1 Introduction

To describe the periodic oscillation in Nicholson’s classic experiments [1] with the Australian sheep blowfly, Lucilia cuprina, Gurney et al. [2] put up the following Nicholson’s blowflies model:

$$ \dot{w}(t)=-dw(t)+b w(t-\vartheta)e^{-\gamma x(t-\vartheta)}, $$
(1.1)

where b is the maximum per capita daily egg production rate, \(\frac{1}{\gamma}\) is the size at which the blowfly population reproduces at its maximum rate, d is the per capita daily adult death rate, and ϑ is the generation time. Due to the immense application of Nicholson’s blowflies model in biology, model (1.1) and its modifications have been extensively discussed by lots of authors (see, e.g., [3,4,5,6] and the references therein). Noticing the periodic change of real environment, many scholars [7,8,9] generalized model (1.1) into the following Nicholson’s blowflies model:

$$ \dot{w}(t)=-d(t)w(t)+\sum_{j=1}^{m}b_{j}(t) w\bigl(t-\vartheta_{j}(t)\bigr)e^{-\gamma_{j}(t) w(t-\vartheta_{j}(t))}, $$
(1.2)

where m is a positive integer, \(d: R\rightarrow R\) and \(b_{j}, \gamma_{j}, \vartheta_{j}: R\rightarrow[0,+\infty)\), \(j=1,2,\ldots,m\), are bounded continuous functions, and \(w(t)\) is the size of the population at time t. Noting that the exponential convergent rate can be unveiled [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29], Long [30] investigated the exponential convergence of model (1.2).

Some researchers think that time delays appearing in many biological models are proportional; in other words, the proportional delay function takes the form \(\vartheta(t)=t-at\) (\(0< a<1\) is a constant). In objective world, proportional delay plays a key role in numerous areas such as web quality, current collection [31], biological systems and many nonlinear models [32, 33], electrodynamics [34], and probability principle [35]. So it is valuable to study the global exponential convergence of Nicholson’s blowflies model with proportional delays. But so far there are no manuscripts about the global exponential convergence of Nicholson’s blowflies model with proportional delays.

Stimulated by the above analysis, it is important for us to analyze the global exponential convergence on Nicholson’s blowflies model with proportional delays. In this paper, we focus on the following Nicholson’s blowflies model with proportional delays:

$$ \dot{w}(t)=-d(t)w(t)+\sum_{j=1}^{m}b_{j}(t) w(a_{j}t)e^{-\gamma_{j}(t) w(a_{j}t)}, $$
(1.3)

where m is a positive integer, \(d: R\rightarrow R\) and \(b_{j}, \gamma_{j}, \vartheta_{j}: R\rightarrow[0,+\infty)\), \(j=1,2,\ldots,m\), are bounded continuous functions, \(a_{j}\) is the proportional delay factor such that \(0< a_{j}<1\), \(a_{j}t=t-(1-a_{j})t\), and \((1-a_{j})t\rightarrow+\infty\) as \(t\rightarrow+\infty\).

The initial condition of model (1.3) takes the form

$$ w(s)=\psi(s),\quad s\in[a_{0} t_{0}, t_{0}], t_{0}>0, $$
(1.4)

where \(a_{0}=\min_{i=1,2,\ldots,m}\{a_{i}\}\), and ψ is a real-valued continuous function on \([a_{0} t_{0}, t_{0}]\).

For convenience, we denote \(l^{+}=\sup_{t\in[t_{0},+\infty)}|l(t)|\) and \(l^{-}=\inf_{t\in [t_{0},+\infty)}|l(t)|\) for a bounded continuous function l on \([t_{0},+\infty)\).

Throughout this paper, we also make the following assumptions:

  1. (K1)

    There exist a bounded continuous function: \(d^{*}:[t_{0},+\infty)\rightarrow(0,+\infty)\) and a positive constant μ such that \(e^{-\int_{s}^{t}d(\theta)\,d\theta}\leq\mu e^{-\int_{s}^{t}d^{*}(\theta)\,d\theta}\) for all \(t,s\in R\) and \(t-s\geq0\).

  2. (K2)

    \(\sup_{t\geq t_{0}} \{-d^{*}(t)+\mu\sum_{j=1}^{m}|b_{j}(t)| \}<0\).

  3. (K3)

    \(ma>1\), where \(a=\max_{1\leq i\leq m}\{a_{i}\}\).

The key task of this paper is finding a sufficient condition that ensures the global exponential convergence of all solutions of (1.3). The key contributions of this paper are the following: (i) For the first time, the new Nicholson’s blowflies model with proportional delays is presented; (ii) A new sufficient condition that guarantees the global exponential convergence of Nicholson’s blowflies model with proportional delays is established; (iii) Until now, the global exponential convergence for Nicholson’s blowflies model with proportional delays has not been studied.

2 Main findings

Now we will discuss the global exponential convergence of model (1.3)

Lemma 2.1

Let \(d^{*-}>0\) and \(\sigma\ge0\) be constants such that

$$\int_{s}^{t}\bigl(d^{*}(v)-d(v)\bigr)\,dv\leq \sigma\quad \textit{for all } t,s\in R \textit{ and } t-s\geq0. $$

Then for any \(t_{0}\in R\), the solution \(w(t;t_{0},\psi)\) of system (1.3) with the initial value (1.4) satisfies \(w(t;t_{0},\psi)>0\) for all \(t\in[t_{0}, \eta(\psi))\) and \(\eta(\psi)=+\infty\), where \([t_{0}, \eta(\psi))\) is the maximal right interval of the existence of a solution \(w(t;t_{0},\psi)\).

In view of the proof of Lemma 2.1 in Long [30], we can easily prove Lemma 2.1.

Theorem 2.1

For system (1.3), under the assumptions of Lemma 2.1, if (K1)–(K3) hold, then there exists a constant \(\xi>0\) such that \(w(t)=O(e^{-\xi t})\) as \(t\rightarrow+\infty\).

Proof

Assume that \(w(t)\) is an arbitrary solution of model (1.3). By (1.3) we have

$$ \dot{w}(t)+d(t)w(t)=\sum_{j=1}^{m}b_{j}(t) w(a_{j}t)e^{-\gamma_{j}(t) w(a_{j}t)}. $$
(2.1)

Define the continuous function

$$ \varPhi(\omega)=\sup_{t\geq t_{0}} \Biggl\{ \omega-d^{*}(t)+\mu \Biggl[\sum_{j=1}^{m} \bigl\vert b_{j}(t) \bigr\vert +\omega \Biggr] \Biggr\} ,\quad \omega\in[0, +\infty). $$
(2.2)

It follows from (K2) that

$$ \varPhi(0)=\sup_{t\geq t_{0}} \Biggl\{ -d^{*}(t)+\mu \Biggl[\sum_{j=1}^{m} \bigl\vert b_{j}(t) \bigr\vert \Biggr] \Biggr\} < 0. $$
(2.3)

In view of the continuity of \(\varPhi(\omega)\), we can choose a constant \(\xi\in(0, \inf_{t\geq t_{0}}d^{*}(t))\) such that

$$ \varPhi(\xi)=\sup_{t\geq t_{0}} \Biggl\{ \xi-d^{*}(t)+\mu \Biggl[\sum_{j=1}^{m} \bigl\vert b_{j}(t) \bigr\vert +\xi \Biggr] \Biggr\} < 0. $$
(2.4)

Let

$$ \Vert \psi \Vert _{\varrho}=\max_{t\in[a_{0} t_{0},t_{0}]} \bigl\vert \psi(t) \bigr\vert . $$
(2.5)

For all \(\epsilon>0\), we get

$$ \bigl\vert w(t) \bigr\vert < \bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t-t_{0})}< P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t-t_{0})} $$
(2.6)

for \(t\in[a_{0} t_{0},t_{0}]\), where \(P>\mu+1\). We will further prove that

$$ \bigl\vert w(t) \bigr\vert < P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t-t_{0})} $$
(2.7)

fir \(t\geq t_{0}\). Otherwise, there exists \(t^{*}>t_{0}\) such that

$$ \bigl\vert w\bigl(t^{*}\bigr) \bigr\vert =P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t^{*}-t_{0})} $$
(2.8)

and

$$ \bigl\vert w(t) \bigr\vert < P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t-t_{0})} $$
(2.9)

for \(t\in[a_{0} t_{0},t^{*}]\). Note that

$$ \dot{w}(s)+d(t)w(s)=\sum_{j=1}^{m}b_{j}(s) w(a_{j}s)e^{-\gamma_{j}(s) w(a_{j}s)} $$
(2.10)

for \(s\in[t_{0},t]\) and \(t\in[t_{0},t^{*}]\). By (2.10) we get

$$ {w}(t)=w(t_{0})e^{-\int_{t_{0}}^{t}d(v)\,dv}+ \int_{t_{0}}^{t}e^{-\int _{s}^{t}d(v)\,dv} \Biggl[\sum _{j=1}^{m}b_{j}(s) w(a_{j}s)e^{-\gamma_{j}(s) w(a_{j}s)} \Biggr]\,ds, t\in\bigl[t_{0},t^{*} \bigr]. $$
(2.11)

By (2.4) and (K3) we have

$$\begin{aligned} \bigl\vert {w}(t) \bigr\vert =& \Biggl\vert w(t_{0})e^{-\int_{t_{0}}^{t}d(v)\,dv}+ \int_{t_{0}}^{t}e^{-\int _{s}^{t}d(v)\,dv} \Biggl[\sum _{j=1}^{m}b_{j}(s) w(a_{j}s)e^{-\gamma_{j}(s) w(a_{j}s)} \Biggr]\,ds \Biggr\vert \\ \leq&\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)\mu e^{-\int_{t_{0}}^{t^{*}} d^{*}(v)\,dv}+ \int_{t_{0}}^{t^{*}} e^{-\int_{s}^{t^{*}} d^{*}(v)\,dv} \mu \Biggl[\sum _{j=1}^{m} \bigl\vert b_{j}(s) \bigr\vert \bigl\vert w(a_{j}s) \bigr\vert \Biggr]\,ds \\ \leq&\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)\mu e^{-\int_{t_{0}}^{t^{*}} d^{*}(v)\,dv} \\ &{}+ \int_{t_{0}}^{t^{*}} e^{-\int_{s}^{t^{*}} d^{*}(v)\,dv} \mu \Biggl[\sum _{j=1}^{m} \bigl\vert b_{j}(s) \bigr\vert P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(a_{j}s-t_{0})} \Biggr]\,ds \\ \leq&\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)\mu e^{-\int_{t_{0}}^{t^{*}} d^{*}(v)\,dv}+ \int_{t_{0}}^{t^{*}} e^{-\int_{s}^{t^{*}} d^{*}(v)\,dv} e^{-\xi(s-t_{0})}\bigl(d^{*}(s)-\xi\bigr) \,ds \\ &{}\times e^{-\xi(t^{*}-t_{0})}P\bigl( \Vert \psi \Vert _{\varrho}+ \epsilon\bigr) \\ =&\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t^{*}-t_{0})} \mu e^{-\int_{t_{0}}^{t^{*}} (d^{*}(v)-\xi)\,dv}+ \int_{t_{0}}^{t^{*}} e^{-\int_{s}^{t^{*}} (d^{*}(v)-\xi)\,dv} \bigl(d^{*}(s)- \xi\bigr)\,ds \\ & \times e^{-\xi(t^{*}-t_{0})}P\bigl( \Vert \varphi \Vert _{\varrho}+ \epsilon\bigr) \\ =&P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t^{*}-t_{0})} \biggl[\frac{e^{-\int_{t_{0}}^{t^{*}} (d^{*}(v)-\xi)\,dv}\mu}{P}+1-e^{-\int_{t_{0}}^{t^{*}} (d^{*}(v)-\xi)\,dv} \biggr] \\ =& P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t^{*}-t_{0})} \biggl[ \biggl(\frac{\kappa}{P}-1 \biggr) e^{-\int_{t_{0}}^{t^{*}} (d^{*}(v)-\xi)\,dv}+1 \biggr] \\ < &P\bigl( \Vert \psi \Vert _{\varrho}+\epsilon\bigr)e^{-\xi(t^{*}-t_{0})}, \end{aligned}$$
(2.12)

which contradicts (2.8). Then (2.7) is true. Thus \(w(t)=O(e^{-\xi t})\) as \(t\rightarrow+\infty\). The theorem is proved. □

Remark 2.1

In [36, 37] the authors dealt with neural networks with proportional delays, but they did not consider the global exponential convergence of involved models. In [10, 38] the authors studied the exponential convergence of neural networks with proportional delays, but they did not investigate Nicholson’s blowflies models. In this paper, we study the global exponential convergence of Nicholson’s blowflies model with proportional delays. All the derived results in [10, 36,37,38] cannot be applied to model (1.3) to obtain the global exponential convergence of system (1.3). So far, no results about the global exponential convergence of Nicholson’s blowflies model with proportional delays are reported. Therefore our findings on the global exponential convergence of Nicholson’s blowflies model with proportional delays are essentially innovative and supplement earlier publications to a certain extent.

3 Example

Consider the model

$$ \dot{w}(t)=-d(t)w(t)+\sum_{j=1}^{2}b_{j}(t) w(a_{j}t)e^{-\gamma_{j}(t) w(a_{j}t)}, $$
(3.1)

where \(d(t)=0.2(1+0.5\sin t)\), \(b_{1}(t)=0.07+0.07|\cos\sqrt{5t}|\), \(b_{2}(t)=0.05+0.05|\sin\sqrt{5t}|\), \(\gamma_{1}(t)=1+0.1|\cos \sqrt{3t}|\), \(\gamma_{2}(t)=1+0.1|\sin\sqrt{3t}|\), \(a_{1}=0.1\), \(a_{2}=0.6\) Then \(d^{*}(t)=0.2\) and \(\mu=e^{\frac{1}{5}}\), Let \(\sigma=\frac{1}{200}\). Then \(e^{-\int_{s}^{t}d(\theta)\,d\theta}\leq e^{\frac{1}{5}}e^{-(t-s)}\), \(t\geq s\), and \(\int_{s}^{t}(d^{*}(v)-d(v))\,dv\leq\sigma\), \(\sup_{t\geq t_{0}} \{-d^{*}(t)+\mu\sum_{j=1}^{2}|b_{j}(t)| \}\approx-0.6052<0\). Thus all the conditions in Theorem 2.1 are satisfied, and all solutions of model (3.1) converge exponentially to \((0,0)^{T}\). This fact is shown in Fig. 1.

Figure 1
figure 1

The relation of t and \(w(t)\). The initial values are \(w_{0}=0.2,0.3,0.5,0.4,0.45\)

Full size image

4 Conclusions

Exponential convergence is an important dynamical behavior of differential dynamical systems. During the past decades, many researchers payed much attention to it. In this paper, we have discussed Nicholson’s blowflies model with proportional delays. By means of the differential inequality knowledge, we derived a sufficient criterion ensuring the exponential convergence of all solutions for Nicholson’s blowflies model with proportional delays. The sufficiency criterion can be easily checked by simple computation. Up to now, there are no papers that focus on the exponential convergence of Nicholson’s blowflies model with proportional delays, which shows that the results derived in this paper are new and extend earlier publications to some extent.