1 Introduction

In 1988, Wazewska-Czyzewska and Lasota [2] proposed the delayed differential model

$$ \dot{w}(t)=-\varrho(t)w(t)+\sum_{k=1}^{p} \theta_{k}(t)e^{-\eta _{k}(t)w(t-\rho_{k}(t))} $$
(1.1)

to describe the survival of red cells in an animal [1]. In this model, p is a positive integer, \(w(t)\) stands for the number of red blood cells at time t, \(\varrho(t)\) stands for the death rate of the red blood cell, \(\theta_{k}(t)\) and \(\eta_{k}(t)\) are related to the production of red blood cells per unit time, and \(\rho_{k}(t)\) represents the time to produce a red blood cell. For details, see [2].

We know that the death rate or harvesting rate usually change under different seasonal fluctuations. In addition, the actual living environment of species have weighted pseudo almost periodic nature due to the effect of human activities and industrial production, for example, the exhaust emission and reconstruction of rivers. Based on this viewpoint, we think that it is reasonable to suppose that the coefficients in model (1.1) are weighted pseudo almost periodic functions, which can be expressed as an almost periodic component plus the weighted ergodic perturbation. Thus a key problem aries: seek the new sufficient conditions to ensure the existence and exponential stability of the weighted pseudo almost periodic solution for model (1.1).

Unfortunately, until now, no scholars have considered the weighted pseudo almost periodic solutions for model (1.1). To make up for this deficiency and inspired by the previous discussion, in this work, we concentrate on the weighted pseudo almost periodic solutions for model (1.1).

Let \({\mathcal{B}}{\mathcal{C}}(R,R)\) denote the set of all bounded continuous functions from R to R, and let \(\bar{g}=\sup_{t\in R} g(t)\) and \(\underline{g}=\inf_{t\in R} g(t)\) for a bounded continuous function \(g(t)\). The initial condition of system (1.1) is given by

$$ w(s)=\psi(s),\quad s\in[-\bar{\rho}, 0], $$
(1.2)

where \(\psi\in{\mathcal{B}}{\mathcal{C}}([-\bar{\rho}, 0],R^{+})\), \(\rho=\max_{k=1,2,\dots,p} \bar{\rho}_{k}\), and \(R^{+}\) denotes the nonnegative real numbers.

We plan the paper as follows. In Sect. 2, we present some preliminary knowledge on weighted pseudo almost periodic solution. In Sect. 3, we investigate the existence and global exponential stability of weighted pseudo almost periodic solution to model (1.1). In Sect. 4, numerical simulations are put into effect. The conclusion is given to end this paper in Sect. 5.

2 Basic knowledge

Throughout this manuscript, let \({\mathcal{U}}\) denote the collection of functions (weights) \(\upsilon: R\rightarrow(0,+\infty)\), which are locally integrable on R and satisfy

$$\lim_{\gamma\rightarrow+\infty}\upsilon\bigl([-\gamma,\gamma]\bigr)=+\infty,\quad \mbox{where } \upsilon\bigl([-\gamma,\gamma]\bigr)= \int_{-\gamma}^{\gamma}\upsilon(s)\,ds\quad (\gamma>0). $$

Let \(\|w\|=|w|\), \(g^{+}=\sup_{t\in R}|g(t)|\), \(g^{-}=\inf_{t\in R}|g(t)|\), \({\mathcal{U}}_{\infty}=\{\upsilon|\upsilon\in{\mathcal{U}}, \inf_{t\in R}\upsilon(w)=\upsilon_{0}>0\}\), \({\mathcal{U}}_{\infty}^{+}=\{\upsilon|\upsilon\in{\mathcal{U}},\lim_{|w|\rightarrow+\infty}\sup\frac{\upsilon(aw)}{\upsilon(w)}<+\infty, \lim_{\gamma\rightarrow+\infty}\sup\frac{\upsilon([-a\gamma,a\gamma ])}{\upsilon([-\gamma,\gamma])}<+\infty\text{ for }a\in(0,+\infty)\}\) Denote the set of bounded continuous function from R to R by \({\mathcal{B}}{\mathcal{C}}(R,R)\). So \(({\mathcal{B}}{\mathcal {C}}(R,R), \|\cdot\|_{\infty})\) is a Banach space with norm \(\|g\|_{\infty}=\sup_{t\in R}\|g(t)\|\). Denote the set of the almost periodic functions from R to R by \({\mathcal{A}}{\mathcal {P}}(R,R)\). Define

$$PAP_{0}^{\upsilon}(R, R)= \biggl\{ \phi\in{\mathcal{B}} { \mathcal{C}}(R,R) \Big\vert \lim_{\gamma\rightarrow+\infty} \frac{1}{\upsilon([-\gamma,\gamma])} \int _{-\gamma}^{\gamma}\bigr\vert \phi(t)\bigl| \upsilon(t)\,dt=0 \biggr\} . $$

We say that a function \(g\in{\mathcal{B}}{\mathcal{C}}(R,R)\) is weighted pseudo almost periodic if \(g=f+\phi\), where \(f\in \mathit{AP}(R,R)\) and \(\phi\in \mathit{PAP}_{0}^{\upsilon}(R, R)\). Here f is called the almost periodic component, and ϕ is called the weighted ergodic perturbation. We denote by \(\mathit{PAP}^{\upsilon}(R, R)\) the space of all weighted almost periodic functions. For more detail, see [3–15].

Lemma 2.1

([2])

Let\(\bar{\varrho}:R\rightarrow(0,+\infty)\)be a bounded continuous function, and let\(\alpha^{M}\), \(\alpha^{m}\), \(\beta^{M}\), \(\beta^{m}\), σ, χbe positive constants such that

$$\begin{gathered} \alpha^{m} e^{-\int_{s}^{t} \bar{\varrho}(v)\,dv}\leq e^{-\int_{s}^{t} \varrho (v)\,dv}\leq \alpha^{M} e^{-\int_{s}^{t} \bar{\varrho}(v)\,dv},\quad \forall t,s \in R \textit{ and } t-s\geq0, \\ -\beta^{M}= \Biggl(-\bar{\varrho}(t)\chi+\alpha^{M}\sum _{k=1}^{r}\theta _{k}(t) \Biggr)^{+},\quad \chi>\sigma,\\ -\beta^{m}= \Biggl(-\varrho^{*}(t) \sigma+\alpha ^{m}\sum_{k=1}^{r} \theta_{k}(t)e^{-\eta_{k}(t)\chi} \Biggr)^{-}.\end{gathered} $$

Then there exists\(t_{0}\)such that the solution\(w(t)\)of system (1.1) with initial condition (1.2) satisfies\(\sigma< w(t)<\chi\)for\(t>t_{0}\).

We introduce the following assumptions:

\(({\mathcal{A}}_{1})\):

For \(k=1,2,\ldots,p\), \(\theta_{k}, \eta_{k}, \rho _{k} \in{\mathcal{P}}{\mathcal{A}}{\mathcal{P}}^{\upsilon}(R,R)\), \(\varrho \in{\mathcal{A}}{\mathcal{P}}(R,R)\),

$$M[\varrho]=\lim_{\mathcal{T}\rightarrow+\infty}\frac{1}{\mathcal {T}}\int_{t}^{t+\mathcal{T}}\varrho(s)\,ds>0.$$
\(({\mathcal{A}}_{2})\):

\(\upsilon\in{\mathcal{U}}_{\infty}^{+}\), and there exist constants \(\epsilon>0\) and \(\zeta>0\) such that

$$\sup_{t\in R} \Biggl\{ -\bar{\varrho}(t)+\zeta^{-1} \alpha^{M}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert \bigl\vert \eta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma} \Biggr\} < -\epsilon. $$

3 Main results

Lemma 3.1

Define the operator

$$\varGamma(\psi) (t)= \int_{-\infty}^{t}e^{-\int_{s}^{t} \varrho(v)\,dv}\Biggl[\zeta ^{-1}\sum_{k=1}^{p} \theta_{k}(t)e^{-\zeta\eta_{k}(t)\psi(t-\rho_{k}(t))}\Biggr] \, ds,\quad \psi\in{\mathcal{P}} { \mathcal{A}} {\mathcal{P}}^{\upsilon}(R,R). $$

ThenΓmaps\({\mathcal{P}}{\mathcal{A}}{\mathcal{P}}^{\upsilon}(R,R)\)into itself.

Proof

In view of \(({\mathcal{A}}_{1})\) and Lemmas 2.1 and 2.3 of [16], we have that

$$\sum_{k=1}^{p}\theta_{k}(t)e^{-\zeta\eta_{k}(t)\psi(t-\rho_{k}(t))} \in {\mathcal{P}} {\mathcal{A}} {\mathcal{P}}^{\upsilon}(R,R). $$

According to \(({\mathcal{A}}_{1})\)–\(({\mathcal{A}}_{2})\) and applying the proof of Lemma 2.1 of [17], we can see that \(\varGamma(\psi)\in {\mathcal{B}}{\mathcal{C}}(R,R)\). Then there exist \({\mathcal{E}}(t)\in{\mathcal{A}}{\mathcal{P}}(R,R)\) and \({\mathcal{F}}(t)\in{\mathcal{P}}{\mathcal{A}}{\mathcal {P}}_{0}^{\upsilon}(R,R)\) such that

$$\sum_{k=1}^{p}\theta_{k}(t)e^{-\zeta\eta_{k}(t)\psi(t-\rho_{k}(t))}={ \mathcal {E}}(t)+{\mathcal{F}}(t). $$

Since

$$M[\varrho]=\lim_{\mathcal{T}\rightarrow+\infty}\frac{1}{\mathcal {T}} \int_{t}^{t+\mathcal{T}}\varrho(s)\,ds>0, $$

applying the theory of exponential dichotomy of [18], we get that

$$ \int_{-\infty}^{t}e^{-\int_{s}^{t} \varrho(v)\,dv}{\mathcal{E}}(s) \,ds\in {\mathcal{A}} {\mathcal{P}}(R,R) $$
(3.1)

is a solution of the following equation

$$ \dot{u}(t)=-\varrho(t)u(t)+{\mathcal{E}}(t). $$
(3.2)

By the proof of Lemma 2.3 of [19] we have that

$$ \int_{-\infty}^{t}e^{-\int_{s}^{t} \varrho(v)\,dv}{\mathcal{F}}(s) \,ds\in {\mathcal{P}} {\mathcal{A}} {\mathcal{P}}_{0}^{\upsilon}(R,R). $$
(3.3)

In view of (3.1) and (3.3), we conclude that

$$\varGamma(\psi) (t)= \int_{-\infty}^{t}e^{-\int_{s}^{t} \varrho(v)\,dv}\Biggl[\zeta ^{-1}\sum_{k=1}^{p} \theta_{k}(t)e^{-\zeta\eta_{k}(t)\psi(t-\rho_{k}(t))}\Biggr] \, ds\in{\mathcal{P}} { \mathcal{A}} {\mathcal{P}}^{\upsilon}(R,R) $$

and Γ maps \({\mathcal{P}}{\mathcal{A}}{\mathcal{P}}^{\upsilon}(R,R)\) into itself. This completes the proof. □

Theorem 3.1

In addition to the condition of Lemma 2.1, suppose that\(({\mathcal {A}}_{1})\)–\(({\mathcal{A}}_{2})\)are fulfilled. Then model (1.1) has a unique weighted pseudo almost periodic solution, which is globally exponentially stable.

Proof

Let \(u(t)=\zeta^{-1}w(t)\). Then system (1.1) becomes

$$ \dot{u}(t)=-\varrho(t)u(t)+\zeta^{-1}\sum _{k=1}^{p}\theta_{k}(t)e^{-\zeta \eta_{k}(t)u(t-\rho_{k}(t))}. $$
(3.4)

For \(\psi, \phi\in{\mathcal{P}}{\mathcal{A}}{\mathcal{P}}^{\upsilon}(R,R)\), in view of \(({\mathcal{A}}_{1})\) and \(({\mathcal{A}}_{2})\), we have

$$\begin{aligned} & \bigl\vert (\varGamma\psi) (t)-(\varGamma\phi) (t) \bigr\vert \\ &\quad= \Biggl\vert \int_{-\infty}^{t}e^{-\int_{s}^{t} \varrho(v)\,dv} [\zeta ^{-1}\sum_{k=1}^{p} \theta_{k}(s)\bigl[e^{-\zeta\eta_{k}(s)\psi(s-\rho _{k}(s))}-e^{-\zeta\eta_{k}(s)\phi(s-\rho_{k}(s))} \bigr]\,ds \Biggr\vert \\ &\quad\leq \int_{-\infty}^{t}e^{-\int_{s}^{t} \bar{\varrho}(v)\,dv}\zeta ^{-1}\alpha^{M}\sum_{k=1}^{p} \theta_{k}(s)\eta_{k}(s)e^{-\eta_{k}^{-}\sigma} \bigl\vert \psi\bigl(s-\rho_{k}(s)\bigr)-\phi\bigl(s-\rho_{k}(s) \bigr) \bigr\vert \,ds \\ &\quad\leq\sup_{t\in R} \int_{-\infty}^{t}e^{-\int_{s}^{t} \bar{\varrho}(v)\, dv} \zeta^{-1}\alpha^{M}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert \bigl\vert \eta_{k}(s) \bigr\vert e^{-\eta _{k}^{-}\sigma} \,ds \bigl\Vert \psi(t)-\phi(t) \bigr\Vert _{\infty} \\ &\quad\leq\sup_{t\in R} \int_{-\infty}^{t}e^{-\int_{s}^{t} \bar{\varrho}(v)\, dv}\bigl[\bar{ \varrho}(s)-\epsilon\bigr] \,ds \bigl\Vert \psi(t)-\phi(t) \bigr\Vert _{\infty} \\ &\quad\leq\sup_{t\in R} \int_{-\infty}^{t}e^{-\int_{s}^{t} \bar{\varrho}(v)\,dv}\,d \biggl(- \int_{s}^{t}\bar{\varrho}(v)\,dv \biggr)- \frac{\epsilon}{2} \int _{-\infty}^{t}e^{-\int_{s}^{t} \bar{\varrho}(v)\,dv} \,ds \bigl\Vert \psi(t)-\phi(t) \bigr\Vert _{\infty} \\ &\quad\leq \biggl(1-\frac{\epsilon}{2\bar{\varrho}^{+}} \biggr) \bigl\Vert \psi(t)-\phi (t) \bigr\Vert _{\infty} \end{aligned}$$
(3.5)

for \(l=1,2,\ldots,r\). By \(({\mathcal{A}}_{2})\) we easily see that \((1-\frac{\epsilon}{2\bar {\varrho}^{+}} )\in(0,1)\), and thus Γ is a contraction mapping. Thus Γ has a unique fixed point \(u^{*}\in{\mathcal{P}}{\mathcal {A}}{\mathcal{P}}^{\upsilon}(R,R)\) and satisfies \(\varGamma u^{*}= u^{*}\). Based on this analysis, we conclude that model (1.1) has a unique weighted pseudo almost periodic solution \(w^{*}=\zeta u^{*}\in {\mathcal{P}}{\mathcal{A}}{\mathcal{P}}^{\upsilon}(R,R)\).

Next, we will prove the exponential stability of the solution \(w^{*}\). Let \(w(t)\) be an arbitrary solution of (1.1) with initial condition \(\psi(t)\) satisfying (1.2). Let \(v(t)=\zeta^{-1}(w(t)-w^{*}(t))\). Then we have

$$ \dot{v}(t)=-\varrho(t)v(t)+\zeta^{-1}\sum _{k=1}^{p}\theta_{k}(t) \bigl[e^{-\eta _{k}(t)w(t-\rho_{k}(t))}-e^{-\eta_{k}(t)w^{*}(t-\rho_{k}(t))}\bigr]. $$
(3.6)

Let

$$ \varPhi(\varphi)=\sup_{t\in R} \Biggl\{ \varphi-\bar{ \varrho}(t)+\zeta ^{-1}\alpha^{M}\sum _{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert \bigl\vert \eta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma }e^{\varphi\rho_{k}(t)} \Biggr\} ,\quad \varphi\in\bigl[0, \bar{ \varrho}^{-}\bigr]. $$
(3.7)

Then

$$ \varPhi(0)=\sup_{t\in R} \Biggl\{ -\bar{\varrho}(t)+ \zeta^{-1}\alpha^{M}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert \bigl\vert \eta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma} \Biggr\} \leq -\epsilon< 0. $$
(3.8)

Since \(\varPhi(\varphi)\) is continuous, we can choose a constant \(\mu \in(0, \bar{\varrho}^{-})\) such that

$$ \varPhi(\mu)=\sup_{t\in R} \Biggl\{ \mu-\bar{\varrho}(t)+ \zeta^{-1}\alpha ^{M}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert \bigl\vert \eta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma} \Biggr\} ,\quad \varphi\in \bigl[0, \bar{\varrho}^{-}\bigr]. $$
(3.9)

Set

$$ \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}=\sup_{t\in(-\bar{\rho},0]} \zeta^{-1} \bigl\vert \psi(t)-w(t) \bigr\vert $$
(3.10)

and choose \({\mathcal{N}}>\alpha^{M}+1\). Then we obtain that

$$ \bigl\Vert v(t) \bigr\Vert \leq\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon\bigr)e^{-\mu t}< ({\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon\bigr)e^{-\mu t},\quad t\in(-\bar{\rho},0]. $$
(3.11)

Next, we will prove that

$$ \bigl\Vert v(t) \bigr\Vert \leq\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon\bigr) e^{-\mu t}< ({\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon\bigr)e^{-\mu t},\quad t>0. $$
(3.12)

If (3.12) were not true, then there would exist \(\xi>0\) such that

$$ \left \{ \textstyle\begin{array}{lc} \vert v(\xi) \vert ={\mathcal{N}}( \Vert \psi-w^{*} \Vert _{\zeta}+\epsilon)e^{-\mu\xi},\\ \vert v(t) \vert ={\mathcal{N}}( \Vert \psi-w^{*} \Vert _{\zeta}+\epsilon)e^{-\mu t},\quad t\in (-\bar{\rho},\xi). \end{array}\displaystyle \right .$$
(3.13)

In view of

$$\begin{aligned}[b] &\dot{v}(s)+\varrho(s)v(s)\\&\quad=\zeta^{-1}\sum _{k=1}^{p}\theta_{k}(s) \bigl[e^{-\eta _{k}(s)\zeta w(s-\rho_{k}(s))}-e^{-\eta_{k}(s)\zeta w^{*}(s-\rho_{k}(s))}\bigr],\quad s\in [0,t], t\in[0, \xi],\end{aligned} $$
(3.14)

we have

$$\begin{aligned} {v}(s) =&v(0)e^{-\int_{0}^{t} \varrho(v)\,dv}+ \int_{0}^{t}e^{-\int_{s}^{t} \varrho (v)\,dv} \\ &{}\times \Biggl\{ \zeta^{-1}\sum_{k=1}^{p} \theta_{k}(s)\bigl[e^{-\eta_{k}(s)\zeta w(s-\rho_{k}(s))}-e^{-\eta_{k}(s)\zeta w^{*}(s-\rho_{k}(s))}\bigr] \Biggr\} \,ds,\quad t\in [0,\xi]. \end{aligned}$$
(3.15)

Hence

$$\begin{aligned} \bigl\vert {v}(s) \bigr\vert =& \Biggl\vert v(0)e^{-\int_{0}^{t} \varrho(v)\,dv}+ \int_{0}^{t}e^{-\int_{s}^{t} \varrho(v)\,dv} \\ &{}\times \Biggl\{ \zeta^{-1}\sum_{k=1}^{p} \theta_{k}(s)\bigl[e^{-\eta_{k}(s)\zeta w(s-\rho_{k}(s))}-e^{-\eta_{k}(s)\zeta w^{*}(s-\rho_{k}(s))}\bigr] \Biggr\} \,ds \Biggr\vert \\ \leq&{\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+ \epsilon\bigr)\alpha^{M}e^{-\int_{0}^{\xi}\bar{\varrho}(v)\,dv}+ \int_{0}^{\xi}e^{-\int_{s}^{\xi}\bar{\varrho}(v)\, dv} \alpha^{M} \\ &{}\times \Biggl[\zeta^{-1}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma }\bigl|v(s- \rho_{k}(s)\bigr| \Biggr]\,ds \\ \leq&{\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+ \epsilon\bigr)\alpha^{M}e^{-\int_{0}^{\xi}\bar{\varrho}(v)\,dv}+ \int_{0}^{\xi}e^{-\int_{s}^{\xi}\bar{\varrho}(v)\, dv} \alpha^{M} \\ &{}\times \Biggl[\zeta^{-1}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma }{ \mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon \bigr)e^{-\mu(s-\rho_{k}(s))} \Biggr]\,ds \\ \leq&{\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+ \epsilon\bigr)e^{-\mu\xi} \alpha ^{M}e^{-\int_{0}^{\xi}(\bar{\varrho}(v)-\mu)\,dv}+ \int_{0}^{\xi}e^{-\int_{s}^{\xi}(\bar{\varrho}(v)-\mu)\,dv} \alpha^{M} \\ &{}\times \Biggl[\zeta^{-1}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma }{ \mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon \bigr)e^{\mu\rho_{k}(s)} \Biggr]\,ds {\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon\bigr)e^{-\mu\xi} \\ \leq&{\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+ \epsilon\bigr)e^{-\mu\xi} \alpha ^{M}e^{-\int_{0}^{\xi}(\bar{\varrho}(v)-\mu)\,dv} \\ &{}+ \int_{0}^{\xi}e^{-\int_{s}^{\xi}(\bar{\varrho}(v)-\mu)\,dv}\bigl(\bar{ \varrho }(v)-\mu\bigr)\,ds {\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon\bigr)e^{-\mu\xi } \\ \leq&{\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+ \epsilon\bigr)e^{-\mu\xi} \biggl[1- \biggl(1-\frac{\alpha^{M}}{{\mathcal{N}}} \biggr) e^{-\int_{s}^{\xi}(\bar {\varrho}(v)-\mu)\,dv} \biggr] \\ \leq&{\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+ \epsilon\bigr)e^{-\mu\xi}. \end{aligned}$$
(3.16)

By (3.13) we get that (3.12) holds. Let \(\epsilon\rightarrow0^{+}\). Then

$$ \bigl\Vert {v}(t) \bigr\Vert < {\mathcal{N}}\bigl( \bigl\Vert \psi-w^{*} \bigr\Vert _{\zeta}+\epsilon\bigr)e^{-\mu t},\quad t>0. $$
(3.17)

Therefore we conclude that the pseudo almost periodic solution of model (1.1) is globally exponentially stable. The proof is finished. □

Remark 3.1

Shao [1] studied the pseudo almost periodic solution of model (1.1). In this paper, we considered the weighted pseudo almost periodic solution. All the assumptions in this paper are different from those in [1]. All the derived results in [1] cannot be applicable to system (1.1) to establish the existence and globally exponential stability of weighted pseudo almost periodic solution for model (1.1). Based on this viewpoint, our results on the existence and globally exponential stability of weighted pseudo almost periodic solution for model (1.1) are essentially new and complement earlier works to a certain extent.

4 Computer simulations

Consider the system

$$\begin{aligned} \dot{w}(t)=-\varrho(t)w(t)+\sum_{k=1}^{2} \theta_{k}(t)e^{-\eta _{k}(t)w(t-\rho_{k}(t))}, \end{aligned}$$
(4.1)

where \(\varrho(t)=17+10\cos1000t\), \(\theta_{1}(t)=2(1+0.5|\sin1.7 t|)\), \(\theta_{2}(t)=2(0.8+0.2|\sin1.7 t|)\), \(\eta_{1}(t)=-1+0.2|\sin5t|\), \(\eta _{2}(t)=-0.9+0.5|\sin30t|\), \(\rho_{1}(t)=e^{0.2|\sin t|}\), \(\rho _{2}(t)=e^{0.3|\sin t|}\). Let \(\bar{\varrho}(t)=17\), \(M[\varrho]=17\), \(\alpha^{M}=\frac{1}{200}\), \(\alpha ^{m}=-\frac{1}{200}\), \(\chi=0.8\), \(\sigma=0.2\), \(\zeta=2\), \(\epsilon=3\). Then

$$\alpha^{m} e^{-\int_{s}^{t} \bar{\varrho}(v)\,dv}\leq e^{-\int_{s}^{t} \varrho (v)\,dv}\leq \alpha^{M} e^{-\int_{s}^{t} \bar{\varrho}(v)\,dv}\quad\mbox{for } t,s \in R \mbox{ such that } t-s\geq0 $$

and

$$\sup_{t\in R} \Biggl\{ -\bar{\varrho}(t)+\zeta^{-1} \alpha^{M}\sum_{k=1}^{p} \bigl\vert \theta_{k}(s) \bigr\vert \bigl\vert \eta_{k}(s) \bigr\vert e^{-\eta_{k}^{-}\sigma} \Biggr\} \approx -3.087< -3=-\epsilon. $$

Thus all the hypotheses of Theorem 3.1 are satisfied, and so system (4.1) has a unique weighted pseudo almost periodic solution, which is globally exponentially stable. Figure 1 reveals this fact.

Figure 1
figure 1

Numerical results of model (4.1): \(t-w(t)\)

Full size image

5 Conclusions

In this paper, we have discussed the existence and globally exponential stability of weighted pseudo almost periodic solutions for a Lasota–Wazewska system. Using the fixed point theory and differential inequalities, we establish new sufficient criteria ensuring the existence and globally exponential stability of weighted pseudo almost periodic solutions for the Lasota–Wazewska model. The derived results complement some earlier publications to some extent. Up to now, to the best of our knowledge, it is the first time to deal with this aspect. In the near future, we will investigate the pseudo almost automorphic solutions for the Lasota–Wazewska model.