness of the positive solution to problem (1.1). For the proof, see [15,30,31]. Lemma 2.2. Suppose that (T1) holds. For any φ ∈ C + , the problem (1.1) has a unique positive solution In addition, we have the following estimate: Moreover, the semiflow Φ t admits a compact global attractor, which attracts every bounded set in C + .
The following lemma can be easily proved (see also the proof of Theorem 1.1 in [29]). Lemma 2.3. Let φ ∈ C be a given initial condition, x(φ) be the solution of problem (1.1) and x (φ), > 0 be the solution of problem (1.1) when replacing f by f ± . Then We focus now on the global stability of the trivial equilibrium. For this purpose, we suppose that Lemma 2.4. Assume that (T1) and condition (2.2) hold. For a given φ ∈ provided that one of the following hypotheses holds: (i) g is a nondecreasing function over (0, x * 1 ).
(ii) f is a nondecreasing function over (0, x * 1 ). Proof. Without loss of generality, we assume that φ(0) < x * 1 . Suppose that (i) holds. We first claim that x(t) < x * 1 for all t ≥ 0. Let x (φ) be the solution of problem (1.1) by replacing f by f + . We prove that x (t) := x (t, φ) ≤ x * 1 for all t > 0. Suppose, on the contrary, that there exists t 1 > 0 such that which leads to a contradiction. Now, applying Lemma 2.3, we obtain x(t) ≤ x * 1 for all t ≥ 0. Next, set X(t) = x * 1 − x(t). We then have Since x(t) ≤ x * 1 for all t ≥ 0 and g is a nondecreasing function, we have, for t ≥ τ, where L is the Lipschitz constant of f. Consequently, X(t) ≥ X(0)e −Lt and the claim is proved. Finally, to prove inequality (2.3), we suppose, on the contrary, that there exist an increasing sequence (t n ) n , t n → ∞, t 0 ≥ τ and a nondecreasing sequence (x(t n )) n , such that x(t n ) → x * 1 as t n → ∞, x(t n ) < x * 1 for some n and x(t n ) = max t∈[0,t n ] x(t). Then, in view of condition (2.2), the equation of x(t n ) satisfies which is a contradiction.
Then, again by contradiction, suppose that there exists t 1 > 0 such that x (t 1 ) = x * 1 , x (t) ≤ x * 1 for all t ≤ t 1 and x (t 1 ) ≥ 0. Then, arguing as above, we obtain Since f is a nondecreasing function and g + is the smallest nondecreasing function that is greater than g, we obtain f (s) > g + (s) for all s ∈ (0, x * 1 ). This contradicts with inequality (2.4). Further, by combining Theorem 2.1 and Lemma 2.3, we get x(t) ≤ x * 1 for all t ≥ 0. Finally, following the same arguments as in the first part of this proof, the lemma is proved.
We now prove the global asymptotic stability of the trivial equilibrium.
Theorem 2.5. Assume that (T1) and condition (2 The trivial equilibrium is globally asymptotically stable if one of the following hypotheses holds: (i) g is a nondecreasing function over (0, x * 1 ).
Proof. Suppose that (i) holds. Let V be the Lyapunov functional defined by From condition (2.2) and Lemma 2.4, we have dV(x t )/dt < 0 and thus, the result is reached by classical Lyapunov theorem (see [6]). Next, suppose that (ii) holds. Let the function V be defined in (2.5) by replacing g(s) by g + (s) := max , we can employ the same argument as above to get dV(x t )/dt < 0. Finally the result is obtained by applying Theorem 2.1 and Lemma 2.4. This completes the proof.
Now, suppose that Using the same proof as in Theorem 2.5, we immediately obtain the following theorem.
3. Global stability of the positive equilibrium x * 2 The following theorem concerns the global stability of x * 2 in the case where g is a nondecreasing function.
Proof. Without loss of generality, suppose that φ(0) > x * 1 . We first claim that lim inf t→∞ x(t) > x * 1 . For this, let x (t) := x (t; φ) be the solution of problem (1.1) when replacing f by f − . To reach the claim, we begin by proving that This reaches a contradiction and thus where L is the Lipschitz constant of f . Consequently x(t) > x * 1 for all t > 0. Now, suppose that there exist an increasing sequence (t n ) n , t n → ∞, t 0 ≥ τ and a nonincreasing sequence (x(t n )) n , such that x(t n ) → x * 1 as t n → ∞, x(t n ) > x * 1 for some n and x(t n ) = min x(t). In view of (T4), the equation of x(t n ) satisfies which is a contradiction. The claim is proved.
To prove that x * 2 is globally asymptotically stable, we consider the following Lyapunov functional Note that ψ(τ) = 0. In view of (T4), we have dV(x t )/dt ≤ 0. If g is an increasing function, then the result is reached by using a classical Lyapunov theorem (see, e.g., [30]). If g is a nondecreasing function, then the result is proved by using the same argument as in the proof of Theorem 2.6 in [31]. This completes the proof.
We focus now on the case where g is non-monotone. Suppose that there existsĜ(x) such that x be the solution of problem (1.1). Then, the following assertions hold: . Since f is an increasing function, we have f (Ĝ(x * 1 )) − > g + (Ĝ(x * 1 )). Accordingly, the functionĜ(x * 1 ) is a super-solution of problem (3.1). Finally, in view of Theorem 2.1, we obtain x (t) ≤ y (t) ≤Ĝ(x * 1 ) for all t ≥ 0. We now prove that x (t) > x * 1 . Suppose, on the contrary, that there exists t 1 > 0 such that 1 for all t ≤ t 1 , and thus, x (t 1 ) ≤ 0. Then, the equation of x (t 1 ) satisfies ]. This reaches a contradiction. Further, from Lemma 2.3, we obtain x(t) ≥ x * 1 . The claim is proved.

(3.2)
Since f is an increasing function andĜ(x * 1 ) > M, we have Now, the assertion x * 2 < M implies that g(M) < f (M), which leads to a contradiction. When x * 2 ≥ M, inequality (3.2) gives a contradiction by hypothesis. The Lemma is proved. Using Lemma 3.2, we next prove the following lemma.
Assume also that (T1)-(T4) hold. Let x be the solution of problem (1.1). Then, we have lim inf provided that one of the following assertions holds: (ii) x * 2 ≥ M and f (Ĝ(x * 1 )) > g(M).
Assume also that (T1)-(T4) hold. Then, the positive equilibrium x * 2 is globally asymptotically stable. Proof. We first claim that there exists T > 0 such that x(t) ≤ M for all t ≥ T. Indeed, let x := x (φ) be the solution of problem (1.1) when replacing f by f + . First, suppose that there exists T > 0 such that x (t) ≥ M for all t ≥ T . So, since x * 2 < M, we have from (T4) that g(M) < f (M). Combining this with (T2), the equation of x satisfies which contradicts with x (t) ≥ M. Hence there exists T > 0 such that x (T ) < M. We show that x (t) < M for all t ≥ T. In fact, at the contrary, if there exists t 1 > T such that x (t 1 ) = M and so x (t 1 ) ≥ 0 then, which is a contradiction. Further, according to Lemma 2.3, there exists T > 0 such that x(t) ≤ M for all t ≥ T and the claim is proved. Finally, since g is a nondecreasing function over (0, M), the global asymptotic stability of x * 2 is proved by applying Theorem 3.1. Remark 3.5. In the case where g(s) > g(x * 1 ) for all s > x * 1 , the above theorem holds true. In fact, it suffices to replaceĜ( We focus now on the case where x Proof. We begin by claiming that x * 2 ≤ A. On the contrary, suppose that x * 2 > A ≥ M. Then, due to (T2) and (T3), we have g( , which is a contradiction. The claim is proved. Next, for a given φ ∈ C [x * 1 ,Ĝ(x * 1 )] \ {x * 1 }, let x := x (φ) be the solution of problem (1.1) when replacing f by f + . Since x converges to x as tends to zero, we only need to prove that there exists T > 0 such that x (t) < A for all t ≥ T . To this end, let y := y (φ) be the solution of problem (1.1), when replacing f by f + and g by g + with g + (s) = max σ∈[0,s] g(σ). By Theorem 2.1, we have x (t) ≤ y (t) for all t ≥ 0, and thus, we only need to show that y (t) < A for all t ≥ T . On the contrary, we suppose that y (t) ≥ A for all t > 0. Then, combining the equation of y and the fact that g + (s) = g(M) for all s ≥ M, we obtain which is a contradiction. Then, there exists T > 0 such that y (T ) < A. We further claim that y (t) < A for all t ≥ T. Otherwise, there exists t 1 > T such that y (t 1 ) = A and y (t 1 ) ≥ 0. Substituting y (t 1 ) in Eq (1.1), we get which is a contradiction. Consequently, by passing to the limit in , we obtain that x(t) ≤ A for all t ≥ T.
We focus now on the lower bound of x. First, define the function where x * 1 <m < M, which is the constant satisfying g(m) = f (M). Note that, due to (T2)-(T4) and condition (3.5), the function g is nondecreasing over (0, A) and satisfies Let y(φ) be the solution of problem (1.1) when replacing g by g. From Theorem 2.1, we have y(t; φ) ≤ x(t; φ) for all t > 0, and from Theorem 3.1, we have lim In addition, if, for all T > 0, there exists (t n ) n such that t n > T, y(t n ) = M,m < y(s) ≤ A for all 0 < s ≤ t n and y (t n ) < 0, then, for t n > T + τ, we introduce the following function: Let y(φ) be the solution of problem (1.1) when replacing g by g. Since g(s) ≤ g(s) for all x * 1 ≤ s ≤ x * 2 , we have y(t) ≤ x(t) for all t > 0. Further, in view of Theorem 3.1, we obtain The local stability is obtained by using the same idea as in the proof of Theorem 2.6 in [31]. Now, for x ∞ ≥ x * 2 , we introduce the function and let y(φ) be the solution of problem (1.1) when replacing g byḡ. As above, we have x(t) ≤ y(t) for all t > 0 and y(t) converges to x * 2 as t goes to infinity. Next, we suppose that x ∞ < x * 2 < x ∞ and we prove that it is impossible. Indeed, according to Lemma 3.6, we know that, for every solution x of Now, using the fluctuation method (see [28,32]), 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.
Substituting x(t n ) in problem (1.1), it follows that Since g is nonincreasing over [M, A], we obtain, by passing to the limit in Eq (3.8), that Similarly, we obtain Multiplying the expression (3.9) by g(x ∞ ) and combining with inequality (3.10), we get This fact, together with the hypothesis (H1), gives x ∞ ≤ x ∞ , which is a contradiction. In a similar way, we can conclude the contradiction for (H2). Now, suppose that (H3) holds. First, notice that G makes sense, that is, for all s ∈ In view of inequalities (3.9) and (3.10) and the monotonicity off , we arrive at and x ∞ ≥ G(x ∞ ), (3.12) with G(s) =f −1 (g(s)). Now, applying the function G to inequalities (3.11) and (3.12), we find (3.13) Due to (H3), it ensures that x * 2 ≤ x ∞ , which is impossible. Using the same arguments as above, we obtain a contradiction for (H4). The theorem is proved.
Remark 3.8. In the case where g(s) > g(A) for all s > M, the two above results hold true. In fact, it suffices to replace A in Lemma 3.6 and Theorem 3.7 by B defined in (T1).
For the tangential case where two positive equilibria x * 1 and x * 2 are equal, we have the following theorem Theorem 3.9. Suppose that (T1)-(T3) hold. Suppose that, in addition to the trivial equilibrium, problem (1.1) has a unique positive equilibrium x * 1 . Then , then x * 1 attracts every solution of problem (1.1) and x * 1 is unstable. Proof. The uniqueness of the positive equilibrium implies that x * 1 ≤ M and g(x) < f (x) for all x x * 1 . For (i), suppose that there exists t 0 > 0 such that x(t) ≥ M for all t ≥ t 0 . Then, by substituting x in Eq (1.1), we get This is impossible and then there exists T > 0 such that Consequently (i) holds. We argue as in the proof of Lemma 3.2 (i), to show (ii). Concerning (iii), we consider the following Lyapunov functional where ψ(a) = τ a h(σ)dσ. As in the proof of Theorem 3.1, the derivative of V along the solution of Since g(s) < f (s) for all s x

Application to a model with Allee effect and distributed delay
In this section, we apply our results to the following distributed delay differential equation: where µ, k are positive constants. The variable x(t) stands for the maturated population at time t and τ > 0 is the maximal maturation time of the species under consideration. h(a) is the maturity rate at age a. In this model, the death function f (x) = µx and the birth function g(x) = kx 2 /(1 + 2x 3 ) reflect the so called Allee effect. Obviously, the functions f and g satisfy the assumptions (T1)-(T3) and g reaches the maximum value k/3 at the point M = 1. The equilibria of Eq (4.1) satisfies the following equation: Analyzing Eq (4.2), we obtain the following proposition. (iii) if 0 < µ < 2 1 3 3 k, then Eq (4.1) has exactly two positive equilibria x * 1 < x * 2 . Moreover, Using Theorem 2.6, we obtain . (iii) There exists two heteroclinic orbits X (1) and X (2) connecting 0 to x * 1 and x * 1 to x * 2 , respectively. Proof. (i) and (ii) directly follow from Proposition 4.1, (iii)-a and Theorems 2.5 and 3.4. For (iii), we follow the same proof as in Theorem 4.2 in [5]. See also [18]. For the sake of completeness, we rewrite it. Let K = {x * 1 }. Clearly, K is an isolated and unstable compact invariant set in C [0,x * 1 ] and C [x * 1 ,Ĝ(x * 1 )] . By applying Corollary 2.9 in [33] to Φ |R + ×C[0,x * 1 ] and Φ |R + ×C[x * 1 ,Ĝ(x * 1 )] , respectively, there exist two precompact full orbits X (1) : such that α(X 1 ) = α(X 2 ) = K. This together with statements (i) and (ii) gives ω(X (1) ) = {0} and ω(X (2) ) = {x * 2 }. In other words, there exist two heteroclinic orbits X (1) and X (2) , which connect 0 to x * 1 and x * 1 to x * 2 , respectively. This completes the proof.  First, let µ = 0.5. In this case, µ > 2 1 3 3 k and thus, by Theorem 4.2, the trivial equilibrium is globally asymptotically stable. In fact, Figure 2 shows that x(t) converges to 0 as t increases for different initial data.  Next, let µ = 0.37. In this case, k 3 < µ < 2 1 3 3 k and x * 1 ≈ 0.43, x * 2 ≈ 0.89 andĜ(x * 1 ) ≈ 3.09 ( Figure 3). Hence, by Theorem 4.3, we see that the trivial equilibrium is globally asymptotically In fact, Figure 4 shows such a bistable situation. Moreover, heteroclinic orbits X (1) and X (2) , which were stated  in Theorem 4.3 (iii), are shown in Figure 5.

Conclusions
In this paper, we studied the bistable nonlinearity problem for a general class of functional differential equations with distributed delay, which includes many mathematical models in biology and ecology. In contrast to the previous work [5], we considered both cases x * 2 < M and x * 2 ≥ M, and obtained sufficient conditions for the global asymptotic stability of each equilibrium. The general results were applied to a model with Allee effect in Section 4, and numerical simulation was performed in Section 5. It should be pointed out that our theoretical results are robust for the variation of the form of the distribution function h. This might suggest us that the distributed delay is not essential for the dynamical system of our model. We conjecture that our results still hold for τ = ∞, and we leave it for a future study. Extension of our results to a reaction-diffusion equation could also be an interesting future problem.