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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 differential 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 effect.

    Citation: Toshikazu Kuniya, Tarik Mohammed Touaoula. Global stability for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7332-7352. doi: 10.3934/mbe.2020375

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  • The present work is devoted to the global stability analysis for a class of functional differential 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 effect.


    In this paper, we study the following general class of functional differential equations with distributed delay and bistable nonlinearity,

    {x(t)=f(x(t))+τ0h(a)g(x(ta))da,t>0,x(t)=ϕ(t),τt0. (1.1)

    Many mathematical models issued from ecology, population dynamics and other scientific fields take the form of Eq (1.1) (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] and references therein). When a spatial diffusion is considered, many models are studied in the literature (see, e.g., [16,17,18,19,20,21,22]).

    For the case where g(x)=f(x) has not only the trivial equilibrium but also a unique positive equilibrium, Eq (1.1) is said to be a problem with monostable nonlinearity. In this framework, many authors studied problem (1.1) with various monostable nonlinearities such as Blowflies equations, where f(x)=μx and g(x)=βxeαx, and Mackey-Glass equations, where f(x)=μx and g(x)=βx/(α+xn) (see, e.g., [1,3,6,7,9,10,11,12,13,21,23,24,25,26,27]).

    When g(x) allows Eq (1.1) to have two positive equilibria x1 and x2 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(tτ)).

    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)=βx2eαx. We point out that the authors in [5] only proved the global stability for x2M, where g(M)=maxsR+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 x1 and x2 in the both cases x2<M and x2M.

    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 x2. 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

    τ0h(a)da=1.

    We give now some standard assumptions.

    (T1) f and g are Lipschitz continuous with f(0)=g(0)=0 and there exists a number B>0 such that maxv[0,s]g(v)<f(s) for all s>B.

    (T2) f(s)>0 for all s0.

    (T3) g(s)>0 for all s>0 and there exists a unique M>0 such that g(s)>0 for 0<s<M, g(M)=0 and g(s)<0 for s>M.

    (T4) There exist two positive constants x1 and x2 such that g(x)<f(x) if x(0,x1)(x2,) and g(x)>f(x) if x(x1,x2) and g(x1)>f(x1).

    Let C:=C([τ,0],R) be the Banach space of continuous functions defined in [τ,0] with ||ϕ||=supθ[τ,0]|ϕ(θ)| and C+={ϕC;ϕ(θ)0,τθ0} is the positive cone of C. Then, (C,C+) is a strongly ordered Banach space. That is, for all ϕ,ψC, we write ϕψ if ϕψC+, ϕ>ψ if ϕψC+{0} and ϕ>>ψ if ϕψInt(C+). We define the following ordered interval:

    C[ϕ,ψ]:={ξC;ϕξψ}.

    For any χR, we write χ for the element of C satisfying χ(θ)=χ for all θ[τ,0]. The segment xtC of a solution is defined by the relation xt(θ)=x(t+θ), where θ[τ,0] and t0. In particular, x0=ϕ. The family of maps

    Φ:[0,)×C+C+,

    such that

    (t,ϕ)xt(ϕ)

    defines a continuous semiflow on C+ [28]. For each t0, the map Φ(t,.) is defined from C+ to C+ which is denoted by Φt:

    Φt(ϕ)=Φ(t,ϕ).

    The set of equilibria of the semiflow, which is generated by (1.1), is given by

    E={χC+;χRandg(χ)=f(χ)}.

    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:

    {x(t)=F(xt),t>0,x(t)=ϕ(t),τt0, (2.1)

    where F:ΩR 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 f1, f2 :ΩR be continuous, Lipschitz on each compact subset of Ω, and assume that either f1 or f2 is a nondecreasing function, with f1(ϕ)f2(ϕ) for all ϕΩ. Then

    x(t,ϕ,f1)x(t,ϕ,f2),

    holds for all t0, 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 ϕC+, the problem (1.1) has a unique positive solution x(t):=x(t,ϕ) on [0,) satisfying x0=ϕ, provided ϕ(0)>0. In addition, we have the following estimate:

    0lim suptx(t)B.

    Moreover, the semiflow Φt 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 ϕC be a given initial condition, x(ϕ) be the solution of problem (1.1) and xϵ(ϕ), ϵ>0 be the solution of problem (1.1) when replacing f by f±ϵ. Then

    xϵ(t,ϕ)x(t,ϕ) as ϵ0, for all t[0,).

    We focus now on the global stability of the trivial equilibrium. For this purpose, we suppose that

    f(s)>g(s) for all s(0,x1) and f(x1)=g(x1). (2.2)

    Lemma 2.4. Assume that (T1) and condition (2.2) hold. For a given ϕC[0,x1]{x1}, the solution x(ϕ) of problem (1.1) satisfies

    lim suptx(t)<x1, (2.3)

    provided that one of the following hypotheses holds:

    (i) g is a nondecreasing function over (0,x1).

    (ii) f is a nondecreasing function over (0,x1).

    Proof. Without loss of generality, we assume that ϕ(0)<x1. Suppose that (ⅰ) holds. We first claim that x(t)<x1 for all t0. Let xϵ(ϕ) be the solution of problem (1.1) by replacing f by f+ϵ. We prove that xϵ(t):=xϵ(t,ϕ)x1 for all t>0. Suppose, on the contrary, that there exists t1>0 such that xϵ(t1)=x1, xϵ(t)x1 for all tt1 and xϵ(t1)0. Then, using the equation of xϵ(t1), we have

    0xϵ(t1)=f(xϵ(t1))ϵ+τ0h(a)g(xϵ(t1a))da,<f(x1)+g(x1)=0,

    which leads to a contradiction. Now, applying Lemma 2.3, we obtain x(t)x1 for all t0. Next, set X(t)=x1x(t). We then have

    X(t)=f(x(t))f(x1)+τ0h(a)(g(x1)g(x(ta)))da.

    Since x(t)x1 for all t0 and g is a nondecreasing function, we have, for tτ,

    X(t)LX(t),

    where L is the Lipschitz constant of f. Consequently, X(t)X(0)eLt and the claim is proved. Finally, to prove inequality (2.3), we suppose, on the contrary, that there exist an increasing sequence (tn)n, tn, t0τ and a nondecreasing sequence (x(tn))n, such that x(tn)x1 as tn, x(tn)<x1 for some n and x(tn)=maxt[0,tn]x(t). Then, in view of condition (2.2), the equation of x(tn) satisfies

    0x(tn)=f(x(tn))+τ0h(a)g(x(tna))da,f(x(tn))+g(x(tn))<0,

    which is a contradiction.

    Suppose now that () holds. Let xϵ(t) be the solution of problem (1.1) by replacing f(s) by f(s)+ϵ and g(s) by g+(s)=maxσ[0,s]g(σ).

    Then, again by contradiction, suppose that there exists t1>0 such that xϵ(t1)=x1, xϵ(t)x1 for all tt1 and xϵ(t1)0. Then, arguing as above, we obtain

    0xϵ(t1)=f(xϵ(t1))ϵ+τ0h(a)g+(xϵ(ta))da,<f(x1)+g+(x1). (2.4)

    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,x1). This contradicts with inequality (2.4). Further, by combining Theorem 2.1 and Lemma 2.3, we get x(t)x1 for all t0. 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 (2.2) hold. Suppose also that ϕC[0,x1]{x1}. The trivial equilibrium is globally asymptotically stable if one of the following hypotheses holds:

    (i) g is a nondecreasing function over (0,x1).

    (ii) f is a nondecreasing function over (0,x1).

    Proof. Suppose that () holds. Let V be the Lyapunov functional defined by

    V(s)=s+τ0ψ(a)g(ϕ(a))da, (2.5)

    where ψ(a)=τah(σ)dσ. The derivative of V along the solution of problem (1.1) gives

    dV(xt)dt=g(x(t))f(x(t))t>0.

    From condition (2.2) and Lemma 2.4, we have dV(xt)/dt<0 and thus, the result is reached by classical Lyapunov theorem (see [6]). Next, suppose that () holds. Let the function V be defined in (2.5) by replacing g(s) by g+(s):=maxσ[0,s]g(σ). Since g+(s)<f(s) for all s(0,x1), we can employ the same argument as above to get dV(xt)/dt<0. Finally the result is obtained by applying Theorem 2.1 and Lemma 2.4. This completes the proof.

    Now, suppose that

    f(s)>g(s) for all s>0. (2.6)

    Using the same proof as in Theorem 2.5, we immediately obtain the following theorem.

    Theorem 2.6. Assume that (T1) and condition (2.6) hold. The trivial equilibrium is globally asymptotically stable for all ϕC+.

    The following theorem concerns the global stability of x2 in the case where g is a nondecreasing function.

    Theorem 3.1. Suppose that ϕC[x1,sups[τ,0]ϕ]{x1} and g is a nondecreasing function. Assume also that (T1) and (T4) hold. Then, the positive equilibrium x2 is globally asymptotically stable.

    Proof. Without loss of generality, suppose that ϕ(0)>x1. We first claim that lim inftx(t)>x1. For this, let xϵ(t):=xϵ(t;ϕ) be the solution of problem (1.1) when replacing f by fϵ. To reach the claim, we begin by proving that xϵ(t)>x1 for all t0. Otherwise, there exists t1>0 such that xϵ(t1)=x1, xϵ(t)x1 for all tt1 and xϵ(t1)0. Then, the equation of xϵ(t1) satisfies

    0xϵ(t1)=f(xϵ(t1))+ϵ+τ0h(a)g(xϵ(t1a))da>f(x1)+g(x1)=0.

    This reaches a contradiction and thus xϵ(t)>x1 for all t>0. This result, together with Lemma 2.3, gives x(t)x1 for all t0.

    Next, for X(t)=x(t)x1 and since f(x1)=g(x1), the equation of X(t) satisfies

    X(t)=f(x1)f(x(t))+τ0h(a)(g(x(ta))g(x1))da,

    this leads to

    X(t)LX(t),

    where L is the Lipschitz constant of f. Consequently x(t)>x1 for all t>0.

    Now, suppose that there exist an increasing sequence (tn)n, tn, t0τ and a nonincreasing sequence (x(tn))n, such that x(tn)x1 as tn, x(tn)>x1 for some n and x(tn)=mint[0,tn]x(t). In view of (T4), the equation of x(tn) satisfies

    0x(tn)=f(x(tn))+τ0h(a)g(x(tna))da,f(x(tn))+g(x(tn))>0,

    which is a contradiction. The claim is proved.

    To prove that x2 is globally asymptotically stable, we consider the following Lyapunov functional

    V(ϕ)=ϕ(0)x2[g(s)g(x2)]ds+12τ0ψ(a)[g(ϕ(a))g(x2)]2da,

    with ψ(a)=τah(σ)dσ. By a straightforward computation, the derivative of V along the solution of problem (1.1) gives, for all t>0,

    dV(xt)dt=[g(x(t))g(x2)]x(t)12τ0ψ(a)a[g(x(ta))g(x2)]2da=[g(x(t))g(x2)][f(x(t))+τ0h(a)g(x(ta))da]12{ψ(τ)[g(x(tτ))g(x2)]2ψ(0)[g(x(t))g(x2)]2τ0ψ(a)[g(x(ta))g(x2)]2da}=[g(x(t))g(x2)][f(x(t))+g(x(t))]+[g(x(t))g(x2)][g(x(t))+τ0h(a)g(x(ta))da]+12τ0h(a)da[g(x(t))g(x2)]212τ0h(a)[g(x(ta))g(x2)]2da=[g(x(t))g(x2)][f(x(t))+g(x(t))]+12τ0h(a){2[g(x(t))g(x2)][g(x(t))+g(x(ta))]+[g(x(t))g(x2)]2[g(x(ta))g(x2)]2}da=[g(x(t))g(x2)][f(x(t))+g(x(t))]12τ0h(a)[g(x(t))g(x(ta))]2da.

    Note that ψ(τ)=0. In view of (T4), we have dV(xt)/dt0. 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 exists ˆG(x) such that ˆG(x)=ˆg1og(x), where ˆg(.) denotes the restriction of g to the interval [M,). Then, ˆG(x)=x for x[M,) and ˆG(x)>M>x for x[0,M).

    Lemma 3.2. Assume that (T1)–(T4) hold. For a given ϕC[x1,ˆG(x1)]{x1}, let x be the solution of problem (1.1). Then, the following assertions hold:

    (i) if x2<M, then x1<x(t)<ˆG(x1) for all t>0.

    (ii) if x2M and f(ˆG(x1))>g(M), then x1<x(t)<ˆG(x1) for all t>0.

    Proof. Without loss of generality, suppose that x1<ϕ(0)ˆG(x1). First, observe that, for a given ϕC[x1,ˆG(x1)]{x1}, we have f(ˆG(x1))>f(x1)=g(x1)=g(ˆG(x1)), and thus, there exists ϵ>0 such that f(ˆG(x1))ϵ>g(ˆG(x1)). We begin by proving that x1x(t)ˆG(x1) for all t>0. To this end, in view of Lemma 2.3, we only need to prove that x1<xϵ(t):=xϵ(t;ϕ)ˆG(x1) for all t>0, where xϵ(t):=xϵ(t;ϕ) is the solution of problem (1.1) when replacing f by fϵ. Let yϵ:=yϵ(ϕ) be the solution of

    {yϵ(t)=f(yϵ(t))+ϵ+τ0h(a)g+(yϵ(ta))da,t>0,yϵ(t)=ϕ(t),τt0. (3.1)

    with g+(s)=maxσ[0,s]g(σ). Since f is an increasing function, we have f(ˆG(x1))ϵ>g+(ˆG(x1)). Accordingly, the function ˆG(x1) is a super-solution of problem (3.1). Finally, in view of Theorem 2.1, we obtain xϵ(t)yϵ(t)ˆG(x1) for all t0.

    We now prove that xϵ(t)>x1. Suppose, on the contrary, that there exists t1>0 such that xϵ(t1)=x1, xϵ(t)x1 for all tt1, and thus, xϵ(t1)0. Then, the equation of xϵ(t1) satisfies

    xϵ(t1)=f(xϵ(t1))+ϵ+τ0h(a)g(xϵ(t1a))da>f(x1)+g(x1)=0,

    since g(s)g(x1) for all s[x1,ˆG(x1)]. This reaches a contradiction. Further, from Lemma 2.3, we obtain x(t)x1. The claim is proved.

    Next, let X(t)=x(t)x1. Since f(x1)=g(x1), the equation of X(t) satisfies

    X(t)=f(x1)f(x(t))+τ0h(a)(g(x(ta))g(x1))da.

    We know that g(s)g(x1) for all s[x1,ˆG(x1)]. We then have

    X(t)LX(t),

    where L is the Lipschitz constant of f. Consequently x(t)>x1 for all t>0.

    Next, we prove that x(t)<ˆG(x1) for all t>0. Suppose, on the contrary, that there exists t1>0 such that x(t1)=ˆG(x1), x(t)ˆG(x1) for all tt1 and x(t1)0. Then

    0x(t1)=f(x(t1))+τ0h(a)g(x(t1a))da,f(ˆG(x1))+g(M). (3.2)

    Since f is an increasing function and ˆG(x1)>M, we have

    0x(t1)f(M)+g(M). (3.3)

    Now, the assertion x2<M implies that g(M)<f(M), which leads to a contradiction.

    When x2M, inequality (3.2) gives a contradiction by hypothesis. The Lemma is proved.

    Using Lemma 3.2, we next prove the following lemma.

    Lemma 3.3. Suppose that ϕC[x1,ˆG(x1)]{x1}. Assume also that (T1)–(T4) hold. Let x be the solution of problem (1.1). Then, we have

    lim inftx(t)>x1,

    provided that one of the following assertions holds:

    (i) x2<M.

    (ii) x2M and f(ˆG(x1))>g(M).

    Proof. Firstly, from Lemma 3.2, both assertions imply that x1<x(t)<ˆG(x1) for all t0. Next, observe that there exists ϵ>0 such that

    g(s)>f(x1+ϵ)for alls[x1+ϵ,θϵ],

    with θϵ(M,ˆG(x1)) and g(θϵ)=g(x1+ϵ). Indeed, in view of (T3), minσ[x1+ϵ,θϵ]g(σ)=g(θϵ)=g(x1+ϵ) and from (T4), we have g(x1+ϵ)>f(x1+ϵ).

    Now, consider the following nondecreasing function:

    g_(s)={g(s),for 0<s<ˉm,f(x1+ϵ),for ˉm<s<ˆG(x1), (3.4)

    where x1<ˉm<x1+ϵ, which is a constant satisfying g(ˉm)=f(x1+ϵ). From (T2) and (T4), we get g_(s)>f(s) for x1<s<x1+ϵ and g_(s)<f(s) for s>x1+ϵ. Let y(ϕ) be the solution of problem (1.1) when replacing g by g_. Then, according to Theorem 2.1, we have y(t;ϕ)x(t;ϕ). In addition, using Theorem 3.1, we obtain

    x1<limty(t;ϕ)=x1+ϵlim inftx(t;ϕ).

    The lemma is proved.

    Under Lemma 3.3, we prove the following theorem on the global asymptotic stability of the positive equilibrium x2<M.

    Theorem 3.4. Suppose that ϕC[x1,ˆG(x1)]{x1} and x2<M. Assume also that (T1)–(T4) hold. Then, the positive equilibrium x2 is globally asymptotically stable.

    Proof. We first claim that there exists T>0 such that x(t)M for all tT. Indeed, let xϵ:=xϵ(ϕ) be the solution of problem (1.1) when replacing f by f+ϵ. First, suppose that there exists T>0 such that xϵ(t)M for all tT. So, since x2<M, we have from (T4) that g(M)<f(M). Combining this with (T2), the equation of xϵ satisfies

    xϵ(t)f(M)ϵ+g(M)ϵ,

    which contradicts with xϵ(t)M. Hence there exists T>0 such that xϵ(T)<M. We show that xϵ(t)<M for all tT. In fact, at the contrary, if there exists t1>T such that xϵ(t1)=M and so xϵ(t1)0 then,

    xϵ(t1)=f(M)ϵ+g(M)<0,

    which is a contradiction. Further, according to Lemma 2.3, there exists T>0 such that x(t)M for all tT and the claim is proved. Finally, since g is a nondecreasing function over (0,M), the global asymptotic stability of x2 is proved by applying Theorem 3.1.

    Remark 3.5. In the case where g(s)>g(x1) for all s>x1, the above theorem holds true. In fact, it suffices to replace ˆG(x1) in C[x1,ˆG(x1)]{x1} by sups[τ,0]ϕ(s).

    We focus now on the case where x2M. To this end, suppose that there exists a unique constant A such that

    AM and g(A)=f(M). (3.5)

    Lemma 3.6. Assume that (T1)–(T4) hold. Suppose also that x2M and f(A)g(M). For a given ϕC[x1,ˆG(x1)]{x1} and the solution x:=x(ϕ) of problem (1.1), there exists T>0 such that

    Mx(t)A for all tT.

    Proof. We begin by claiming that x2A. On the contrary, suppose that x2>AM. Then, due to (T2) and (T3), we have g(x2)<g(A)=f(M)<f(x2)=g(x2), which is a contradiction. The claim is proved.

    Next, for a given ϕC[x1,ˆG(x1)]{x1}, let xϵ:=xϵ(ϕ) be the solution of problem (1.1) when replacing f by f+ϵ. Since xϵ converges to x as ϵ tends to zero, we only need to prove that there exists T>0 such that xϵ(t)<A for all tT. To this end, let yϵ:=yϵ(ϕ) be the solution of problem (1.1), when replacing f by f+ϵ and g by g+ with g+(s)=maxσ[0,s]g(σ). By Theorem 2.1, we have xϵ(t)yϵ(t) for all t0, and thus, we only need to show that yϵ(t)<A for all tT. On the contrary, we suppose that yϵ(t)A for all t>0. Then, combining the equation of yϵ and the fact that g+(s)=g(M) for all sM, we obtain

    yϵ(t)g(M)f(A)ϵ<0,

    which is a contradiction. Then, there exists T>0 such that yϵ(T)<A. We further claim that yϵ(t)<A for all tT. Otherwise, there exists t1>T such that yϵ(t1)=A and yϵ(t1)0. Substituting yϵ(t1) in Eq (1.1), we get

    0yϵ(t1)=f(yϵ(t1))ϵ+τ0h(a)g+(yϵ(t1a))da,<f(A)+g(M)0,

    which is a contradiction. Consequently, by passing to the limit in ϵ, we obtain that x(t)A for all tT.

    We focus now on the lower bound of x. First, define the function

    g_(s)={g(s),for 0<s<ˉm,f(M),for ˉm<sA, (3.6)

    where x1<ˉm<M, which is the constant satisfying g(ˉm)=f(M). Note that, due to (T2)–(T4) and condition (3.5), the function g_ is nondecreasing over (0,A) and satisfies

    {g_(s)g(s),0sA,g_(s)>f(s),x1<s<M,g_(s)<f(s),M<sA.

    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 limty(t;ϕ)=M. Consequently, lim inftx(t)M. In addition, if, for all T>0, there exists (tn)n such that tn>T, y(tn)=M, ˉm<y(s)A for all 0<stn and y(tn)<0, then, for tn>T+τ,

    y(tn)=f(M)+f(M)=0,

    which is a contradiction. The lemma is established.

    Denote ˉf=f|[M,A] the restriction function of f over [M,A] and G(s):=ˉf1(g(s)) for s[M,A]. Now, we are ready to state our main theorem related to x2M.

    Theorem 3.7. Under the assumptions of Lemma 3.6, the positive equilibrium x2 of problem (1.1) is globally asymptotically stable, provided that one of the following conditions holds:

    (H1) fg is a nondecreasing function on [M,A].

    (H2) f+g is a nondecreasing function over [M,A].

    (H3) (GoG)(s)s is a nonincreasing function over [M,x2],

    (H4) (GoG)(s)s is a nonincreasing function over [x2,A],

    Proof. Denote x:=lim inftx(t) and x:=lim suptx(t). First, suppose that either xx2 or xx2. For xx2, we introduce the following function:

    g_(s)={minσ[s,x2]g(σ)forx1<s<x2,g(x2)forx2sA.

    Let y(ϕ) be the solution of problem (1.1) when replacing g by g_. Since g_(s)g(s) for all x1sx2, we have y(t)x(t) for all t>0. Further, in view of Theorem 3.1, we obtain

    limty(t)=x2lim suptx(t)=xx2.

    The local stability is obtained by using the same idea as in the proof of Theorem 2.6 in [31].

    Now, for xx2, we introduce the function

    ˉg(s)={minσ[s,x2]g(σ)forx1<sx2,maxσ[x2,s]g(σ)forx2<sA, (3.7)

    and let y(ϕ) be the solution of problem (1.1) when replacing g by ˉg. As above, we have x(t)y(t) for all t>0 and y(t) converges to x2 as t goes to infinity. Next, we suppose that x<x2<x and we prove that it is impossible. Indeed, according to Lemma 3.6, we know that, for every solution x of problem (1.1), we have MxxA.

    Now, using the fluctuation method (see [28,32]), there exist two sequences tn and sn such that

    limnx(tn)=x, x(tn)=0, n1,

    and

    limnx(sn)=x, x(sn)=0, n1.

    Substituting x(tn) in problem (1.1), it follows that

    0=f(x(tn))+τ0h(a)g(x(tna))da. (3.8)

    Since g is nonincreasing over [M,A], we obtain, by passing to the limit in Eq (3.8), that

    f(x)g(x). (3.9)

    Similarly, we obtain

    f(x)g(x). (3.10)

    Multiplying the expression (3.9) by g(x) and combining with inequality (3.10), we get

    f(x)g(x)f(x)g(x).

    This fact, together with the hypothesis (H1), gives xx, 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[M,A], the range of g is contained in [ˉf(M),ˉf(A)] since ˉf is strictly increasing over [M,A]. In fact, for all s[M,A] and since g is non-increasing over [M,A], we have g(A)g(s)g(M). Now, using the fact that f(A)f(M), we show that

    ˉf(M)g(s)ˉf(A)for alls[M,A].

    Therefore, the function G is nonincreasing and maps [M,A] to [M,A]. In view of inequalities (3.9) and (3.10) and the monotonicity of ˉf, we arrive at

    xG(x), (3.11)

    and

    xG(x), (3.12)

    with G(s)=ˉf1(g(s)). Now, applying the function G to inequalities (3.11) and (3.12), we find

    xG(x)(GoG)(x),

    which gives

    (GoG)(x)x1=(GoG)(x2)x2. (3.13)

    Due to (H3), it ensures that x2x, 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 x1 and x2 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 x1. Then

    (i) for ϕC+, there exists T>0 such that 0x(t)M for all tT.

    (ii) ϕC[x1,ˆG(x1)]{x1} implies that x1<x(t)M for all tT.

    (iii) if ϕC[x1,ˆG(x1)], then x1 attracts every solution of problem (1.1) and x1 is unstable.

    Proof. The uniqueness of the positive equilibrium implies that x1M and g(x)<f(x) for all xx1. For (), suppose that there exists t0>0 such that x(t)M for all tt0. Then, by substituting x in Eq (1.1), we get

    x(t)f(M)+g(M)<0.

    This is impossible and then there exists T>0 such that x(T)<M. Next, if there exists t1>T such that x(t1)=M and x(t)M for all tt1, then

    0x(t1)f(M)+g(M)<0.

    Consequently () holds. We argue as in the proof of Lemma 3.2 (ⅰ), to show (). Concerning (), we consider the following Lyapunov functional

    V(ϕ)=ϕ(0)x1(g(s)g(x1))ds+12τ0ψ(a)(g(ϕ(a))g(x1))2da,

    where ψ(a)=τah(σ)dσ. As in the proof of Theorem 3.1, the derivative of V along the solution of problem (1.1) gives

    dV(xt)dt=12τ0h(a)[g(x(t))g(x(ta))]2da+[g(x(t))g(x1)][g(x(t))f(x(t))],t>0.

    Since g(s)<f(s) for all sx1 and g(s)g(x1) for all s[x1,M], we have dV(xt)/dt0. By LaSalle invariance theorem, x1 attracts every solution x(ϕ) of problem (1.1) with ϕC[x1,ˆG(x1)]. From Theorems 2.5 and 3.4, we easily show that x1 is unstable.

    In this section, we apply our results to the following distributed delay differential equation:

    x(t)=μx(t)+τ0h(a)kx2(ta)1+2x3(ta)da, for t0, (4.1)

    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)=kx2/(1+2x3) 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:

    μx=kx21+2x3. (4.2)

    Analyzing Eq (4.2), we obtain the following proposition.

    Proposition 4.1. Equation (4.1) has a trivial equilibrium x=0. In addition

    (i) if μ>2133k, then Eq (4.1) has no positive equilibrium;

    (ii) if μ=2133k, then Eq (4.1) has exactly one positive equilibrium x1;

    (iii) if 0<μ<2133k, then Eq (4.1) has exactly two positive equilibria x1<x2. Moreover,

    (iii)-a if k3<μ<2133k, then 0<x1<x2<1;

    (iii)-b if 0<μk3, then 0<x1<1x2.

    Using Theorem 2.6, we obtain

    Theorem 4.2. If μ>2133k, then the trivial equilibrium of Eq (4.1) is globally asymptotically stable for all ϕC+.

    For the case (ⅲ)-a in Proposition 4.1 we have the following result

    Theorem 4.3. Suppose that k3<μ<2133k. Then

    (i) The trivial equilibrium of Eq (4.1) is globally asymptotically stable for all ϕC[0,x1]{x1}.

    (ii) The positive equilibrium x1 is unstable and the positive equilibrium x2 is globally asymptotically stable for all ϕC[x1,ˆG(x1)]{x1}, where ˆG(x1)[1,) satisfies g(ˆG(x1))=g(x1).

    (iii) There exists two heteroclinic orbits X(1) and X(2) connecting 0 to x1 and x1 to x2, 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={x1}. Clearly, K is an isolated and unstable compact invariant set in C[0,x1] and C[x1,ˆG(x1)].

    By applying Corollary 2.9 in [33] to Φ|R+×C[0,x1] and Φ|R+×C[x1,ˆG(x1)], respectively, there exist two pre-compact full orbits X(1):RC[0,x1]{0} and X(2):RC[x1,ˆG(x1)]{0} such that α(X1)=α(X2)=K. This together with statements (i) and (ii) gives ω(X(1))={0} and ω(X(2))={x2}. In other words, there exist two heteroclinic orbits X(1) and X(2), which connect 0 to x1 and x1 to x2, respectively. This completes the proof.

    For the case (ⅲ)-b in Proposition 4.1, we first show that

    f(A)g(M) with g(A)=f(M) and AM. (4.3)

    Lemma 4.4. Suppose that f(s)=μs. Condition (4.3) holds if and only if M1μg(1μg(M)).

    Proof. It follows that f(A)g(M) if and only if Af1(g(M))M. Since g is a decreasing function over [M,), we have g(A)g(f1(g(M)))=g(1μ(g(M))). Moreover, since g(A)=f(M), we have M1μg(1μg(M)). The lemma is proved.

    Lemma 4.5. For Eq (4.1), condition (4.3) holds if and only if

    0<μk3. (4.4)

    Proof. By a straightforward computation, we have f1(x)=x/μ. For G(x)=g(x)/μ, we obtain

    (GoG)(x)=x(2px6+px3)8x9+(12+2p)x6+6x3+1,

    where p=k3/μ3. By applying Lemma 4.4, it is readily to see that 1(GoG)(1) holds if and only if

    13p2p+27,

    and thus, 0<μk/3.

    Theorem 4.6. Suppose that 0<μk3. Then, the positive equilibrium x2 of Eq (4.1) is globally asymptotically stable for all ϕC[x1,ˆG(x1)]{x1}, where ˆG(x1)[1,) satisfies g(ˆG(x1))=g(x1).

    Proof. Observe that, in view of Lemmas 4.4 and 4.5, all hypotheses of Lemma 3.6 are satisfied. Now, in order to apply Theorem 3.7, we only show that (GoG)(x)x is nonincreasing over [M,A]. Indeed, by a simple computation, we have

    ((GoG)(x)x)=48px1448px116p2x8+12px5+3px2[8x9+(12+2p)x6+6x3+1]2,

    where p=k3/μ3. Finally, observe that

    ((GoG)(x)x)<0,

    for x1. In this case, both of (H3) and (H4) in Theorem 3.7 hold. This completes the proof.

    Finally, for the tangential case, we have

    Theorem 4.7. Suppose that μ=2133k. Then

    (i) The trivial equilibrium of Eq (4.1) is globally asymptotically stable for all ϕC[0,x1]{x1}.

    (ii) If ϕC[x1,ˆG(x1)]{x1}, then the unique positive equilibrium x1 is unstable and attracts every solution of Eq (4.1).

    Proof. The proof of this theorem follows immediately from Theorems 2.5 and 3.9.

    In this section, we perform numerical simulation that supports our theoretical results. As in Section 4, we assume that

    f(x)=μx,g(x)=kx21+2x3,

    and confirm the validity of Theorems 4.2, 4.3, 4.6 and 4.7. In what follows, we fix

    τ=1,k=1,h(a)=e(a0.5τ)2×102τ0e(σ0.5τ)2×102dσ (5.1)

    and change μ and ϕ. Note that 2133k0.42 and h is a Gaussian-like distribution as shown in Figure 1.

    Figure 1.  Function h in definition (5.1).

    First, let μ=0.5. In this case, μ>2133k 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.

    Figure 2.  Time variation of function x for μ=0.5>2133k0.42.

    Next, let μ=0.37. In this case, k3<μ<2133k and x10.43, x20.89 and ˆG(x1)3.09 (Figure 3). Hence, by Theorem 4.3, we see that the trivial equilibrium is globally asymptotically stable for ϕC[0,x1]{x1}, whereas the positive equilibrium x2 is so for ϕC[x1,ˆG(x1)]{x1}. 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.

    Figure 3.  Functions f and g for μ=0.37.
    Figure 4.  Time variation of function x for μ=0.37(k3,2133k)(0.33,0.42).
    Figure 5.  Heteroclinic orbits X(1) and X(2) in the x-x plane for μ=0.37.

    Thirdly, let μ=0.27<k30.33. In this case, we have x10.28, x21.2 and ˆG(x1)6.52 (Figure 6). Thus, by Theorem 4.6, we see that the positive equilibrium x2 is globally asymptotically stable for ϕC[x1,ˆG(x1)]{x1}. In fact, Figure 7 shows that x(t) converges to x2 as t increases for different initial data.

    Figure 6.  Functions f and g for μ=0.27.
    Figure 7.  Time variation of function x for μ=0.27<k30.33.

    Finally, let μ=2133k0.42. In this case, we have x10.63 and ˆG(x1)1.72 (Figure 8). By Theorem 4.7, we see that the trivial equilibrium is globally asymptotically stable for ϕC[0,x1]{x1}, whereas x1 is unstable and attracts all solutions for ϕC[x1,ˆG(x1)]{x1}. In fact, Figure 9 shows such two situations.

    Figure 8.  Functions f and g for μ=2133k0.42.
    Figure 9.  Time variation of function x for μ=2133k0.42.

    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 x2<M and x2M, 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.

    We deeply appreciate the editor and the anonymous reviewers for their helpful comments to the earlier version of our manuscript. The first author is partially supported by the JSPS Grant-in-Aid for Early-Career Scientists (No.19K14594). The second author is partially supported by the DGRSDT, ALGERIA.

    All authors declare no conflicts of interest in this paper.



    [1] E. Braverman, D. Kinzebulatov, Nicholson's blowflies equation with distributed delay, Can. Appl. Math. Quart., 14 (2006), 107-128.
    [2] C. Foley, M. C. Mackey, Dynamics hematological disease: a review, J. Math. Biol., 58 (2009), 285-322. doi: 10.1007/s00285-008-0165-3
    [3] K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Springer, Dordrecht, 1992.
    [4] W. S. C. Gurney, S. P. Blythe, R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0
    [5] C. Huang, Z. Yang, T. Yi, X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Diff. Equat., 256 (2014), 2101-2114. doi: 10.1016/j.jde.2013.12.015
    [6] Y. Kuang, Delay Differential Equations: With Application in Population Dynamics, Academic Press, INC, 1993.
    [7] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956. doi: 10.1182/blood.V51.5.941.941
    [8] E. Liz, A. Ruis-Herrera, Delayed population models with Allee effects and exploitation, Math. Biosci. Eng., 12 (2015), 83-97. doi: 10.3934/mbe.2015.12.83
    [9] E. Liz, M. Pinto, V. Tkachenko, S. Tromichuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70. doi: 10.1090/S0033-569X-05-00951-3
    [10] E. Liz, G. Röst, On the global attractor of delay differential equations with unimodal feedback, Disc. Cont. Dyn. Syst., 24 (2009), 1215-1224. doi: 10.3934/dcds.2009.24.1215
    [11] E. Liz, V. Tkachenko, S. Trofimchuk, Mackey-Glass type delay differential equations near the boundary of absolute stability, J. Math. Anal. Appl., 275 (2002), 747-760. doi: 10.1016/S0022-247X(02)00416-X
    [12] E. Liz, V. Tkachenko, S. Tromichuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222
    [13] G. Röst, J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. A, 463 (2007), 2655-2669. doi: 10.1098/rspa.2007.1890
    [14] Y. Yuan, J. Belair, Stability and Hopf bifurcation analysis for functional differential equation with distributed delay, SIAM J. Appl. Dyn. Syst., 10 (2011), 551-581. doi: 10.1137/100794493
    [15] Y. Yuan, X. Q. Zhao, Global stability for non monotone delay equations (with application to a model of blood cell production), J. Diff. Equat., 252 (2012), 2189-2209. doi: 10.1016/j.jde.2011.08.026
    [16] N. Bessonov, G. Bocharov, T. M. Touaoula, S. Trofimchuk, V. Volpert, Delay reaction-diffusion equation for infection dynamics, Disc. Cont. Dyn. Syst. Series B, 24 (2019), 2073-2091.
    [17] K. Deng, Y. Wu, On the diffusive Nicholso's blowflies equation with distributed delay, Appl. Math. Lett., 50 (2015) 126-132. doi: 10.1016/j.aml.2015.06.013
    [18] W. Hu, Y. Zhou, Stability of a nonlocal delayed reaction-diffusion equation with a non-monotone bistable nonlinearity, Appl. Math. Lett., 79 (2018), 20-26. doi: 10.1016/j.aml.2017.11.018
    [19] R. H. Martin, H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
    [20] R. H. Martin, H. L. Smith, Reaction-diffusion systems with time delay: monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.
    [21] T. M. Touaoula, Global dynamics for a class of reaction-diffusion equations with distributed delay and Neumann condition, Commun. Pur. Appl. Anal., 19 (2020), 2473-2490. doi: 10.3934/cpaa.2020108
    [22] T. M. Touaoula, M. N. Frioui, N. Bessonov, V. Volpert, Dynamics of solutions of a reactiondiffusion equation with delayed inhibition, Disc. Cont. Dyn. Syst. Series S, 13 (2020), 2425-2442.
    [23] E. Braverman, S. Zhukovskiy, Absolute and delay-dependent stability of equations with a distributed delay, Disc. Cont. Dyn. Syst., 32 (2012), 2041-2061. doi: 10.3934/dcds.2012.32.2041
    [24] L. Berezansky, E. Braverman, L. Idels, Nicholson's blowflies differential equations revisited: main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417. doi: 10.1016/j.apm.2009.08.027
    [25] L. Berezansky, E. Braverman, L. Idels, Mackey-Glass model of hematopoiesis with non-monotone feedback: stability, oscillation and control, Appl. Math. Compt., 219 (2013), 6268-6283. doi: 10.1016/j.amc.2012.12.043
    [26] H. A. El-Morshedy, Global attractivity in a population model with nonlinear death rate and distributed delays, J. Math. Anal. Appl., 410 (2014), 642-658. doi: 10.1016/j.jmaa.2013.08.060
    [27] I. Gyori, S. Trofimchuk, Global attractivity in x'(t) = -δx(t) + pf (x(t - h)), Dynam. Syst. Appl., 8 (1999), 197-210.
    [28] H. L. Smith, H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Providence, 2011.
    [29] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, 1995.
    [30] J. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
    [31] T. M. Touaoula, Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Disc. Cont. Dyn. Syst., 38 (2018), 4391-4419. doi: 10.3934/dcds.2018191
    [32] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.
    [33] T. Yi, Q. Li, Unstable sets, heteroclinic orbits and generic quasi-convergence for essentially strongly order-preserving semiflows, Proc. Edinb. Math. Soc., 52 (2009), 797-807. doi: 10.1017/S0013091506000320
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