# New results on global asymptotic stability for a nonlinear density-dependent mortality Nicholson’s blowflies model with multiple pairs of time-varying delays

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## Abstract

This paper is concerned with a class of Nicholson’s blowflies model involving nonlinear density-dependent mortality terms and multiple pairs of time-varying delays. By using differential inequality techniques and the fluctuation lemma, we establish a delay-independent criterion on the global asymptotic stability of the addressed model, which improves and complements some existing ones. The effectiveness of the obtained result is illustrated by some numerical simulations.

## Introduction

Just as pointed out by Berezansky and Braverman , in the study of mathematical biology, many models of population dynamics can be characterized by the following delayed differential equation:

$$x '(t)=\sum^{m}_{j=1}F_{j} \bigl(t, x\bigl(t-\tau _{1}(t)\bigr),\ldots ,x\bigl(t-\tau _{l}(t)\bigr)\bigr)-G\bigl(t, x(t)\bigr),\quad t \geq t_{0},$$
(1.1)

where m and l are positive integers, $$F_{j}$$ and G are nonnegative continuous functions. Here the functions $$F_{j}$$ describe productions incorporating delay, and G corresponds to the instantaneous mortality. Clearly, (1.1) includes the modified Nicholson’s blowflies model with a nonlinear density-dependent mortality term

$$x' (t)= -\frac{a(t)x(t)}{b(t)+x(t)} +\sum _{j=1}^{m}\beta _{j}(t) x \bigl(t-h_{j} (t) \bigr)e^{-\gamma _{j}(t) x(t-g_{j}(t))},\quad t \geq t_{0},$$
(1.2)

which in the case $$h_{j}\equiv g_{j}$$ coincides with the classical models ; $$\frac{a(t)x(t)}{b(t)+x(t)}$$ is the death rate of the population which depends on time t and the current population level $$x (t)$$, $$\beta _{j}(t) x (t-h_{j} (t) )e^{-\gamma _{j}(t) x(t-g_{j}(t))}$$ is the time-dependent birth function which involves maturation delay $$h_{ j}(t)$$ and incubation delay $$g_{ j}(t)$$, and reproduces at its maximum rate $$\frac{1}{\gamma _{ j}(t)}$$; $$a(t)$$, $$b(t)$$, $$\beta _{j}(t)$$, $$g _{j}(t)$$, $$h _{j}(t)$$, and $$\gamma _{j}(t)$$ are all nonnegative, continuous and bounded functions; $$a(t)$$, $$b(t)$$ and $$\gamma _{j}(t)$$ are bounded below by positive constants, and $$j\in I:=\{1,2,\ldots , m \}$$.

For the past decade or so, for the special case of (1.1) with $$h_{j}\equiv g_{j} (j\in I)$$, the existence of positive solutions, permanence, oscillation, periodicity, and stability of such equations and similar models have been studied extensively . In particular, the authors in  illustrated that two or more delays involved in the same nonlinear function $$F_{j}$$ can lead to chaotic oscillations, and they also gave some examples to show that having two delays instead of one can produce sustainable oscillations. In fact, if two or more delays occur, the time delay feedback function $$F_{j}$$ should be considered as a function of several variables. This will add difficulty when studying the dynamics of (1.1) and (1.2). So far, results of global stability analysis for models (1.1) and (1.2) involving two or more delays are very few, we only find that the global stability results of Mackey–Glass equation with two different delays

$$x '(t)=r(t)\biggl[\frac{ax(t-h(t))}{1+x^{\nu }(t-g(t))}-x(t)\biggr],\quad a>1, \nu > 0, t \geq t_{0},$$
(1.3)

are established under the additional technical conditions on the delay terms .

Most recently, Győri et al.  established the permanence in the following two constant-delay differential equation:

$$x '(t) = \alpha (t)H\bigl(x(t-\sigma ), x(t - \tau )\bigr)-\beta (t)f\bigl( x(t)\bigr).$$
(1.4)

On the other hand, El-Morshedy and Ruiz-Herrera  used the classical approach of “decomposing + embedding” to derive some criteria to guarantee the global attraction to a positive equilibrium for the autonomous equation

$$x '(t) = -\beta x(t) + \beta F\bigl(x(t-\sigma ), x(t - \tau ) \bigr),$$
(1.5)

where $$\beta , \sigma , \tau \in (0, +\infty )$$, and $$\sigma \leq \tau$$. However, as in , the above two works shed no light on the global stability on the modified Nicholson’s blowflies model (1.2).

For convenience, given a bounded continuous function g defined on $$\mathbb{R}$$, let $$g^{+}$$ and $$g^{-}$$ be defined as

$$g^{+}=\sup_{t\in [t_{0}, +\infty )}g(t),\qquad g^{-}=\inf _{t\in [t_{0}, +\infty )}g(t).$$

It should be mentioned that some delay-independent criteria ensuring the global asymptotic stability of for the Nicholson’s blowflies model (1.2) with $$h_{j}\equiv g_{j} (j\in I)$$ have been established in . More precisely, the author in  obtained the main result as follows.

### Theorem 1.1

Suppose that

$$\textstyle\begin{cases} \max_{ j \in I}\gamma _{j}^{+} \leq 1 , \\ \sup_{t\in R} \sum_{j=1}^{m} \frac{\beta _{j} (t )}{ \gamma _{j} (t ) } < \frac{a ^{-}}{\max \{1, b^{+} \}}, \\ \varlimsup_{t\rightarrow +\infty }\sum_{j=1}^{m}\frac{ \beta _{j}(t )}{\gamma _{j} (t )} \frac{1}{e}< \frac{a^{-} }{b^{+ }+1}. \end{cases}$$
(1.6)

Then 0 is a globally asymptotically stable equilibrium point on $$C([-\tau , 0], (0, +\infty ))$$, where $$\tau :=\max \{\max_{1\leq j \leq m} g_{j}^{+}, \max_{1\leq j \leq m} h _{j}^{+} \} >0$$.

Unfortunately, there are some mistakes in the proof of main results in . In fact, in lines 3–4 of page 856 in , letting $$t\rightarrow \eta (\varphi )$$ cannot lead to $$\varlimsup_{t\rightarrow +\infty } \sum_{j=1} ^{m} \frac{\beta _{j }(t)}{\gamma _{j}(t)a(t)} \frac{1}{e} \geq 1$$ since $$\eta (\varphi )=+\infty$$ has not been proved. We find that the conclusion of Theorem 1.1 is correct, and the above mistake can be corrected. This has been done in the first half of the proof of Lemma 2.1; please see Sect. 2.

Inspired by the above discussions, in this paper, we consider the nonlinear density-dependent mortality Nicholson’s blowflies model with multiple pairs of time-varying delays described in (1.2). Here, we develop an approach based on differential inequality techniques coupled with an application of the Fluctuation Lemma to establish a delay-independent criterion to ensure the global asymptotic stability of (1.2) in the important, yet difficult case where the two delays are asymptotically apart, i.e., $$h_{j}\not \equiv g_{j}$$ ($$j\in I$$). The obtained results have not been investigated till now. Moreover, the proposed results extend and improve all known ones in , and the error mentioned above has been corrected. In particular, our analysis can also be applied to the nonautonomous Mackey–Glass equation, and our work partially solves an open problem posed for the Mackey–Glass equation in .

## Preliminary results

We first recall some notions. Let $$C= C([-\tau , 0], \mathbb{R})$$ be the Banach space of all continuous functions from $$[-\tau , 0]$$ to $$\mathbb{R}$$ equipped with the supremum norm $$\|\cdot \|$$ and $$C_{+}= C([-\tau , 0], [0, +\infty ))$$. Let $$t_{0}\in \mathbb{R}$$. Then for a continuous function $$x:[t_{0}-\tau ,t_{0}+\sigma )\to \mathbb{R}$$ with $$\sigma >0$$ and $$t\in [t_{0},t_{0}+\sigma )$$, $$x_{t}\in C$$ is defined by $$x_{t}(\theta )=x(t+\theta )$$ for $$\theta \in [-\tau , 0]$$. Denote by $$x_{t}( t_{0}, \varphi )$$ ($$x(t; t _{0}, \varphi )$$) a solution of (1.2) with the initial condition

$$x_{t_{0}}=\varphi ,\quad \varphi \in C_{+}.$$
(2.1)

In addition, let $$[t_{0},\eta (\varphi ))$$ be the maximal right-interval of the existence of $$x_{t}(t_{0}, \varphi )$$.

### Lemma 2.1

Assume that

$$\varlimsup_{t\rightarrow +\infty } \sum_{j=1}^{m} \frac{ \beta _{j }(t)}{\gamma _{j}(t)a(t)} \frac{1}{e} < 1,$$
(2.2)

as well as

$$g_{j}(t)\geq h_{j}(t), \quad \textit{and}\quad \lim _{t\rightarrow +\infty }\bigl( g_{j}(t)-h_{j}(t)\bigr) e^{\int _{t_{0}}^{t}(\sum _{j=1}^{m}\beta _{j}(v))\,dv}=0,$$
(2.3)

where $$t\in [t_{0},+\infty )$$, $$j\in I$$, hold. Then, the solution $$x (t)=x(t; t_{0}, \varphi )\geq 0$$for all $$t\in [t_{0}, \eta (\varphi ))$$, the set of $$\{x_{t}(t_{0},\varphi ): t\in [t_{0}, \eta (\varphi ))\}$$is bounded, and $$\eta (\varphi )=+\infty$$.

### Proof

From Theorem 5.2.1 in , we obtain $$x (t)=x(t; t _{0}, \varphi )\geq 0$$ for all $$t \in [t_{0}, \eta (\varphi ))$$. Now, we show that $$\eta (\varphi )=+\infty$$. For all $$t\in [t_{0}, \eta (\varphi ))$$, defining $$y(t)=\max_{t_{0}-\tau \leq s \leq t} x (s)$$, we gain

$$x' (t)\leq \sum_{j=1}^{m} \beta _{j}(t) x \bigl(t-h_{j} (t) \bigr) \leq \sum _{j=1}^{m}\beta _{j} (t) y (t),$$

and

$$x (t)\leq x (t_{0}) + \int _{t_{0}}^{t} \Biggl[ \sum _{j=1}^{m}\beta _{j} (v) y (v) \Biggr]\,dv \leq \Vert \varphi \Vert + \int _{t_{0}}^{t}\Biggl(\sum _{j=1} ^{m}\beta _{j} (v)\Biggr)y (v) \,dv,$$

which suggests that

$$y (t)\leq \Vert \varphi \Vert + \int _{t_{0}}^{t}\Biggl(\sum _{j=1}^{m}\beta _{j} (v)\Biggr) y (v) \,dv,\quad \forall t\in \bigl[t_{0}, \eta (\varphi )\bigr).$$

Hence, by the Gronwall–Bellman inequality, we obtain

$$x (t)\leq y (t)\leq \Vert \varphi \Vert e^{\int _{t_{0}}^{t}(\sum _{j=1} ^{m}\beta _{j} (v)) \,dv} ,\quad \forall t\in \bigl[t_{0}, \eta (\varphi )\bigr).$$

This follows from Theorem 2.3.1 in  that $$\eta ( \varphi )=+\infty$$, and then

$$x (t)\leq y (t)\leq \Vert \varphi \Vert e^{\int _{t_{0}}^{t}(\sum _{j=1} ^{m}\beta _{j} (v)) \,dv} , \quad \forall t\in [t_{0}, +\infty ).$$
(2.4)

Furthermore, for each $$t\in [t_{0}-\tau , +\infty )$$, we define

$$M(t)=\max \Bigl\{ \xi :\xi \leq t, x (\xi )=\max_{t_{0}-\tau \leq s\leq t} x (s) \Bigr\} .$$

Next, we show that $$x (t)$$ is bounded on $$[t_{0}, \eta (\varphi ))$$. Assume on the contrary that

$$\lim_{t\rightarrow +\infty }x \bigl(M(t)\bigr)=+\infty \quad \mbox{and}\quad \lim _{t\rightarrow +\infty }M(t)=+\infty .$$
(2.5)

Let $$T^{*}\ge t_{0}$$ be such that $$M(t)\ge t_{0}+\tau$$ for $$t\ge T^{*}$$. Note that for $$t\ge t_{0}$$, it follows from (1.2) that

$$x'(s) \le \sum_{j=1}^{m} \beta _{j}(s)x\bigl(M(t)\bigr)= \sum_{j=1} ^{m}\beta _{j}(s)y \bigl(M(t)\bigr),$$
(2.6)

for all $$s\in [t_{0}, t]$$ and $$t\in [t_{0}, + \infty )$$.

This, combined with (1.2), (2.3), (2.4), and the fact that $$\sup_{w\ge 0} we^{-w}=\frac{1}{e}$$, gives us

\begin{aligned} 0 &\leq x'\bigl(M(t)\bigr) \\ &=-\frac{a(M(t))x(M(t))}{b(M(t))+x(M(t))} \\ &\quad {} +\sum_{j=1}^{m}\beta _{j}\bigl(M(t)\bigr)x\bigl(M(t)-g_{j}\bigl(M(t)\bigr) \bigr)e^{-\gamma _{j}(M(t))x(M(t)-g_{j}(M(T)))} \\ &\quad {} +\sum_{j=1}^{m}\beta _{j}\bigl(M(t)\bigr) \int _{M(t)-g_{j} (M(t))}^{M(t)-h _{j} (M(t))}x '(s)\,ds e^{-\gamma _{j}(M(t)) x(M(t)-g_{j}(M(t)))} \\ &\leq a\bigl(M(t)\bigr)\Biggl\{ \Biggl[-\frac{ x (M(t))}{b(M(t))+x (M(t))} \\ &\quad {} +\sum_{j=1}^{m} \frac{\beta _{j }(M(t))}{\gamma _{j}(M(t))a(M(t))} \gamma _{j}\bigl(M(t)\bigr) x \bigl(M(t)-g _{j} \bigl(M(t)\bigr)\bigr)e^{-\gamma _{j}(M(t)) x(M(t)-g_{j}(M(t)))}\Biggr] \\ &\quad {} +\frac{1}{a(M(t))}\sum_{j=1}^{m} \beta _{j }\bigl(M(t)\bigr) \int _{M(t)-g_{j} (M(t))}^{M(t)-h_{j} (M(t))}\sum_{j=1}^{m} \beta _{j}(s)y \bigl(M(t)\bigr)\,ds\Biggr\} \\ &\leq a\bigl(M(t)\bigr)\Biggl\{ \Biggl[-\frac{x(M(t))}{b(M(t))+x(M(t))}+\sum _{j=1}^{m}\frac{ \beta _{j}(M(t))}{\gamma _{j}(M(t))a(M(t))} \frac{1}{e}\Biggr] \\ &\quad {} +\frac{1}{a(M(t))}\sum_{j=1}^{m} \beta _{j}\bigl(M(t)\bigr)\bigl[\bigl(g_{j}\bigl(M(t) \bigr)-h _{j}\bigl(M(t)\bigr)\bigr)e^{\int _{t_{0}}^{M(t)}(\sum _{j=1}^{m}\beta _{j}(v))\,dv}\bigr] \\ &\quad {} \times \Biggl[\Biggl(\sum_{j=1}^{m} \beta _{j}^{+}\Biggr) \Vert \varphi \Vert \Biggr] \Biggr\} \end{aligned}

and

\begin{aligned} 0 &\leq \Biggl[-\frac{x(M(t))}{b(M(t))+x(M(t))}+\sum_{j=1}^{m} \frac{\beta _{j}(M(t))}{\gamma _{j}(M(t))a(M(t))}\frac{1}{e}\Biggr] \\ &\quad {} +\frac{1}{a(M(t))}\sum_{j=1}^{m} \beta _{j}\bigl(M(t)\bigr)\bigl[\bigl(g_{j}\bigl(M(t) \bigr)-h _{j}\bigl(M(t)\bigr)\bigr)e^{\int _{t_{0}}^{M(t)}(\sum _{j=1}^{m}\beta _{j}(v))\,dv}\bigr] \\ &\quad {} \times \Biggl[\Biggl(\sum_{j=1}^{m} \beta _{j}^{+}\Biggr) \Vert \varphi \Vert \Biggr], \end{aligned}
(2.7)

where $$M(t)>2\tau +t_{0}$$.

Letting $$t\rightarrow +\infty$$, due to the facts

$$\lim_{t\rightarrow +\infty }\bigl( g_{j}(t)-h_{j}(t) \bigr) e^{\int _{t _{0}}^{t}(\sum _{j=1}^{m}\beta _{j} (v)) \,dv}=0 \quad \mbox{and}\quad \lim_{t\rightarrow +\infty }M(t)=+\infty ,$$

inequality (2.7) yields

$$0 \leq -1 +\varlimsup_{t\rightarrow +\infty }\sum _{j=1}^{m} \frac{\beta _{j }(t)}{\gamma _{j}(t)a(t)} \frac{1}{e},$$

which contradicts assumption (2.2). This implies that $$x (t)$$ is bounded on $$[t_{0}, +\infty )$$, and ends the proof of Lemma 2.1. □

## Main result

### Theorem 3.1

Assume that (2.3) and

$$\textstyle\begin{cases} \max_{1\leq j \leq m}\varlimsup_{t\rightarrow + \infty }\gamma _{j}(t) \leq 1,\qquad \sup_{t\in [t_{0}, +\infty )} \sum_{j=1}^{m} \frac{\beta _{j }(t)\max \{1, b(t ) \}}{ a(t)} < 1, \\ \varlimsup_{t\rightarrow +\infty } \sum_{j=1}^{m} \frac{\beta _{j} (t )(1+ b (t ))}{ a (t )\gamma _{j} (t ) } \frac{1}{e}< 1,\qquad \varlimsup_{t\rightarrow +\infty } \sum_{j=1}^{m} \frac{ \beta _{j} (t )\max \{1, b (t )\}}{ a (t )\gamma _{j} (t ) } < 1 . \end{cases}$$
(3.1)

are satisfied. Then 0 is a globally asymptotically stable equilibrium point on $$C_{+}$$.

### Proof

Let $$x (t)=x(t; t_{0}, \varphi )$$. From Lemma 2.1, one can see that the set $$\{x_{t}(t_{0}, \varphi ): t\in [t_{0}, +\infty )\}$$ is bounded, and $$0\leq u = \limsup_{t\rightarrow +\infty } x (t ) <+\infty$$.

We now prove that 0 is a stable equilibrium point. Without loss of generality, let $$0<\epsilon <1$$ satisfy

$$\sup_{t\in [t_{0}, +\infty )}\sum_{j=1}^{m} \frac{ \beta _{j }(t)\max \{1, b(t ) \}}{ a(t)} < e^{-\epsilon } .$$
(3.2)

Choosing $$0<\delta <\epsilon$$, we claim that, for $$\|\varphi \|< \delta$$,

$$x(t)=x(t; t_{0}, \varphi )< \epsilon \quad \mbox{for all } t\in [t _{0}, +\infty ).$$
(3.3)

We can pick $$t_{*}\in (t_{0}, +\infty )$$ such that

$$x(t_{*})=\epsilon , x(t)< \epsilon\quad \mbox{for all } t\in [t_{0}-\tau , t_{*}) .$$
(3.4)

Noting that

$$b(t)+x \leq \max \bigl\{ 1, b(t) \bigr\} e^{ x}\quad \mbox{for all } (t, x) \in [t_{0}, +\infty )\times [0, +\infty ) ,$$
(3.5)

(1.2), (3.2) and (3.4) result in

\begin{aligned} 0 &\leq x'(t_{*}) \\ &= -\frac{a(t_{*})x(t_{*})}{b(t_{*})+x(t_{*})} +\sum_{j=1} ^{m} \beta _{j} (t_{*}) x \bigl(t_{*}-h_{j}(t_{*}) \bigr)e^{- \gamma _{j} (t_{*}) x (t_{*}-g_{j}(t_{*}))} \\ &\leq a(t_{*}) \Biggl\{ - \frac{1}{\max \{1, b(t_{*}) \}} x(t_{*})e ^{-x(t_{*})} +\sum_{j=1}^{m} \frac{\beta _{j} (t_{*})}{a(t_{*})} \Biggr\} \\ &= a(t_{*}) \Biggl\{ - \frac{1}{\max \{1, b(t_{*}) \}} \epsilon e^{- \epsilon } +\sum_{j=1}^{m} \frac{\beta _{j} (t_{*})}{a(t_{*}) } \epsilon \Biggr\} \\ &= a(t_{*})\frac{1}{\max \{1, b(t_{*}) \}} \Biggl\{ - e^{- \epsilon } + \sum_{j=1}^{m} \frac{\beta _{j }(t_{*})\max \{1, b(t_{*}) \}}{ a(t _{*})} \Biggr\} \epsilon \\ &< 0, \end{aligned}

which is a contradiction, so that (3.3) holds. Thus, 0 is a stable equilibrium point.

Hereafter, it is sufficient to show that $$u =\limsup_{t\rightarrow +\infty } x (t ) =0$$. By the Fluctuation Lemma [20, Lemma A.1], there exists a sequence $$\{t_{k} \} _{k\geq 1}$$ such that

$$t_{k}\to +\infty ,\quad x (t_{k})\to u, \qquad x '(t_{k})\to 0\quad \mbox{as }k\to +\infty .$$

Moreover, from (3.1) and the boundedness of the coefficients and delay functions in (1.2), without loss of generality, we can assume that

\begin{aligned} &\lim_{k\rightarrow + \infty }a(t_{k})=a^{*}\in \bigl[a^{-},a^{+}\bigr],\qquad \lim_{k\rightarrow + \infty }b(t_{k})=b^{*} \in \bigl[b^{-},b^{+}\bigr], \\ &\lim_{k\rightarrow + \infty }\beta _{j}(t_{k})=\beta _{j}^{*} \in \bigl[\beta _{j}^{-}, \beta _{j}^{+}\bigr],\qquad \lim_{k\rightarrow + \infty }\gamma _{j}(t_{k})=\gamma _{j}^{*} \in \bigl[\gamma _{j}^{-},\gamma _{j}^{+} \bigr],\quad j \in I, \\ &\lim_{k\rightarrow + \infty }g_{j}(t_{k})=g_{j}^{*} \in \bigl[g _{j}^{-},g_{j}^{+} \bigr],\qquad \lim_{k\rightarrow + \infty }h_{j}(t_{k})=h _{j}^{*}\in \bigl[h_{j}^{-},h_{j}^{+} \bigr],\quad j\in I, \\ &\lim_{k \rightarrow + \infty }\gamma _{j}(t_{k}) x \bigl(t_{k}-g _{j} (t_{k})\bigr)= \mu _{j}^{*} \in [ 0, u ], \quad j\in I, \\ &\lim_{k \rightarrow + \infty } \sum_{j=1}^{m} \frac{ \beta _{j }(t_{k})(1+b(t_{k})) }{\gamma _{j}(t_{k})a(t_{k})} \frac{1}{e}=\sum_{j=1}^{m} \frac{\beta ^{*} _{j}( b ^{*}+1 )}{ \gamma ^{*} _{j} a^{*}}\frac{1}{e}\leq \varlimsup_{t\rightarrow +\infty } \sum_{j=1}^{m} \frac{\beta _{j} (t )(1+ b (t ))}{ a (t )\gamma _{j} (t ) } \frac{1}{e}< 1, \end{aligned}

and

$$\lim_{k \rightarrow + \infty } \sum_{j=1}^{m} \frac{\beta _{j}(t _{k})\max \{1, b (t_{k}) \}}{\gamma _{j}(t_{k})a(t_{k})}=\sum_{j=1}^{m} \frac{\beta ^{*} _{j}\max \{1, b ^{*} \}}{a^{*}\gamma ^{*} _{j}} \leq \varlimsup_{t\rightarrow +\infty } \sum _{j=1}^{m} \frac{\beta _{j} (t )\max \{1, b (t )\}}{ a (t ) \gamma _{j} (t ) } < 1.$$

Furthermore, from (1.2), (2.3), (2.4), we get

\begin{aligned} x '(t_{k}) &= -\frac{a(t_{k})x(t_{k})}{b(t_{k})+x(t_{k})} \\ &\quad {} +\sum_{j=1}^{m} \beta _{j }(t_{k}) x \bigl(t_{k}-g_{j} (t _{k})\bigr)e^{-\gamma _{j}(t_{k}) x(t_{k}-g_{j}(t_{k}))} \\ &\quad {} +\sum_{j=1}^{m} \beta _{j }(t_{k}) \int _{t_{k}-g_{j} (t_{k})}^{t_{k}-h_{j} (t_{k})}x '(s)\,ds e^{-\gamma _{j}(t_{k}) x(t_{k}-g_{j}(t_{k}))} \\ &\leq a(t_{k})\Biggl[-\frac{x(t_{k})}{b(t_{k})+x(t_{k})} \\ &\quad {} +\sum_{j=1}^{m} \frac{\beta _{j }(t_{k})}{\gamma _{j}(t _{k})a(t_{k})} \gamma _{j}(t_{k}) x \bigl(t_{k}-g_{j} (t_{k})\bigr)e^{-\gamma _{j}(t _{k}) x(t_{k}-g_{j}(t_{k}))}\Biggr] \\ &\quad {} +\sum_{j=1}^{m} \beta _{j }(t_{k}) \int _{t_{k}-g_{j} (t_{k})}^{t_{k}-h_{j} (t_{k})}\sum_{j=1}^{m} \beta _{j}(s)y (t_{k})\,ds \\ &\leq a(t_{k})\Biggl[-\frac{x(t_{k})}{b(t_{k})+x(t_{k})} \\ &\quad {} +\sum_{j=1}^{m} \frac{\beta _{j }(t_{k})}{\gamma _{j}(t _{k})a(t_{k})} \gamma _{j}(t_{k}) x \bigl(t_{k}-g_{j} (t_{k})\bigr)e^{-\gamma _{j}(t_{k}) x(t_{k}-g_{j}(t_{k}))}\Biggr] \\ &\quad {}+\sum_{j=1}^{m} \beta _{j }(t_{k}) \bigl[\bigl(g_{j}(t_{k})-h_{j}(t_{k}) \bigr) e^{\int _{t_{0}}^{t_{k}}(\sum _{j=1}^{m}\beta _{j}(v))\,dv}\bigr] \Biggl[\Biggl( \sum_{j=1}^{m} \beta _{j} ^{+} \Biggr) \Vert \varphi \Vert \Biggr], \end{aligned}

and

\begin{aligned} \frac{1}{a(t_{k})}x '(t_{k}) \leq{} &\Biggl[- \frac{x(t_{k})}{b(t_{k})+x(t_{k})} \\ &{}+\sum_{j=1}^{m} \frac{\beta _{j }(t_{k}) }{\gamma _{j}(t_{k})a(t _{k})} \gamma _{j}(t_{k}) x \bigl(t_{k}-g_{j} (t_{k})\bigr)e^{-\gamma _{j}(t_{k}) x(t_{k}-g_{j}(t_{k}))}\Biggr] \\ &{}+\frac{1}{a(t_{k})}\sum_{j=1}^{m} \beta _{j }(t_{k}) \bigl[\bigl(g_{j}(t _{k})-h_{j}(t_{k})\bigr) e^{\int _{t_{0}}^{t_{k}}(\sum _{j=1}^{m}\beta _{j}(v))\,dv}\bigr] \\ &{} \times \Biggl[\Biggl(\sum_{j=1}^{m} \beta _{j} ^{+} \Biggr) \Vert \varphi \Vert \Biggr], \end{aligned}
(3.6)

where $$t_{k}>2\tau +t_{0}$$.

If $$u\geq 1$$, from (2.3), (3.1), (3.6) and the facts that $$\frac{u}{b ^{*}+u}\geq \frac{1}{b ^{*}+1}$$ and $$\sup_{u\geq 0} ue^{- u}=\frac{1}{ e}$$, letting $$k\rightarrow +\infty$$ leads to

$$0 \leq -\frac{1}{b ^{*}+1} +\sum_{j=1}^{m} \frac{\beta ^{*} _{j}}{\gamma ^{*} _{j} a^{*}} \frac{1}{e} = \frac{1}{b ^{*}+1} \Biggl\{ -1 + \sum_{j=1}^{m}\frac{\beta ^{*} _{j}( b ^{*}+1 )}{\gamma ^{*} _{j} a^{*}} \frac{1}{e} \Biggr\} < 0,$$

which a contradiction, so that $$0\leq u<1$$.

If $$0< u< 1$$, from (2.3), (3.1), (3.5), (3.6), and the fact that $$xe^{-x}$$ is monotone increasing on $$[0,1]$$, we have

\begin{aligned} & \frac{1}{a(t_{k})}x '(t_{k}) \\ &\quad \leq \Biggl[- \frac{1}{\max \{1, b(t_{k} ) \}}x(t_{k})e^{-x(t_{k})} \\ &\qquad {} +\sum_{j=1}^{m} \frac{\beta _{j }(t_{k})}{\gamma _{j}(t_{k})a(t _{k})} \gamma _{j}(t_{k}) x \bigl(t_{k}-g_{j} (t_{k})\bigr)e^{-\gamma _{j}(t_{k}) x(t_{k}-g_{j}(t_{k}))}\Biggr] \\ &\qquad {} +\frac{1}{a(t_{k})}\sum_{j=1}^{m} \beta _{j }(t_{k}) \bigl[ \bigl(g_{j}(t _{k})-h_{j}(t_{k})\bigr) e^{\int _{t_{0}}^{t_{k}}(\sum _{j=1}^{m}\beta _{j}(v))\,dv}\bigr] \Biggl[ \Biggl(\sum_{j=1}^{m} \beta _{j} ^{+} \Biggr) \Vert \varphi \Vert \Biggr] \\ &\quad = \frac{1}{\max \{1, b(t_{k} ) \}} [- x(t_{k})e^{-x(t_{k})} \\ &\qquad {} +\frac{1}{a(t_{k})}\sum_{j=1}^{m} \beta _{j }(t_{k}) \bigl[ \bigl(g_{j}(t _{k})-h_{j}(t_{k})\bigr) e^{\int _{t_{0}}^{t_{k}}(\sum _{j=1}^{m}\beta _{j}(v))\,dv}\bigr] \Biggl[ \Biggl(\sum_{j=1}^{m} \beta _{j} ^{+} \Biggr) \Vert \varphi \Vert \Biggr] , \end{aligned}

where $$t_{k}>2\tau +t_{0}$$, and

\begin{aligned} 0 &\leq \frac{1}{\max \{1, b ^{*} \}} \Biggl[ -ue^{-u} +\sum _{j=1} ^{m}\frac{\beta ^{*}_{j}\max \{1,b ^{*}\}}{a^{*}\gamma ^{*} _{j}} \mu ^{*}_{j} e^{-\mu ^{*}_{ j}} \Biggr] \\ & \leq \frac{1}{\max \{1, b ^{*} \}} \Biggl[ -u e^{-u} +\sum _{j=1} ^{m}\frac{\beta ^{*} _{j}\max \{1, b ^{*} \}}{a^{*}\gamma ^{*} _{j}} ue ^{-u} \Biggr] \\ &= \frac{1}{\max \{1, b ^{*} \}} \Biggl[ -1 +\sum_{j=1}^{m} \frac{ \beta ^{*} _{j}\max \{1, b ^{*} \}}{a^{*}\gamma ^{*} _{j}} \Biggr]ue^{-u} \\ &\leq 0, \end{aligned}

which is also a contradiction, proving that $$u=0$$. This completes the proof of Theorem 3.1. □

By applying Theorem 3.1, we can obtain the following result.

### Corollary 3.1

Suppose that (3.1) holds, and $$g_{j}(t)\equiv h_{j}(t)$$, for all $$t\in [t_{0}, +\infty )$$, $$j\in I$$. Then 0 is a globally asymptotically stable equilibrium point on $$C_{+}$$.

### Remark 3.1

It is obvious that all results in Theorem 1.1 are special cases of Corollary 3.1 since the assumptions adopted are weaker than those ones in Theorem 1.1.

## A numerical example

This section presents an example with graphical illustration to show the applicability of the analytical results derived in this article.

### Example 4.1

Consider the following nonautonomous Nicholson’s blowflies equation with two pairs of different time-varying delays:

\begin{aligned} x' (t) = {}&{-}\frac{(3 + \vert \cos \sqrt{2} t \vert ) x (t)}{(5.1 + \frac{t^{4}}{t ^{4}+1} ) +x (t)} \\ &{}+\frac{1}{200}\bigl(1 + \sin ^{2} t \bigr) x \bigl(t-2e^{ \vert \arctan t \vert }\bigr) \\ &{} \times e^{-\frac{1+23e^{-t}}{12}(1+ \vert \arctan t \vert )x (t-2e^{ \vert \arctan t \vert }-100e^{-2t})} \\ &{}+\frac{1}{200}\bigl(1 + \sin ^{2} 2t \bigr) x \bigl(t-2e^{ \vert \arctan t \vert }\bigr) \\ &{} \times e^{-\frac{1+25e^{-t}}{13}(1+ \vert \arctan t \vert )x (t-2e^{ \vert \arctan t \vert }-150e^{-2t})}, \quad t\geq t_{0}=0. \end{aligned}
(4.1)

Obviously, it is easy to check that assumptions (2.3) and (3.1) are satisfied in (4.1). Hence, from Theorem 3.1, we have that the zero equilibrium point for the model (3.1) is globally asymptotically stable on $$C_{+} = C([-(2e^{ \frac{\pi }{2}}+150), 0], [0, +\infty ) )$$. Figure 1 supports this result with the numerical solutions involving different initial values.

### Remark 4.1

It should be mentioned that the global asymptotic stability on the Nicholson’s blowflies model involving nonlinear density-dependent mortality terms and multiple pairs of time-varying delays has not been touched in the previous literature. As for [117, 2151], the authors still give no clues on the global asymptotic stability of the Nicholson’s blowflies model involving multiple pairs of time-varying delays. One can see that all the results in the above mentioned references cannot be applied to prove that all the solutions of model (4.1) converge to the zero equilibrium.

## Conclusions

In this article, the global asymptotic stability of the zero equilibrium for a Nicholson’s blowflies model involving nonlinear density-dependent mortality terms and multiple pairs of time-varying delays is established. To the best of our knowledge, this is the first paper to study the global dynamics for the nonlinear density-dependent mortality Nicholson’s blowflies model involving multiple pairs of time-varying delays and the obtained results are new. Some sufficient conditions set up here are easily verified and these conditions are independent of the multiple pairs of time-varying delays in the addressed model, which improves and complements some existing results mentioned in the Introduction. Moreover, the method used in this paper provides a possible approach for studying the global asymptotic stability of other population dynamic models involving multiple pairs of time-varying delays.

## References

1. 1.

Berezansky, L., Braverman, E.: A note on stability of Mackey–Glass equations with two delays. J. Math. Anal. Appl. 450, 1208–1228 (2017)

2. 2.

Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)

3. 3.

Yao, L.: Global attractivity of a delayed Nicholson-type system involving nonlinear density-dependent mortality terms. Math. Methods Appl. Sci. 41, 2379–2391 (2018)

4. 4.

Wang, W.: Positive periodic solutions of delayed Nicholson’s blowflies models with a nonlinear density-dependent mortality term. Appl. Math. Model. 36, 4708–4713 (2012)

5. 5.

Liu, B., Gong, S.: Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms. Nonlinear Anal., Real World Appl. 12, 1931–1937 (2011)

6. 6.

Liu, S., Chen, Y., Huang, Y., Zhou, J.: An efficient two grid method for miscible displacement problem approximated by mixed finite element methods. Comput. Math. Appl. 77(3), 752–764 (2019)

7. 7.

Huang, C., Zhang, H., Cao, J., Hu, H.: Stability and Hopf bifurcation of a delayed prey–predator model with disease in the predator. Int. J. Bifurc. Chaos 29(7), 1950091 (2019)

8. 8.

Huang, C., Zhang, H., Huang, L.: Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun. Pure Appl. Anal. 18(6), 3337–3349 (2019)

9. 9.

Liu, P., Zhang, L., Liu, S., et al.: Global exponential stability of almost periodic solutions for Nicholson’s blowflies system with nonlinear density-dependent mortality terms and patch structure. Math. Model. Anal. 22(4), 484–502 (2017)

10. 10.

Wang, F., Huang, C., Huang, L.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019)

11. 11.

Berezansky, L., Braverman, E.: Mackey–Glass equation with variable coefficients. Comput. Math. Appl. 51, 1–16 (2006)

12. 12.

Berezansky, L., Braverman, E., Idels, L.: The Mackey–Glass model of respiratory dynamics: review and new results. Nonlinear Anal. 75, 6034–6052 (2012)

13. 13.

Berezansky, L., Braverman, E., Idels, L.: Mackey–Glass model of hematopoiesis with monotone feedback revisited. Appl. Math. Comput. 219, 4892–4907 (2013)

14. 14.

Berezansky, L., Braverman, E., Idels, L.: Mackey–Glass model of hematopoiesis with non-monotone feedback: stability, oscillation and control. Appl. Math. Comput. 219, 6268–6283 (2013)

15. 15.

Győri, I., Hartung, F., Mohamady, N.A.: Permanence in a class of delay differential equations with mixed monotonicity. Electron. J. Qual. Theory Differ. Equ. 2018, 53 (2018)

16. 16.

El-Morshedy, H.A., Ruiz-Herrera, A.: Global convergence to equilibria in non-monotone delay differential equations. Proc. Am. Math. Soc. 147, 2095–2105 (2019)

17. 17.

Tang, Y.: Global asymptotic stability for Nicholson’s blowflies model with a nonlinear density-dependent mortality term. Appl. Math. Comput. 250, 854–859 (2015)

18. 18.

Smith, H.L.: Monotone Dynamical Systems. Math. Surveys Monogr. Am. Math. Soc., Providence (1995)

19. 19.

Hale, J., Verduyn Lunel, S.: Introduction to functional differential equations. In: Applied Mathematical Sciences, vol. 99. Springer, New York (1993)

20. 20.

Smith, H.L.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York (2011)

21. 21.

Wang, J., Chen, X., Huang, L.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019)

22. 22.

Hu, H., Yuan, X., Huang, L., Huang, C.: Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks. Math. Biosci. Eng. 16(5), 5729–5749 (2019)

23. 23.

Rajchakit, G., Pratap, A., Raja, R., Cao, J., Alzabut, J., Huang, C.: Hybrid control scheme for projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks with mixed delays. Mathematics 7(8), 759 (2019). https://doi.org/10.3390/math7080759

24. 24.

Song, C., Fei, S., Cao, J., Huang, C.: Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control. Mathematics 7(7), 599 (2019). https://doi.org/10.3390/math7070599

25. 25.

Yang, X., Wen, S., Liu, Z., Li, C., Huang, C.: Dynamic properties of foreign exchange complex network. Mathematics 7(9), 832 (2019). https://doi.org/10.3390/math7090832

26. 26.

Kumari, S., Chugh, R., Cao, J., Huang, C.: Multi fractals of generalized multivalued iterated function systems in b-metric spaces with applications. Mathematics 7(10), 967 (2019). https://doi.org/10.3390/math7100967

27. 27.

Li, W., Huang, L., Ji, J.: Periodic solution and its stability of a delayed Beddington–DeAngelis type predator–prey system with discontinuous control strategy. Math. Methods Appl. Sci. 42(13), 4498–4515 (2019)

28. 28.

Hu, H., Yi, T., Zou, X.: On spatial-temporal dynamics of Fisher–KPP equation with a shifting environment. Proc. Am. Math. Soc. 148(1), 213–221 (2020)

29. 29.

Zhang, J., Lu, C., Li, X., Kim, H.-J., Wang, J.: A full convolutional network based on DenseNet for remote sensing scene classification. Math. Biosci. Eng. 16(5), 1–13 (2019)

30. 30.

Huang, Y., Chen, X., Zhu, H., Huang, C., Tian, Z.: The heterogeneous effects of FDI and foreign trade on CO2 emissions: evidence from China. Math. Probl. Eng. 2019, Article ID 9612492 (2019). https://doi.org/10.1155/2019/9612492

31. 31.

Li, X., Liu, Z., Li, J., Tisdell, C.: Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces. Acta Math. Sci. 39(1), 229–242 (2019)

32. 32.

Tan, Y., Huang, C., Sun, B., Wang, T.: Dynamics of a class of delayed reaction–diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018)

33. 33.

Huang, C., Qiao, Y., Huang, L., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, 186 (2018). https://doi.org/10.1186/s13662-018-1589-8

34. 34.

Duan, L., Fang, X., Huang, C.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(5), 1954–1965 (2018)

35. 35.

Chen, T., Huang, L., Yu, P., Huang, W.: Bifurcation of limit cycles at infinity in piecewise polynomial systems. Nonlinear Anal., Real World Appl. 41, 82–106 (2018)

36. 36.

Cai, Z., Huang, J., Huang, L.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146(11), 4667–4682 (2018)

37. 37.

Yang, C., Huang, L., Li, F.: Exponential synchronization control of discontinuous nonautonomous networks and autonomous coupled networks. Complexity 2018, Article ID 6164786 (2018)

38. 38.

Liu, J., Yan, L., Xu, F., Lai, M.: Homoclinic solutions for Hamiltonian system with impulsive effects. Adv. Differ. Equ. 2018, 326 (2018)

39. 39.

Hu, H., Zou, X.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(11), 4763–4771 (2017)

40. 40.

Duan, L., Huang, C.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017)

41. 41.

Zhang, H. : Global exponential stability of periodic solutions in a nonsmooth model of hematopoiesis with time-varying delays. Discrete Contin. Dyn. Syst. 39(11), 6669–6682 (2017)

42. 42.

Huang, C., Cao, J., Wen, F., Yang, X.: Stability analysis of SIR model with distributed delay on complex networks. PLoS ONE 11(8), e0158813 (2016)

43. 43.

Huang, C., Yang, Z., Yi, T., Zou, X.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014)

44. 44.

Huang, C., Yang, X., Cao, J.: Stability analysis of Nicholson’s blowflies equation with two different delays. Math. Comput. Simul. (2019). https://doi.org/10.1016/j.matcom.2019.09.023

45. 45.

Cao, Q., Wang, G., Qian, C.: New results on global exponential stability for a periodic Nicholson’s blowflies model involving time-varying delays. Adv. Differ. Equ. (2019). https://doi.org/10.1186/s13662-020-2495-4

46. 46.

Long, X., Gong, S.: New results on stability of Nicholson’s blowflies equation with multiple pairs of time-varying delays. Appl. Math. Lett. 100, 106027 (2020). https://doi.org/10.1016/j.aml.2019.106027

47. 47.

Huang, C., Long, X., Huang, L., Fu, S.: Stability of almost periodic Nicholson’s blowflies model involving patch structure and mortality terms. Can. Math. Bull. (2019). https://doi.org/10.4153/S0008439519000511

48. 48.

Li, J., Du, C.: Existence of positive periodic solutions for a generalized Nicholson’s blowflies model. J. Comput. Appl. Math. 221, 226–233 (2008)

49. 49.

Wang, F., Yao, Z.: Approximate controllability of fractional neutral differential systems with bounded delay. Fixed Point Theory 17, 495–507 (2016)

50. 50.

Feng, Q., Yan, J.: Global attractivity and oscillation in a kind of Nicholson’s blowflies. J. Biomath. 17, 21–26 (2002)

51. 51.

Qian, C., Hu, Y.: Novel stability criteria on nonlinear density-dependent mortality Nicholson’s blowflies systems in asymptotically almost periodic environments. J. Inequal. Appl. (2020). https://doi.org/10.1186/s13660-019-2275-4

### Acknowledgements

We would like to thank the anonymous referees and the editor for very helpful suggestions and comments which led to improvements of our original paper.

### Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

### Funding

This work is supported by the Foundation of Hunan Provincial Education Department of China(Grant No. 14C0806), the Natural Scientific Research Fund of Hunan Provincial of China (Grant No. 2016JJ6104), and the Natural Scientific Research Fund of Zhejiang Province of China under Grant No. LY18A010019.

## Author information

The four authors contributed equally to this work. All authors read and approved the final manuscript.

Correspondence to Shuhua Gong.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests. 