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\jyear{2021}%

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\begin{document}

\title[Global dynamics for a class of nonautonomous mosquito models]{Global dynamics for a class of nonautonomous mosquito models with time delay and switching control strategy}

%%=============================================================%%
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\author[1]{\fnm{}  \sur{Qingwen Yu}}%\email{yqw1236688@163.com}

\author*[1]{\fnm{}  \sur{Dongshu Wang}}\email{wds-03@163.com}
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\author[2]{\fnm{} \sur{Yaqiong Cheng}}%\email{312842580@qq.com}

\affil*[1]{\orgdiv{School of Mathematical Sciences}, \orgname{Huaqiao University}, \orgaddress{\street{} \city{Quanzhou, Fujian}, \postcode{362021}, \country{China}}}

\affil[2]{\orgdiv{Department of Social Sciences and Public Affairs}, \orgname{Quanzhou Medical Vocational College}, \orgaddress{\street{Street}, \city{Quanzhou, Fujian}, \postcode{362100}, \country{China}}}
%\equalcont{ }
%\affil[3]{\orgdiv{Department}, \orgname{Organization}, \orgaddress{\street{Street}, \city{City}, \postcode{610101}, \state{State}, \country{Country}}}

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\abstract{%The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. Authors are advised to check the author instructions for the journal they are submitting to for word limits and if structural elements like subheadings, citations, or equations are permitted.
Mosquitoes serve as the primary vectors for several life-threatening pathogens, including malaria, dengue fever, and Zika virus. To effectively control the transmission of these diseases, a variety of integrated vector management strategies are currently employed to reduce mosquito population densities. This paper focuses on the sterile insect technique (SIT) and incompatible insect technique (IIT), which involves the release of sterile male mosquitoes to suppress wild mosquito populations, and delves into the impact of different release strategies on mosquito population dynamics. We established a switched mosquito population suppression model with time delay. We first investigate the  existence and stability of the switching ordinary differential equation model, and derive some criteria for determining whether $T$-periodic solutions are stable or unstable. By taking the initial function as a solution to the delay-free model, we obtain sufficient conditions for existence of a unique or exactly two periodic solutions. Furthermore, we gain necessary and sufficient conditions for nonexistence of periodic solutions. Finally, the numerical results have been shown to further highlighted the effectiveness of our proposed model.}

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% \abstract{\textbf{Purpose:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
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% \textbf{Methods:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
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% \textbf{Results:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.
%
% \textbf{Conclusion:} The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. The abstract must not include subheadings (unless expressly permitted in the journal's Instructions to Authors), equations or citations. As a guide the abstract should not exceed 200 words. Most journals do not set a hard limit however authors are advised to check the author instructions for the journal they are submitting to.}

\keywords{Time delay, Switching dynamical systems, Density dependence, Stability, Mosquito population suppresion, Periodic solutions,  Sterile mosquitoes}

%%\pacs[JEL Classification]{D8, H51}

%%\pacs[MSC Classification]{35A01, 65L10, 65L12, 65L20, 65L70}

\maketitle
\section{Introduction}
Mosquito borne diseases are a major public health issue worldwide, posing a serious threat to human life safety \cite{ref1,ref2}. Currently, there are no specific drugs or licensed vaccines for mosquito borne diseases. Therefore, the most direct way to prevent and control these mosquito borne diseases is to control vector mosquitoes. Sterile insect technique (SIT) and Incompatible insect technique (IIT) are two biological methods for controlling mosquitoes. The former uses radiation to destroy mosquito germ cells and make them infertile, while the latter infects insects with intracellular symbiotic bacteria of the $Wolbachia$ genus to achieve the same goal \cite{ref11,ref12}. Release these infertile male mosquitoes into the natural environment and mate with wild female mosquitoes. The offspring produced by mating either do not lay eggs or do not hatch, thus achieving the goal of mosquito control.

In order to study the dynamic interaction between wild mosquitoes and sterile male mosquitoes, it is assumed that in the absence of interaction, the population dynamics of wild mosquitoes and sterile mosquitoes follow logistic growth, and the birth rate of sterile mosquitoes is their release rate. Assuming that mosquitoes are homogeneous and their gender is also evenly distributed. There are mathematical models in the literature formulated to study the interactive dynamics of mosquito populations or the control of mosquitoes \cite{ref3,ref4,ref5,ref6,ref7,ref10,ref14,ref17,ref18,ref19,ref20,ref21}.

\numberwithin{equation}{section}
Ordinary differential equations can reflect the development trend of things, the development trend of things depends not only on the current state, but also on the state of past or future time. Let $\tau >0$ represent the maturation period of wild mosquitoes from parent mating to the emergence of reproductive progenies. Considering that the life cycle of wild mosquitoes is $5-37$ days \cite{ref8}, while male mosquitoes infected with $Wolbachia$ release $2-3$ times a week \cite{ref5}, assuming that in the mature period $\tau$ , there are $m$ batches of male mosquitoes are released, $m\ge 1$ is an integer, $\tau =mT $. Hereafter, we have
\begin{equation}\label{eq5}
g(t-\tau )=\left\{\begin{matrix}
 0, & t\in (0,\tau ]=(0,mT]; & \\
 c, & t\in (iT,iT+\bar{T} ]; & \\
 0, & t\in (iT+\bar{T},(i+1)T], &
\end{matrix}\right.
\end{equation}
for $i=m,m+1,\cdots $. Here $T$ is the waiting period between two consecutive releases, $T>0$. $\bar{T} $ represent the duration of mating competitiveness of sterile male mosquitoes, We focus on the case $T>\bar{T} $. In such a case, the released males are sexually active for $t\in(iT,iT+\bar{T} ]$ , and inactive for $t\in (iT+\bar{T},(i+1)T]$. The release strategy is a $T$-periodic and piecewise constant function, releasing a constant of $c$ mosquitoes during the sexually active period and not releasing mosquitoes during the sexually inactive period.

Motivated by the implementation of release strategy in Guangzhou \cite{ref12} and following the lines in \cite{ref6,ref7,ref14}, we consider the switching system with a time delay described by
\begin{equation}\label{eq6}
\frac{dw}{dt} =a w (t-\tau ) (1-\eta w(t-\tau ))-\mu w(t), t\in (iT,(i+1)T ],
\end{equation}
for $i=0,1,\cdots,m-1 $; and
\begin{equation}\label{eq7}
\frac{dw}{dt} =a\frac{w ^{2} (t-\tau)}{w(t-\tau )+c} (1-\eta w(t-\tau ))-\mu w(t), t\in (iT,iT+\bar{T} ],
\end{equation}
\begin{equation}\label{eq8}
\frac{dw}{dt} =a w(t-\tau ) (1-\eta w(t-\tau ))-\mu w(t), t\in (iT+\bar{T},(i+1)T],
\end{equation}
for $i=m,m+1,\cdots $.
Here $w(t)$ and $g(t)$ are the numbers of wild and sterile mosquitoes at time t, respectively. $a$ is the maximum number of survived offspring produced per mosquito; $\mu $ denotes the natural mortality of the wild mosquitoes; and $B(\cdot )$ is the release rate of the sterile mosquitoes. $\eta $ represents the carrying capacity parameters of wild mosquitoes, $1-\eta \omega (t)$ describes the impact of density on the survival probability of larvae during the larval stage. Define the intrinsic growth rate of wild mosquito population by $r=\frac{a}{\mu } $. If $r\le 1$, then the wild mosquitoes eventually die out naturally without the need of releasing sterile mosquitoes. It is observed that when r is very large, the wild mosquito population will experience explosive growth. At this point, simply relying on the release of male mosquitoes infected with $Wolbachia$ is insufficient to control the number of wild mosquitoes. Therefore, this paper conducts research under the conditions of limited resources, moderate population density, and good ecological niche adaptation. At this time, the growth rate of wild mosquitoes is moderate, neither explosive nor extremely slow, that is, the situation where r is bounded. For convenience, it is assumed that $1\le r\le 2$.

Define three threshold values
$$g^{*}=\frac{(a-\mu )^{2} }{4\mu\eta a} ,~T^{*} =\frac{a}{a-\mu_{1}}\bar{T},~c^{*} =\frac{a-\mu}{\mu\eta }.$$

Notably, the existing literature on the release quantity within the range $c\in (0,g^{*})$ is not sufficiently comprehensive. While some literature has provided necessary and sufficient conditions for stability \cite{ref7}, these conditions are mathematically rigorous but do not fully capture the complexity of wild mosquito population dynamics. In reality, the key factors influencing wild mosquito populations include not only the number of sterile male mosquitoes released but also the release frequency and its interactions with the natural environment. Therefore, it is essential to incorporate the release cycle as a critical factor in stability analysis. The main contributions of our results can be summarized as following:

(i) Inspired by reference \cite{ref13}, we yield the stability of periodic solutions of ordinary differential equations \eqref{eqa1}-\eqref{eqa2} under the case $c\ge c^{*} $, which is different from existing literature \cite{ref2, ref3, ref4, ref5, ref6, ref7, ref8}.

(ii) Developmental lag significantly influences mosquito population dynamics. Inspired by References \cite{ref5} and \cite{ref26}, we establish a framework that connects ordinary differential equations with delay differential equations, and analyze the stability of $T$-periodic solutions in time-delay switched dynamical systems \eqref{eq7}-\eqref{eq8}.

(iii) Deriving existence and uniqueness conditions for periodic solutions is mathematically challenging, and establishing both necessary and sufficient conditions is even more complex. While previous works \cite{ref5,ref6} have provided sufficient conditions for the global asymptotic stabilization of the origin in the ranges $c\in (g^{*},c^{*})$ and $c\ge c^{*}$, we offer both necessary and sufficient conditions.

(iv) The methodology presented here is applicable to the analysis of dynamics in other single-species delay differential equations with seasonality.

The remaining portions of the article are arranged as follows. In Section 2, preliminaries to prove the main theorem are provided. In Section 3, we provide proof of the main results. Numerical examples and brief discussions on our theoretical results are provided in Section 4. Section 5 summarizes the results and provides some directions for future research.

\section{Preliminaries}

We investigated Equations \eqref{eq7} and \eqref{eq8} for exploring the dynamical behaviors of wild mosquitoes under the interferences of released sterile mosquitoes.
Define set $\Omega$ as
$$\Omega =\left \{ \omega \mid 0\le \omega \le \frac{a-\mu }{a\eta }  \right \} ,$$
then $\Omega$ is a positive invariant set of systems \eqref{eq7}-\eqref{eq8}. Concerning with the asymptotic behavior of model dynamics, we neglect the dynamics dictated by \eqref{eq6} for $t\in [0,mT]$. Without loss of generality, we select $t_{0} \in [mT,(m+1)T)$, where \eqref{eq7}-\eqref{eq8} is $T$-periodic. The initial function $\phi (t)$ is thus defined as:
$$\phi \in C([t_{0}-\tau ,t_{0}];(0,\frac{a-\mu }{a\eta } ) ), $$
where $[t_{0} -\tau ,t_{0}]\subset [0,t_{0}]$. Then we define the solution $\omega (t)=\omega (t;g,t_{0} ,\phi )$ to the system \eqref{eq7}-\eqref{eq8} through $(t_{0} ,\phi )$.

The solution $\omega (t)$ satisfies Equation \eqref{eq7} for $t\in (t_{0} ,t_{0} +\bar{T} ]$ with initial condition $\omega (t_{0}^{+} )=\phi (t_{0})$. Then by defining $\omega ((t_{0} +\bar{T})^{+} )=\textstyle \lim_{t \to (t_{0} +\bar{T})^{+}} \omega (t)$, it satisfies Equation \eqref{eq8} for $t\in (t_{0} +\bar{T} ,(m+1)T]$. Repeating this procedure, we can get the solution $\omega (t)$ for all $t\ge t_{0}$, whose dynamics is determined by the switching \eqref{eq7}-\eqref{eq8}. Apparently  $\omega (t)$ is a continuous and piecewise differentiable function defined on $[t_{0} ,+\infty )$.
\vspace{10pt}

For further analysis of model dynamics, some definitions are given as follows.
\vspace{10pt}

\textbf{Definition 2.1.} \cite{ref9} A solution $\omega_{0}(t)$ of \eqref{eq7}-\eqref{eq8} is said to be

$(i)$ Stable if for any $t_{0} \ge 0$ and $\varepsilon>0$, there exists $\delta >0$ such that $\phi \in C([t_{0}-\tau ,t_{0}],(0,\frac{a-\mu}{a\eta }  ) )$ and $\left\lvert \phi(t)-\omega _{0} (t)  \right\rvert<\delta$ for $t\in[t_{0}-\tau ,t_{0}] $ implies
$$\left\lvert \omega (t)-\omega _{0} (t) \right\rvert= \left\lvert \omega (t,g,t_{0},\phi )- \omega _{0} (t) \right\rvert<\varepsilon,~\mathrm{for}~t\ge t_{0} .$$
If $\delta $ is independent of $t_{0}$, then $\omega _{0} (t)$ is uniformly stable.

$(ii)$ Globally asymptotically stable if it is uniformly stable, and for any $t_{0} \ge 0,~\phi \in C([t_{0}-\tau ,t_{0}],(0,\frac{a-\mu}{a\eta }  ) )$
$$\lim_{t\to \infty } (\omega (t,g,t_{0} ,\phi )-\omega _{0}(t) )=0.$$

To prove our main result, we introduce a special kind of solution whose initial functions are given by solutions of the switching ordinary differential equation (ODE) model
\begin{equation}\label{eqa1}
\frac{dv}{dt} =a \frac{v^{2} (t)}{v(t)+c} (1-\eta v(t))-\mu v(t),~t\in (iT,iT+\bar{T }], \end{equation}
\begin{equation}\label{eqa2}
\frac{dv}{dt} =a v(t)(1-\eta v(t))-\mu v(t),~t\in (iT+\bar{T },(i+1)T],
\end{equation}
with $i=0,1,2,\cdots . $

For convenience, we denote $Poincar\acute{e}$ map $h(u)=v(T;0,u)$ and $\bar{h} (u)=v(\bar{T} ;0,u)$. Clearly, $h(u)$ and $\bar{h} (u)$ are two continuously differentiable functions in $u>0$.

Define function series $\left \{ h_{n}  \right \} $ and $\left \{ \bar{h} _{n}  \right \} $ by
\[
h _{n}(u)=v(nT;0,u ), \ \ \makebox{and}\
\bar{h} _{n}(u)=v(nT+\bar{T};0,u ).
\]

For any $t\in [0,T)$, define $$v_{n}(t):=v_{n}(t,u)=v(t+nT;0,u),~n=0,1,2,\dots ,$$
where $v(t;0,u)$ is the solution to \eqref{eqa1}-\eqref{eqa2} initiated from $u>0$. Then we have the following Lemmas.
\vspace{10pt}

\textbf{Lemma 2.1.} For any given initial value $u>0$, we have the the following:

(1) If $h(u)>u$, then the sequences $\left \{ h_{n}(u)  \right \}$ and $\left \{ \bar{h} _{n}(u)  \right \} $ are both strictly increasing.

(2) If $h(u)=u$, then $h_{n}(u)=u, n=0,1,2,\dots $. Furthermore, $v(t;0,u)$ is a $T$-periodic solution of \eqref{eqa1}-\eqref{eqa2}.

(3) If $h(u)<u$, then the sequences $\left \{ h_{n}(u)  \right \}$ and $\left \{ \bar{h} _{n}(u)  \right \} $ are both strictly decreasing.
\vspace{10pt}

\textbf{Proof.} Firstly, we will show that the assert (1) holds. Otherwise, there exists a positive integer $n_{0} \ge 2$ such that
\begin{equation}\label{eqa3}
h_{n} (u)>h_{n-1}(u)~\mathrm{for}~n=1,2,\cdots ,n_{0} -1,~\mathrm{but}~ h_{n_{0} } (u)\le h_{n_{0}-1}(u).
\end{equation}
Since \eqref{eqa1}-\eqref{eqa2} is $T$-periodic, $\bar{v} (t)=v(t+T;0,u)$ is also a solution to \eqref{eqa1}-\eqref{eqa2}. Furthermore, from \eqref{eqa3}, we get
$$v_{n_{0}-2 } (t)=v(t+(n_{0}-2 )T)< v(t+(n_{0}-1)T)=\bar{v} _{n_{0}-2 } (t).$$
and
$$v_{n_{0}-1 } (t)=v(t+(n_{0}-1 )T)\ge v(t+n_{0}T)=\bar{v} _{n_{0}-1 } (t).$$
Therefore, $v(t)$ and $\bar{v}(t)$ must intersect in $[t+(n_{0}-2)T, t+(n_{0}-1)T)$, which contradicts the uniqueness of solutions to \eqref{eqa1}-\eqref{eqa2}. Hence, the assert (1) holds.

For the asserts (2) and (3), the proof are similar as the assert (1). We omit it here. Thus, the proof is completed. $\hfill\square$
\vspace{10pt}

The $T$-periodic solutions of ordinary differential equations and delay differential equations are generally different, and there is a huge gap between them. Hence, in order to find the $T$-periodic solution to \eqref{eq7}-\eqref{eq8}, we consider a special kind of solutions to \eqref{eq7}-\eqref{eq8}, denoted by
\begin{equation}\label{eqa4}
\omega _{u} (t)=\omega (t;t_{0} ,\phi _{u} ),~t_{0} \in [mT,(m+1)T).
\end{equation}
And the initial function $\phi _{u}(t)$ is taken as
\begin{equation}\label{eqa5}
\phi _{u}(t) =v(t;0,u),~t\in [t_{0}-\tau ,t_{0} ]\subset [0,t_{0}].
\end{equation}
It is easy to see that $\phi_{u}(T)=h(u)$, where $u>0$ is a fixed point of the $Poincar\acute{e}$ map $h$. We can prove similar monotonicity in any two consecutive $\tau$ -periodic intervals for solution $\omega _{u} (t)$ to \eqref{eq7}-\eqref{eq8}.

According to the theory of monotone dynamical systems, the right-hand function needs to have certain monotonicity regarding the delay part. For convenience, we first examine the monotonic behavior of the delay component. Let $$f_{1} (\omega)=a\frac{\omega^{2} }{\omega+c}(1-\eta \omega)~\mathrm{and}~g_{1} (\omega)=a\omega(1-\eta \omega), $$
and compute their derivatives, we have $$ {f'_{1}}  (\omega)=\frac{a\omega}{(\omega+c)^{2}}[-2\eta \omega^{2}+(1-3\eta c)\omega+2c ] ,$$
$${g'_{1}}  (\omega)=a-2a\eta \omega.$$
It is evident that the symbols for ${f'_{1}}  (\omega)$ and the quadratic expression  $l(\omega)=-2\eta \omega^{2}+(1-3\eta c)\omega+2c$ are the same. The roots of  $l(\omega)$ are given by:
$$\omega^{(1)} =\frac{1-3\eta c-\sqrt{(1-3\eta c)^{2}+16\eta c } }{4\eta },~\omega^{(2)} =\frac{1-3\eta c+\sqrt{(1-3\eta c)^{2}+16\eta c } }{4\eta }. $$
Since $2c>0$, it follows that $\omega^{(1)}<0$ and $\omega^{(2)}>0 $. Thus $f_{1} (\omega)$ is monotonically increasing on $(0,\omega^{(2)})$. Similarly, differentiating $g_{1} (\omega)$ yields that $g_{1} (\omega)$ is monotonically increasing on $(0,\frac{1}{2\eta })$.
Additionally, we note that $$\sqrt{(1-3\eta c)^{2}+16\eta c }>1+3\eta c, $$ which implies that $\omega^{(2)}>\frac{1}{2\eta }$. Therefore, both $f_{1} (\omega)$ and $g_{1} (\omega)$ are strictly increasing on the interval $(0,\frac{1}{2\eta })$. Since  $\frac{a-\mu }{a\eta } \le \frac{1}{2\eta }$(due to $1\le r\le 2$), both $f_{1} (\omega)$ and $g_{1} (\omega)$ are strictly increasing on the interval $(0,\frac{a-\mu }{a\eta })\subset(0,\frac{1}{2\eta })$.


\vspace{10pt}

\textbf{Lemma 2.2.} For solutions $\omega _{u} (t)$ defined in \eqref{eqa4} to \eqref{eq7}-\eqref{eq8}, the following conclusions hold.

$(1)$ If $\phi  _{u} (T)>u$, then
\begin{equation}\label{eqa6}
\omega  _{u} (t)\ge \omega _{u}(t-\tau ),~t\ge t_{0}.
\end{equation}
and the sequence $\left \{ \omega _{u} (n\tau ) \right \}$ is monotonically increasing.

$(2)$ If $\phi _{u} (T)=u$, then $\omega _{u} (t)=\omega (t;t_{0} ,\phi _{u} )$ is a $\tau $-periodic solution to \eqref{eq7}-\eqref{eq8}.

$(3)$ If $\phi _{u}(T)<u$, then
\begin{equation}\label{eqa7}
\omega  _{u} (t)\le  \omega _{u}(t-\tau ),~t\ge t_{0}.
\end{equation}
and the sequence $\left \{ \omega _{u} (n\tau ) \right \}$ is monotonically decreasing.
\vspace{10pt}

\textbf{Proof.}  It follows from $\phi _{u}(T) >u$ and Lemma $2.1$ that
\begin{equation}\label{eqa8}
\phi _{u}(t_{0} ) >\phi _{u}(t_{0}-\tau  ),~t_{0}\in [mT,(m+1)T).
\end{equation}
Now, we assert that $\omega  _{u} (t)\ge \omega _{u}(t-\tau ),~t\ge t_{0}$. Otherwise, there exists a $\bar{t} >t_{0} $ such that
\begin{equation}\label{eqa9}
\omega _{u} (t)>\omega _{u} (t-\tau ),~\mathrm{for~all}~t\in [mT,\bar{t} ),~\omega _{u}(\bar{t}) =\omega _{u} (\bar{t}-\tau ),
\end{equation}
and
\begin{equation}\label{eqa10}
\omega _{u} (t)<\omega _{u} (t-\tau ),~t\in (\bar{t} ,\bar{t}+\delta  ),
\end{equation}
where $\delta >0$ is sufficiently small. Then we have ${\omega}' _{u} (\bar{t} )\le {\omega}' _{u} (\bar{t}-\tau)$. Moreover, note that $\omega _{u}(t)$ is continuously differentiable,  thus we can choose $t_{1} \in (\bar{t} ,\bar{t}+\delta )$ such that ${\omega}'  _{u} (t_{1} )\le {\omega}'  _{u} (t_{1}-\tau  )$.

Next, we will show that $\bar{t} \ge (m+1)T$. In fact, if $\bar{t} \in (mT,mT+\bar{T} )$, then from \eqref{eq7} and \eqref{eqa1} we have
$${\omega}'  _{u} (t_{1} )=a\frac{\omega _{u} ^{2}(t_{1}-\tau )  }{\omega _{u} (t_{1}-\tau)+c} (1-\eta \omega _{u} (t_{1}-\tau )) -\mu\omega _{u} (t_{1} ),$$
and
$${\omega}'  _{u} (t_{1}-\tau  )=a\frac{\omega _{u} ^{2}(t_{1}-\tau )  }{\omega _{u} (t_{1}-\tau)+c} (1-\eta \omega _{u} (t_{1}-\tau )) -\mu\omega _{u} (t_{1}-\tau  ).$$
Which yields $\omega  _{u} (t_{1})\ge \omega  _{u} (t_{1}-\tau  )$, a contradiction to \eqref{eqa10}.

If $\bar{t} \in [mT+\bar{T} ,(m+1)T),$ then from \eqref{eq8} and \eqref{eqa2}, we get
$${\omega}'  _{u} (t_{1} )=a\omega _{u} (t_{1}-\tau ) (1-\eta \omega _{u} (t_{1}-\tau )) -\mu\omega _{u} (t_{1} ),$$
and
$${\omega}'  _{u} (t_{1}-\tau  )=a \omega _{u} (t_{1}-\tau ) (1-\eta \omega _{u} (t_{1}-\tau ) ))-\mu\omega _{u} (t_{1}-\tau  ).$$
Which also yields $ \omega _{u} (t_{1})\ge  \omega _{u} (t_{1}-\tau )$, a contradiction to \eqref{eqa10} again.

Therefore, we obtain $\bar{t} \ge (m+1)T.$ Without loss of generality, there exists a positive integer $k\ge m+1$ such that $\bar{t} \in [kT,(k+1)T).$

If $\bar{t} \in [kT,kT+\bar{T} )$, then from \eqref{eq7}, we have
$${\omega}'  _{u} (t_{1} )=a\frac{\omega _{u} ^{2}(t_{1}-\tau )  }{\omega _{u} (t_{1}-\tau)+c} (1-\eta \omega _{u} (t_{1}-\tau )) -\mu\omega _{u} (t_{1} ),$$
and
$${\omega}'  _{u} (t_{1}-\tau )=a\frac{\omega _{u} ^{2}(t_{1}-2\tau )  }{\omega _{u} (t_{1}-2\tau)+c} (1-\eta \omega _{u} (t_{1}-2\tau )) -\mu\omega _{u} (t_{1}-\tau  ).$$
Which yields $\omega _{u} (t_{1}-\tau)\le \omega _{u} (t_{1}-2\tau)$, a contradiction to \eqref{eqa9}.

If $\bar{t} \in [kT+\bar{T},(k+1)T )$, then from \eqref{eq8} we get
$${\omega}'  _{u} (t_{1} )=a\omega _{u} (t_{1}-\tau ) (1-\eta \omega _{u} (t_{1}-\tau )) -\mu\omega _{u} (t_{1} ),$$
and
$${\omega}'  _{u} (t_{1}-\tau  )=a\omega _{u} (t_{1}-2\tau ) (1-\eta \omega _{u} (t_{1}-2\tau ) )-\mu\omega _{u} (t_{1}-\tau  ).$$
Which also yields $\omega _{u} (t_{1}-\tau )\le \omega _{u} (t_{1}-2\tau )$, a contradiction to \eqref{eqa9}. Hence, the inequality \eqref{eqa6} is also true in any $T$-periodic interval $(kT,(k+1)T]$ with $k\ge m+1.$

In the following, we will show that the sequence $\left \{ \omega _{u} (n\tau ) \right \}$ is monotonically increasing. Otherwise, noticing from $\phi _{u} (T)>u$ that
$$\omega _{u} (\tau )=\omega _{u} (mT )\ge\omega _{u} ((m-1)T )\ge\cdots \ge\omega _{u} (0 ),$$
then there exists a positive integer $l\ge 1$ such that
\begin{equation}\label{eqa11}
\omega _{u} ((i+1)\tau )\ge \omega _{u} (i\tau ),i=0,1,\cdots ,l-1,~\mathrm{and}~\omega _{u} ((l+1)\tau )<\omega _{u} (l\tau ).
\end{equation}
a contradiction to \eqref{eqa6}, and so sequence $\left \{ \omega _{u} (n\tau ) \right \}$ is monotonically increasing. All analogous results hold with inequalities reversed, or being equaled, which completes the proof of the second and the third parts. $\hfill\square$
\vspace{10pt}

It follows from Lemma 2.1 that $\phi _{u} (t)$ is a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2} if $\phi _{u} (T)=u$. In this case, we claim that $\omega  _{u} (t)$ is also a $T$-periodic solution to \eqref{eq7}-\eqref{eq8}. Furthermore, we have the following result.
\vspace{10pt}

\textbf{Lemma 2.3.} The solution $\phi _{u} (t)$ is a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2} if and only if $\omega  _{u} (t)$ is a $T$-periodic solution to \eqref{eq7}-\eqref{eq8}.
\vspace{10pt}

\textbf{Proof.} It follows from Lemma 2.2 that, $\omega  _{u} (t)$ is a $\tau$-periodic solution to \eqref{eq7}-\eqref{eq8} if $\phi _{u} (t)$ is a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2}. In the following, we will show that $\omega  _{u} (t)$ is a $T$-periodic solution. If $t_{0} \in (mT,mT+\bar{T} ]$, then for $ t\in [t_{0},mT+\bar{T} ]$, we have
$$
\begin{aligned}
{\omega}'  _{u} (t )
&=a\frac{\omega _{u} ^{2}(t-\tau )  }{\omega _{u} (t-\tau)+c} (1-\eta \omega _{u} (t-\tau )) -\mu\omega _{u} (t)\\
&=a\frac{\omega _{u} ^{2}(t )  }{\omega _{u} (t)+c} (1-\eta \omega _{u} (t)) -\mu\omega _{u} (t).
\end{aligned}
$$
Which implies that $\omega _{u} (t)$ of \eqref{eq7}-\eqref{eq8} coincides with $\phi _{u} (t)$ of \eqref{eqa1}-\eqref{eqa2} for $ t\in [t_{0}-\tau ,mT+\bar{T} ]$. Similarly, for $t_{0}\in (mT+\bar{T},(m+1)T ]$, then for $t\in (t_{0},(m+1)T ]$, we have
$$
\begin{aligned}
{\omega}'  _{u} (t )
&=a\omega _{u} (t-\tau ) (1-\eta \omega _{u} (t-\tau )) -\mu\omega _{u} (t )\\
&=a\omega _{u} (t) (1-\eta \omega _{u} (t)) -\mu\omega _{u} (t ).
\end{aligned}
$$
That is, $\omega _{u} (t)$ of \eqref{eq7}-\eqref{eq8} coincides with $\phi  _{u} (t)$ of \eqref{eqa1}-\eqref{eqa2} on $t\in (t_{0}-\tau ,(m+1)T ]$. Following the same procedure, we can prove that $\omega _{u} (t)$ of \eqref{eq7}-\eqref{eq8}  and $\phi  _{u} (t)$ of \eqref{eqa1}-\eqref{eqa2} coincide for all $t\in (kT,(k+1)T]$ with $k\ge m+1$, which guarantees that $\omega _{u} (t)$ is also $T$-periodic.

On the other hand, suppose that $\omega_{u} (t;t_{0} ,\phi_{u} )$ is a $T$-periodic solution. Noticing that $\tau =mT$, it is easy to see $\omega_{u}(t)$ is also a  $\tau$-periodic solution to \eqref{eq7}-\eqref{eq8} with $\phi (t)\in C([t_{0}-\tau , t_{0}],(0,\frac{a-\mu }{a\eta } ))$. For any $t_{0}\in (mT,mT+\bar{T} ]$, then $t\in (t_{0},mT+\bar{T} ]$, we get
$$\begin{aligned}
{\omega}'_{u} (t;\phi)
&=a\frac{\omega _{u} ^{2} (t-\tau;\phi) }{\omega _{u} (t-\tau;\phi)+c} (1-\eta \omega _{u} (t-\tau;\phi)) -\mu\omega _{u} (t;\phi)\\
&=a\frac{\omega _{u}^{2}  (t;\phi)}{\omega _{u} (t;\phi)+c}  (1-\eta \omega _{u} (t;\phi)) -\mu\omega _{u} (t;\phi),
\end{aligned}$$
and for any $t_{0}\in (mT+\bar{T},(m+1)T]$, then $t\in (t_{0},(m+1)T]$, we have
$$\begin{aligned}
{\omega}'  _{u} (t;\phi)
&=a\omega _{u} (t-\tau;\phi) (1-\eta \omega _{u} (t-\tau;\phi)) -\mu\omega _{u} (t;\phi)\\
&=a\omega _{u} (t;\phi) (1-\eta \omega _{u} (t;\phi)) -\mu\omega _{u} (t;\phi).
\end{aligned}$$
The above two equalities imply that $\omega_{u} (t;t_{0} ,\phi_{u} )$ is also a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2}. That is, the initial function of any $T$-periodic solution of \eqref{eq7}-\eqref{eq8} must be a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2}. The proof is complete by following similar procedure in each $T$-periodic interval $(kT,(K+1)T]$ with $k\ge m+1.$ $\hfill\square$
\vspace{10pt}

Firstly, we will illustrate that $\omega _{u}(t)$ is positive. Otherwise, there exists $\bar{t} $ such that $\omega _{u}(\bar{t})=0$, thus ${\omega}' _{u}  (\bar{t})\le 0$. Without loss of generality, there exists $m_{0} $ such that either $\bar{t}\in (m_{0}T,m_{0}T+ \bar{T} ]$ or $\bar{t}\in (m_{0}T+ \bar{T}, (m_{0}+1)T ]$. If $\bar{t}\in (m_{0}T,m_{0}T+ \bar{T} ]$, then we have
\begin{equation}\label{eqa12}
0<\frac{a \omega_{u}^{2}(\bar{t}-\tau ) }{\omega_{u}(\bar{t}-\tau )+c} (1-\eta \omega_{u}(\bar{t}-\tau))\le 0.
\end{equation}
Similarly, we can also lead to contradiction as $\bar{t}\in (m_{0}T+ \bar{T}, (m_{0}+1)T]$. These contradictions verify the positivity of $\omega_{u}(t)$.

Next, we prove that $\omega _{u}(t)$ is boundness. Otherwise, there is an increasing sequence $\left \{ t_{n}  \right \} $ with $\left \{ t_{n}  \right \} \to \infty $ as $n \to \infty$, such that $\left \{ \omega \left \{ t_{n}  \right \}  \right \} $ increases monotonically to infinity as $n\to\infty $, $\omega (t_{n}-\tau  )<\omega (t_{n})$, and ${\omega}'  (t_{n})\ge 0$. Thus
$$a\omega_u(t_{n}-\tau ) (1-\eta \omega_u(t_{n}-\tau) )>a\frac{\omega_u^{2} (t_{n}-\tau)}{\omega_u(t_{n}-\tau)+c}(1-\eta \omega_u(t_{n}-\tau) )>\mu\omega_u(t_{n} )
.$$
which is impossible for large $n$. In fact, on the left is a quadratic polynomial that must have a maximum constant, while on the right is an infinite value with $n$ is infinite. Therefore, all the solutions of \eqref{eq7}-\eqref{eq8} are bounded.

Hence, the boundedness of ${\omega}'  _{u} (t)$ is guaranteed from \eqref{eq7}-\eqref{eq8} due to the uniformly continuous and bundedness of $\omega _{u}(t)$.
If $\phi_{u}(T)=u$, Then it follows from Lemma 2.3 that $\omega _{u}(t)$ is a $T$-periodic solution of \eqref{eq7}-\eqref{eq8}. Meanwhile, from Lemma 2.2 we have the sequence $\left \{ \omega _{u}(n\tau )  \right \} $ is monotonically increasing if $\phi_{u}(T)>u$, and the sequence $\left \{ \omega _{u}(n\tau )  \right \} $ is monotonically decreasing if $\phi_{u}(T)<u$. According to the monotonically bounded principle,  for any given $t\in [0,T)$, the limitation $\lim_{n \to \infty} \omega _{u}(t+n\tau )$ exists. Set
\begin{equation}\label{eqa13}
\psi _{u} (t)= \lim_{n \to \infty} w_{u} (t+n\tau ),
\end{equation}
for each $t\ge 0$. It is easy to see that $\psi  _{u} (t)$ is a $\tau$-periodic function. In the following, we will prove that $\psi _{u} (t)$ is a $T$-periodic solution of \eqref{eqa1}-\eqref{eqa2} as follows.
\vspace{10pt}

\textbf{Lemma 2.4.} The function $\psi  _{u} (t)$ defined in \eqref{eqa13} is continuous and piecewise differentiable on $[0,\infty )$, which is a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2}.
\vspace{10pt}

\textbf{Proof.} For any $t,t^{*}  \ge 0$, it follows from \eqref{eqa13} that for any $\varepsilon >0$, there exist $N_{1} =N_{1}(t,t^{*})$  such that
\begin{equation}\label{eqa14}
\left\lvert \psi _{u}(t)-\omega _{u} (t+n\tau )  \right\rvert <\frac{\epsilon }{3},~\mathrm{and}~\left\lvert \psi _{u}(t^{*} )-\omega _{u} (t^{*} +n\tau )  \right\rvert<\frac{\epsilon }{3},\ \ n\geq N_{1}.
\end{equation}
Since the uniform continuity of $\omega _{u} (t)$, for above $\varepsilon >0$, there exists a $\delta >0$, such that
\begin{equation}\label{eqa15}
\left\lvert \omega _{u} (t+n\tau )-\omega _{u} (t^{*} +n\tau ) \right\rvert <\frac{\varepsilon }{3}, ~\mathrm{for}~\left\lvert t-{t}^{*} \right\rvert <\delta .
\end{equation}
Combining \eqref{eqa14} with \eqref{eqa15}, we have
$$\begin{aligned}
\left\lvert \psi _{u}(t)-\psi _{u}(t^{*}) \right\rvert 
&\le \left\lvert \psi _{u}(t)-\omega _{u} (t+n\tau )  \right\rvert +\left\lvert \omega _{u} (t+n\tau )-\omega _{u} (t^{*}+n\tau ) \right\rvert \\
&+ \left\lvert \omega _{u} (t^{*}+n\tau ) -\psi _{u}(t^{*}) \right\rvert \\
&<\varepsilon ,
\end{aligned}$$
and hence $ \psi _{u}(t)$ is continuous.

Furthermore, for any nonnegative integer $k$ and ${t}^{*} \in (kT,kT+\bar{T})\cup (kT+\bar{T},(k+1)T)$, the limitation
$$
\begin{aligned}
\lim_{t \to {t}^{*} } \frac{\psi _{u}(t)-\psi _{u}({t}^{*} )  }{t-{t}^{*} }
&=\lim_{t \to {t}^{*} }\lim_{n \to \infty}\frac{\psi _{u} (t+n\tau )-\psi _{u} ({t}^{*} +n\tau )}{t-{t}^{*} }\\
&=\lim_{n \to \infty}\lim_{t \to {t}^{*} }\frac{\psi _{u} (t+n\tau )-\psi _{u} ({t}^{*} +n\tau )}{t-{t}^{*} }\\
&=\lim_{n \to \infty} {\psi}^{*}  _{u} ({t}^{*} +n\tau ) \\
&=a\frac{\psi _{u}^{2} ({t}^{*}) }{\psi _{u}({t}^{*})+g({t}^{*}) } (1-\eta\psi _{u}({t}^{*}) )-\mu \psi _{u}({t}^{*}) ,
\end{aligned}
$$
exists, which implies that $\psi _{u}(t)$ is differentiable at point ${t}^{*}$.
Similarly, we can show that $\psi _{u}(t)$ is left differentiable at point $t=kT$. Hence, $\psi _{u}(t)$ is piecewise differentiable on $[0,\infty )$. Clearly, $\psi _{u}(t)$ is a $\tau$-periodic solution to \eqref{eqa1}-\eqref{eqa2}. Hence, from lemma 2.3, it is also a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2}. This concludes the proof.$\hfill\square$
\vspace{10pt}

\textbf{Lemma 2.5.} Let $\omega _{1}(t)=\omega (t;t_{0},\phi _{1} )$ and $\omega _{2}(t)=\omega (t;t_{0},\phi _{2} )$ be two solutions of \eqref{eq7}-\eqref{eq8} with initial function $\phi _{1}$ and $\phi _{2}$ respectively. If $\phi _{1}(t)\le \phi _{2}(t)$ for $t\in [t_{0}-\tau ,t_{0}]$, then
\begin{equation}\label{eqa16}
\omega _{1}(t)\le \omega _{2}(t) ,~\mathrm{for}~t\ge t_{0} .
\end{equation}

\textbf{Proof.} For any $t_{0}\in [mT,(m+1)T) $, if $\omega _{1}(t)$ is defined on $[t_{0}-\tau ,t_{1}]$ for $t_{1}>t_{0}$, then $\omega _{2}(t)$ is also defined on this same interval. We claim that $\omega _{1}(t)< \omega _{2}(t),~t\in [t_{0},t_{1}] $. If false, there exists $t_{0}<\bar{s}\le t_{1} $ such that
\begin{equation}\label{eqa161}
\omega _{1}(t)< \omega _{2}(t), t_{0} \le t<\bar{s}~\mathrm{and}~\omega _{1}(\bar{s})=\omega _{2}(\bar{s}).
\end{equation}
It follows that ${\omega' _{1}} (\bar{s})\ge {\omega' _{2}}(\bar{s})$.

Noticing that $f_{1}(w) $ and $g_{1}(w) $ are strictly increasing if $w(t)\in C^{1}([t_{0},+\infty ),(0,\frac{1}{2\eta } ) )$.

If $[t_{0},t_{1}]\subset [mT,mT+\bar{T} )$, from \eqref{eqa161}, we have
$$\omega _{1}(\bar{s}-\tau  )< \omega _{2}(\bar{s}-\tau )~\mathrm{and}~\omega _{1}(\bar{s})=\omega _{2}(\bar{s}).$$
Thus
$$
\begin{aligned}
{\omega' _{1}} (\bar{s})
&=a\frac{\omega _{1}^{2}(\bar{s}-\tau )}{\omega _{1}(\bar{s}-\tau )+c }(1-\eta \omega _{1}(\bar{s}-\tau ))-\mu\omega _{1}(\bar{s})\\
&< a\frac{\omega _{2}^{2}(\bar{s}-\tau )}{\omega _{2}(\bar{s}-\tau )+c }(1-\eta \omega _{2}(\bar{s}-\tau ))-\mu\omega _{2}(\bar{s})\\
&=  {\omega' _{2}} (\bar{s}).
\end{aligned}
$$
This is a contradiction. That is, $\omega _{1}(t)< \omega _{2}(t)$, for $t\in [t_{0},t_{1}]\subset [mT,mT+\bar{T} ) $.

For the general case, let $\varepsilon >0$ and $\omega _{\varepsilon }(t)$ be the solution of
$${\omega}'(t)=a\frac{\omega ^{2}(t-\tau )}{\omega (t-\tau )+c }(1-\eta \omega(t-\tau ))-\mu\omega (t)-\varepsilon, $$
corresponding to initial data $\omega _{\varepsilon }(t)=\phi(t)-\epsilon ,t\in [t_{0}-\tau , t_{0}].$ By the results of the previous paragraph, we may conclude that $\omega _{1\varepsilon }(t)<\omega _{2}(t)$ for all $t\in [t_{0},t_{1} ]$. It can be shown that for sufficiently small $\varepsilon >0,~\omega _{1\varepsilon }(t)\to \omega _{1}(t) $ as $\varepsilon \to 0$ for all $t\in [t_{0},t_{1} ]$. Consequently, $\omega _{1}(t) =\textstyle \lim_{\varepsilon  \to 0}\omega _{1\varepsilon }(t) \le \omega _{2}(t).$ This proves the general case.

If $[t_{0},t_{1} ]\subset [mT+\bar{T} ,(m+1)T)$, similarly, we can derive the contradiction. It is also true in any $T$-periodic interval $[kT,(k+1)T)$ with $k\ge m+1$, the proof procedure is the same as Lemma 2.2. Therefore, we yields $\omega _{1}(t)< \omega _{2}(t),t\ge t_{0} $.$\hfill\square$
\vspace{10pt}

Before proceeding further, we use $\Phi (t,\cdot )$ with $t\ge 0$ to denote the solution semiflow. That is,
 $$\Phi (t,v(0)):=\phi_{u}(t)=v(t;0,u).$$
Denote
$$f(v)=\frac{av}{v+c} (1-\eta v)-\mu , ~g(v)=a(1-\eta v)-\mu. $$
It is easy to verify that $f(v),g(v)$ satisfy the following condition:

\textbf{(H1)} Both $f$ and $g$ belong to $C^{1}(R_{+},R)$ with $R_{+}=[0,\infty )$. There exists $\bar{v} =\frac{a-\mu }{a\eta }>0$ such that $f(v)<0$ and $g(v)<0$ for all $v>\bar{v}$.

\textbf{(H2)} $f(v)g(v)<0$ for all $v\in [0,\bar{v} )$. Moreover, $f(\bar{v})$ and $g(\bar{v})$ are not simultaneously zero.

It is easy to see that the wild mosquito population is stabilized at $\bar{v}=\frac{a-\mu }{a\eta } $ in the absence of sterile mosquitoes. Obviously, $v(t)\le \bar{v} ,\textstyle \lim \sup_{t \to\infty } v(t)<\bar{v}.$ From \cite{ref13} we calculate
\begin{equation}\label{a}
{h}' (0)=e^{f(0)\bar{T}+g(0)(T-\bar{T})}=e^{(a-\mu)(T-T^{*} )},
\end{equation}
and
\begin{equation}\label{aa}
{\bar{h} }' (0)=e^{f(0)\bar{T}}=e^{-\mu\bar{T} }.
\end{equation}
Clearly, ${h}' (0)=1$ if and only if $T=T^{*} $. Evidently, $c^{*}>g^{*}$, which in fact, follows directly from
$$\begin{aligned}
4\mu\eta a(c^{*}- g^{*})
&=4\mu\eta a[4a(a-\mu)-(a-\mu)^{2} ]\\
&=4\mu\eta a[2(a^{2} -\mu^{2} )+(a-\mu)^{2}]>0.
\end{aligned}$$
If $c>g^{*}$, then $f(v)<0$ for all $v>0$, and hence \textbf{(H2)} is satisfied. Next we analyze the function
$$Q(v):=\frac{f(v)}{g(v)} =\frac{-a\eta v^{2}+(a-\mu)v-\mu c }{-a\eta v^{2}+(a-\mu-a\eta c)v+c(a-\mu) },v\in (0,\bar{v} ).$$
Taking derivative gives
$${Q}' (v)=\frac{Q_{1}(v)}{(-a\eta v^{2}+(a-\mu-a\eta c)v+c(a-\mu) )^{2} },$$
where $Q_{1}(v)= a^{2}\eta ^{2}cv^{2}-2a^{2}\eta cv+ca(a-\mu-\eta\mu c)$. Note that the graph of the quadratic function $Q_{1}(v) $ is concave down with the axis of symmetry $v=\frac{1}{\eta } $. It is easy to know $\bar{v} <\frac{1}{\eta } $ so we have $$Q_{1}(0)>Q_{1}(v)> Q_{1}(\bar{v} ) , v\in (0,\bar{v} ).$$
If $c\ge c^{*} $, a sample calculation gives
$Q_{1}(0)=ca(a-\mu-\eta\mu c )\le 0$ which implies ${Q}' (v)\le 0$ for all $v\in (0,\bar{v} )$, that is $Q(v)$ is decreasing.

Thus by Theorem 4 in \cite{ref13}, we can obtain the following results.
\vspace{10pt}

\textbf{Lemma 2.6.} If $c\ge c^{*} $, then $Q(v)$ is decreasing for $ v\in (0,\bar{v} )$, and the global dynamics is determined by the sign of ${h}'(0)-1 $; namely,

(i)when ${h}'(0)\le 1 (i.e.,T\le T^{*} )$, $\phi_{0}$ is globally asymptotically.

(ii)when ${h}'(0)>1(i.e., T> T^{*})$, there exists a unique periodic solution which is globally asymptotically stable.
\vspace{10pt}

\textbf{Lemma 2.7.} The origin $\phi_{0} $ is asymptotically stable if and only if there is $\delta _{0}>0$ such that
\begin{equation}\label{eqa25}
h(u)<u,~\mathrm{for}~u\in (0,\delta _{0} ).
\end{equation}

\textbf{Proof.} Assume that $\phi_{0} $ is asymptotically stable. We need to show that \eqref{eqa25} is true. Otherwise, for any $\delta>0$, there exists $u_{0}\in (0,\delta )$, such that $h(u_{0})>u_{0}$. It follows from Lemma 2.1 that, the series $\left \{ h_{n} (u_{0}) \right \}$ is strictly increasing. Hence, $v(t;t_{0} ,u_{0})$ does not go to zero as $t\to +\infty $, a contradiction to the asymptotic stability of $\phi_{0} $. So \eqref{eqa25} holds.

On the other hand, assume \eqref{eqa25} is true. We need to prove that $\phi_{0} $ is asymptotically stable. By the virtue \eqref{aa}, we obtain ${\bar{h}}' (0)=e^{-\mu\bar{T }}$. Select $\varepsilon _{0} \in (0,\frac{\bar{v} }{2} )$ such that $\bar{h} (u)>Mu$ for $u\in (0,\varepsilon _{0} )$, where $M=\frac{1}{2} e^{-\mu\bar{T } } $. Let $\delta =\min\left \{ M\varepsilon ,\frac{m^{*}\bar{v} \varepsilon  }{\bar{v} -(1-m^{*}\varepsilon  )}  \right \}$. We will show that for any $0<\varepsilon <\varepsilon _{0} $ and $t \ge t_{0}$, there has
\begin{equation}\label{eqa27}
v(t;t_{0},u_{0})<\varepsilon ,~\mathrm{if}~u_{0} \in (0,\delta ).
\end{equation}
Otherwise, there exists $\bar{t} >t_{0}$ such that
$$ v(t)<\varepsilon,~\mathrm{for}~t_{0}\le t<\bar{t }~\mathrm{and}~v(\bar{t} )=v(\bar{t};t_{0},u_{0})=\varepsilon .$$
Thus, there exists a nonnegative integer $p$ such that $\bar{t}$ must fall into one of the following three categories.
$$ (i) \bar{t} =pT+\bar{T}; ~(ii)~ \bar{t} =pT; ~(iii)~ \bar{t} \in (pT,pT+\bar{T})\cup(pT+\bar{T},(p+1)T). $$

Case (i) $\bar{t} =pT+\bar{T}$. This is  impossible because $v(t)$ is the minimum value on $((p-1)T,pT)$.

Case (ii) $\bar{t} =pT$. In this case, we have
$$v(pT)=v(pT;t_{0},u_{0})=\varepsilon ~\mathrm{and}~v(t)<\varepsilon ~\mathrm{for}~t\in [t_{0},pT).$$
If $t_{0} \le (p-1)T$, then $v(pT)>v((p-1)T)$. From Lemma 2.1, the series $\left \{ h_{n} (u) \right \}$ is strictly increasing. Hence, the limitation $\lim_{n \to \infty}  h_{n} (u)$ exists and it is not zero. This is impossible. On the other hand, if $t_{0}>(p-1)T $, then we can reverse the solution $v(t;t_{0},u_{0})$ to $(p-1)T$. Integrating \eqref{eqa2} from $(p-1)T+\bar{T}$ to $pT$, we have
$$v((p-1)T+\bar{T} )=\frac{m^{*}\bar{v} \varepsilon}{\bar{v} -(1-m^{*})\varepsilon } \ge \delta ,$$
where $m^{*} =e^{-(a-\mu)(T-\bar{T} )} $. Since $v((p-1)T+\bar{T} )$ is the minimum of $v(t)$ for $t\in [t_{0},pT ]$, we have $w(t_{0} )\ge w((p-1)T+\bar{T} )\ge \delta$, a contradiction.

Case (iii) $\bar{t} \in (pT,pT+\bar{T})\cup(pT+\bar{T},(p+1)T)$. Then ${v}'(\bar{t})\ge 0$. This means that $\bar{t} \in (pT+\bar{T},(p+1)T)$. From \eqref{eqa2}, we get
$$v((p-1)T+\bar{T} )=\frac{m^{*}\bar{v} v(pT)}{\bar{v} -(1-m^{*})v(pT)} >\frac{m^{*}\bar{v} \varepsilon}{\bar{v} -(1-m^{*})\varepsilon } .$$
Because of $v(pT)>v(\bar{t} )=\varepsilon $, this leads to $v(t_{0})>\delta  $, a contradiction again. Therefore, \eqref{eqa27} is true, and thus $\phi_{0} $ is stable. We now just need to show that for any $u\in (0,\delta _{0} )$, that $\textstyle \lim_{t \to \infty } v(t;0,u)=0$ which however can be easily proved by using Lemma 2.1. The proof is thus complete.$\hfill\square$
\vspace{10pt}

\textbf{Lemma 2.8.}(Theorem 2 in \cite{ref7}) Assume that $0<c\le g^{*} $. Then, the following two conclusions hold.

(1) Model \eqref{eqa1}-\eqref{eqa2} has a unique globally asymptotically stable $T$-periodic solution, if and only if $\phi_{0} $ is unstable;

(2) Model \eqref{eqa1}-\eqref{eqa2} has exactly two $T$-periodic solutions if and only if $\phi_{0}$ is locally asymptotically stable. For the two $T$-periodic solutions, the one with a larger initial value is asymptotically stable, and the other is unstable.
\vspace{10pt}

\textbf{Lemma 2.9.} (1) Model \eqref{eqa1}-\eqref{eqa2} has exact two $T$-periodic solutions if $0<c\le g^{*},~T<T^{*} $. Moreover, the bigger one is asymptotically stable, the smaller one is unstable. Meanwhile, the origin $\phi_{0} $ is asymptotically stable.

(2) Model \eqref{eqa1}-\eqref{eqa2} has a unique T-periodic solution if $T\ge T^{*}$, $0<c\le g^{*} $, which is globally asymptotically stable.
\vspace{10pt}

\textbf{Proof.} In terms of Lemma 2.7 and Lemma 2.8(2), in order to proof (1) we just need to show that the origin $\phi_{0} $ is asymptotically stable. In fact $h(u)\to 0$ as $u\to 0$, from \eqref{a} we have $\lim_{u \to 0} \frac{h(u) }{u}=e^{(a-\mu)(T-T^{*} ) }<1$, as $T<T^{*} $. Thus (1) holds as $0<c\le g^{*},~T<T^{*} $. From Lemma 2.8(1), to certify (2) only need to certify $\phi_{0} $ is unstable. This can be done. From Lemma 2.7, Lemma 2.8 and \eqref{a}, we have $\phi_{0} $ is unstable as $T\ge T^{*} $. The proof is complete.$\hfill\square$

\section{Main results}

\textbf{Theorem 3.1.}
Assume that the release amount satisfies $c\in (0,g^{*}] $. The following two statements are true.

$(1)$ Model \eqref{eq7}-\eqref{eq8} has exactly two $T$-periodic solutions if $T<T^{*} $, among which the bigger one is asymptotically stable, and the other is unstable. Meanwhile, the origin $\omega _{0} $ is locally asymptotically stable if $T<T^{*} $.

$(2)$ Model \eqref{eq7}-\eqref{eq8} has a unique  $T$-periodic solution if $T\ge  T^{*} $, which is globally asymptotically stable.
\vspace{10pt}

\textbf{Proof.} \textbf{$(1)$}
Firstly, we will prove that the origin $\omega _{0}$ of \eqref{eq7}-\eqref{eq8} is asymptotically stable if $T<T^{*} $. Note  that $c\in (0,g^{*}] $ and $T<T^{*} $, then from Lemma 2.9(1), $\phi _{0}$  of the ODE model \eqref{eqa1}-\eqref{eqa2} is asymptotically stable. Therefore, there exists $u_{1} $, such that $\phi _{u} (T)<u$  for $u\in (0,u_{1} )$.

For any $\varepsilon \in (0,\bar{h}(u_{1} ) )$, set
\begin{equation}\label{eqb1}
\delta =\min\left \{ \frac{\mu }{a+\mu  } \varepsilon ,\frac{\mu }{a+\mu  } \bar{h} (h_{m}(\varepsilon ) )\right \}=\frac{\mu }{a+\mu  } \bar{h} (h_{m}(\varepsilon ) ).
\end{equation}
In the following, we will show the local stability of $\omega _{0}$ for \eqref{eq7}-\eqref{eq8}, that is, for any $\phi \in C([t_{0}-\tau, t_{0} ],(0,\delta ))$,
\begin{equation}\label{eqb2}
\omega(t):=\omega(t;t_{0},\phi  )<\varepsilon ,t\ge t_{0} .
\end{equation}
Next, we will prove it by dividing it into the following two cases.

\textbf{Case I} $t\in (t_{0},(m+1)T ]$. In this case, from \eqref{eq7}-\eqref{eq8}, we have
$$\begin{aligned}
{\omega}'(t)&\le \frac{a \omega^{2} (t-\tau )}{\omega(t-\tau )+c} (1-\eta \omega(t-\tau ) ) -\mu\omega(t)\\
&<a \omega(t-\tau )(1-\eta \omega(t-\tau ) ) -\mu\omega(t)\\
&<a\delta-\mu\omega(t),~t\in [t_{0},(m+1)T ].
\end{aligned} $$
Integrating above inequality from $t_{0}$ to $t\in (t_{0},(m+1)T ]$, we obtain
$$\omega(t)<\omega(t_{0} )e^{-\mu(t-t_{0})}+\frac{1}{\mu}a\delta (1-e^{-\mu(t-t_{0} ) } )
<\delta +\frac{1}{\mu}a \delta =\frac{\mu+a}{\mu}\delta \le \varepsilon  .$$
Thus, the inequality \eqref{eqb2} holds for $t\in (t_{0},(m+1)T ]$.

\textbf{Case II} $t>(m+1)T$. From \eqref{eqb1}, we reach
\begin{equation}\label{eqb3}
\phi _{\varepsilon } (t)\ge \bar{h} (h_{m}(\varepsilon ) )\ge \frac{a +\mu }{\mu}\delta >\delta >\phi (t),t\in [0,(m+1)T ] .
\end{equation}
Which, together with Lemma 2.5 imply that
\begin{equation}\label{eqb4}\omega(t;t_{0},\phi  )\le \omega_{\varepsilon } (t),~t\ge (m+1)T.
\end{equation}
Note that $\phi _{\varepsilon } <\varepsilon $, which together with Lemma 2.2(3) imply that $\omega_{\varepsilon } (t)\le \omega_{\varepsilon } (t-\tau )$ for $t\ge t_{0} $. Then it follows from Lemma 2.5 that
$$\omega(t;t_{0},\phi  )\le \omega_{\varepsilon } (t)\le \omega_{\varepsilon } (t-\tau )=\phi _{\varepsilon }(t-\tau )<\varepsilon ,~t\in ((m+1)T,(m+2)T]. $$
That is, the inequality \eqref{eqb2} is true for $t\in ((m+1)T,(m+2)T ]$. Repeatedly using Lemma 2.5 and \eqref{eqa7}, we can prove that \eqref{eqb2} holds in each $T$-periodic interval $[kT,(k+1)T]$ for all $k\ge m+2$. Thus, the inequality \eqref{eqb2} holds for all $t>(m+1)T$. From Case I and II, we completes the proof of the stability of $\omega_{0} $ to \eqref{eq7}-\eqref{eq8}.

Consider the attractive of $\omega_{0} $. We shall prove that for any $\phi \in C([t_{0}-\tau ,t_{0} ],(0,\delta ))$,
\begin{equation}\label{eqb5}
\lim_{t \to \infty} \omega(t;t_{0},\phi  )=0,
\end{equation}
where $\delta$ defined in \eqref{eqb1}. From \eqref{eqb4}, we only need to prove
\begin{equation}\label{eqb6}
\lim_{t \to \infty} \omega_{\varepsilon } (t)=0.
\end{equation}
For any $\tau \in [0,T)$, from Lemma 2.2(3) the sequence $\left \{ \omega_{\varepsilon }(t+n\tau )  \right \} $ is monotonically decreasing with respect to $n$. Note that $\omega_{\varepsilon }(t+n\tau )$ is positive. Thus, the limitation $\lim_{n \to \infty} w_{\varepsilon } (t+n\tau )$ exists. From Lemma 2.4, the function $\psi _{\varepsilon } (t)=\lim_{n \to \infty} w_{\varepsilon } (t+n\tau )$ is a $T$-periodic solution of \eqref{eqa1}-\eqref{eqa2}, which leads to $\psi _{\varepsilon } (t)\equiv 0$. Thus, \eqref{eqb6} is true as well as \eqref{eqb5}. So the origin $\omega_{0} $ is attractive.

Hence, the origin $\omega _{0} $ is locally asymptotically stable.

Regarding $c\in (0,g^{*}] $ and $T<T^{*} $, it follows from Lemma 2.9(1) that model \eqref{eqa1}-\eqref{eqa2} has exactly two $T$-periodic solutions, denoted by $\phi _{u_{1} }(t) $ and $\phi _{u_{2} }(t) $ with $u_{1} <u_{2} $. Moreover, $\phi _{u_{1} }(t) $ is unstable and  $\phi _{u_{2} }(t) $ is asymptotically stable. From Lemma 2.3, $\omega_{u_{1} }(t) $ and $\omega _{u_{2} }(t) $ are exactly two $T$-periodic solutions of \eqref{eq7}-\eqref{eq8}.

Next, we will prove that the $T$-periodic solution $\omega_{u_{1} } (t)$ is unstable.

Note that
\begin{equation}\label{eqb7}
\phi _{u}(T)<u,u\in (0,u_{1}),~\mathrm{and}~\phi _{u}(T)>u,u\in (u_{1},u_{2}).
\end{equation}
Thus, for all $t\ge t_{0} $, from (3.7) and Lemma 2.2, we have
$$\omega_{u} (t)\le \omega_{u} (t-\tau ),~u\in (0,u_{1} ),~\mathrm{and}~\omega_{u} (t)\ge  \omega_{u} (t-\tau ),~u\in (u_{1},u_{2}).$$
That is, no matter how close the initial function $\phi _{u_{1}} $ is to $\omega_{u_{1} } (t)$ for $t\in [t_{0}-\tau ,t_{0} ]$, the solutions $\omega_{u} (t)$ will drift away from $\omega_{u_{1} } (t)$. Therefore, the solution $\omega_{u_{1} } (t)$ of \eqref{eq7}-\eqref{eq8} is unstable.

Finally, we will prove that the $T$-periodic solution $\omega_{u_{2} } (t)$ of \eqref{eq7}-\eqref{eq8} is asymptotically stable.

We first prove the $T$-periodic solution $\omega_{u_{2} } (t)$ of \eqref{eq7}-\eqref{eq8} is attractive. We only need to prove that, if
$$\phi (t)>\omega_{u_{1} } (t),~\mathrm{and}~\left\lvert \phi (t)-\omega_{u_{2} } (t) \right\rvert <\min_{t\in [0,T]} \left\lvert \omega_{u_{2} } (t)-\omega_{u_{1} } (t) \right\rvert,t\in [t_{0}-\tau , t_{0}] ,$$
then
\begin{equation}\label{eqb8}
\lim_{t \to \infty} [\omega(t;t_{0},\phi )-\omega_{u_{2} }(t) ]=0.
\end{equation}
Denote
$$\eta _{1} =\min_{t\in [t_{0}-\tau , t_{0}]}\phi (t)~\mathrm{and}~ \eta _{2} =\max_{t\in [t_{0}-\tau , t_{0}]}\phi (t).$$
By choosing
\begin{equation}\label{eqb9}
\underline{u} =\min\left \{ \eta _{1},h^{-1} _{m+1}(\eta _{1})   \right \} ~\mathrm{and}~\bar{u} =\max\left \{ \bar{h}^{-1}(\eta _{2}),h^{-1} _{m}(\bar{h}^{-1}(\eta _{2} ) )   \right \} .\end{equation}
Thus we have
$$\phi _{\underline{u} }(t)\le \max\left \{\underline{u},h_{m+1}(\underline{u})   \right \}\le \eta _{1}\le \phi (t),t\in [t_{0}-\tau ,t_{0} ], $$
and
$$\phi _{\bar{u } }(t)\ge \min\left \{\bar{h}(\bar{u} ),\bar{h}(h_{m} (\bar{u} ))  \right \}\ge \eta _{2}\ge  \phi (t),t\in [t_{0}-\tau ,t_{0} ].  $$
Thus, from Lemma 2.5 we have
$$\omega_{\underline{u}} (t)\le \omega(t)\le \omega_{\bar{u} } (t),~t\ge t_{0}. $$
To prove \eqref{eqb8}, we only need to prove
\begin{equation}\label{eqb10}
\lim_{t \to \infty} \left\lvert \omega_{\underline{u}}(t)-\omega_{u_{2} }(t) \right\rvert=0~\mathrm{and}~\lim_{t \to \infty}\left\lvert \omega_{\bar{u} }(t)-\omega_{u_{2} }(t) \right\rvert=0.
\end{equation}
It is easy to see that both $\left \{\omega_{\underline{u}}(t+n\tau ) \right \} $ and $\left \{\omega_{\bar{u} }(t+n\tau ) \right \} $ are monotone. Therefore, from Lemma 2.4 we have
$$\psi _{\underline{u} }(t) =\lim_{n \to \infty} \omega_{\underline{u} }(t+n\tau )  ,\psi _{\bar{u}}(t) =\lim_{n \to \infty} \omega_{\bar{u} }(t+n\tau )$$
are $T$-periodic solutions to \eqref{eqa1}-\eqref{eqa2}. Hence, $\psi _{\underline{u} }(t) =\psi _{\bar{u}}(t) =\omega_{u_{2}  }(t)  $. Hence, \eqref{eqb10} holds as well as \eqref{eqb8}.
which proves that \eqref{eqb10} and hence \eqref{eqb8} holds. Thus we complete the proof of the attractiveness of $\omega_{u_{2} } (t)$.

Now, we will show that the solution $\omega_{u_{2}  }(t)  $ of \eqref{eq7}-\eqref{eq8} is stable. For any $\varepsilon >0$, there exists $\delta =\delta (\varepsilon ,t_{0} )>0$ such that, for $\phi \in C([t_{0}-\tau , t_{0}],(0,\frac{a-\mu}{a\eta } ))$ and $\left\lvert \phi (t) -\omega_{u_{2} }(t) \right\rvert<\delta  $, $t\in [t_{0}-\tau , t_{0}]$,
\begin{equation}\label{eqb11}
\left\lvert \omega(t;t_{0},\phi )-\omega_{u_{2} } (t) \right\rvert<\varepsilon ,~t\ge t_{0}.
\end{equation}
Note that $\omega_{u_{2} }(t)$ is also a stable solution of \eqref{eqa1}-\eqref{eqa2}.  That is, there exists $\delta _{1} =\delta (\varepsilon )\in (0,\frac{u_{2}- u_{1}}{2} )$ such that, for $\left\lvert u-u_{2}  \right\rvert<\delta _{1}$,
\begin{equation}\label{eqb12}
\left\lvert \phi _{u}(t) -\omega_{u_{2}}(t)\right\rvert <\frac{\varepsilon }{2} ,t\ge t_{0}. \end{equation}
Set
\begin{equation}\label{eqb13}
\delta _{2} =\min_{t\in [0, t_{0}]} [\phi _{_{u_{2}} +\delta _{1}  }(t)-\omega_{u_{2}}(t)],~\mathrm{and}~\delta _{3}=\min_{t\in [0, t_{0}]}[\omega_{u_{2}}(t)-\phi _{{u_{2}}-\delta _{1}  }(t)]  .
\end{equation}
It is obvious that $\phi _{u_{2}+\delta _{1}  }(t)\le u_{2}+\delta _{1}$ and $\phi _{u_{2}-\delta _{1}  }(t)\ge  u_{2}-\delta _{1}$ for all $t\in [0,t_{0}]$. Thus $0<\delta _{2} <\delta _{1} $ and $0<\delta _{3} <\delta _{1} $. By choosing $\delta =\min\left \{ \delta _{2},\delta _{3} \right \} $, then from \eqref{eqb13} and $\left\lvert \phi (t)-\omega_{u_{2} }(t) \right\rvert<\delta$ for $t\in [t_{0}-\tau , t_{0}]$, we have
$$ \phi (t)<\omega_{u_{2} }(t)+\delta \le \omega_{u_{2} }(t)+\delta_{2}\le \phi _{u_{2}+\delta _{1} } (t),~t\in [t_{0}-\tau ,t_{0} ],  $$
and
$$ \phi (t)>\omega_{u_{2} }(t)-\delta \ge \omega_{u_{2} }(t)-\delta_{3}\ge  \phi _{u_{2}-\delta _{1} } (t),~t\in [t_{0}-\tau ,t_{0} ].  $$
Thus, it follows that for $t\ge t_{0} $,
$$\omega_{u_{2}-\delta _{1}}(t)\le \omega(t;t_{0},\phi )\le \omega_{u_{2}+\delta _{1}} (t),~\omega_{u_{2}-\delta _{1}}(t)\le \omega_{u_{2} } (t)\le \omega_{u_{2}+\delta _{1}} (t)  .$$
Therefore,
$$\left\lvert \omega(t;t_{0},\phi )- \omega_{u_{2}}(t) \right\rvert \le \omega_{u_{2}+\delta _{1}}(t)-\omega_{u_{2}-\delta _{1}}(t),~t\ge t_{0}.$$
In order to show \eqref{eqb11}, we only need to verify that
\begin{equation}\label{eqb14}
 \omega_{u_{2}+\delta _{1}}(t)-\omega_{u_{2}-\delta _{1}}(t)<\varepsilon ,~\mathrm{for}~t\ge t_{0}.
\end{equation}
Since
$$ \phi _{u_{2}+\delta _{1}}(T)<u_{2}+\delta _{1},~\mathrm{and}~\phi _{u_{2}-\delta _{1}}(T)>u_{2}-\delta _{1}. $$
It follows from Lemma 2.2 that
$$\omega_{u_{2}+\delta _{1}}(t_{0})\le \omega_{u_{2}+\delta _{1}}(t_{0}-\tau ),~\mathrm{and}~ \omega_{u_{2}-\delta _{1}}(t_{0})\ge  \omega_{u_{2}-\delta _{1}}(t_{0}-\tau ).$$
Then, for $t\in [t_{0},(m+1)T ]$, from Lemma 2.3, we have
$$ \omega_{u_{2}+\delta _{1}}(t)\le \omega_{u_{2}+\delta _{1}}(t-\tau )=\phi _{u_{2}+\delta _{1}} (t-\tau),~t_{0}\in [mT,(m+1)T). $$
and
$$\omega_{u_{2}-\delta _{1}}(t)\ge  \omega_{u_{2}-\delta _{1}}(t-\tau )=\phi _{u_{2}-\delta _{1}} (t-\tau) .$$
Hence, for any $t\in [t_{0},(m+1)T ]$, from \eqref{eqb12} we have
$$\omega_{u_{2}+\delta _{1}}(t)-\omega_{u_{2}}(t)\le\phi _{u_{2}+\delta _{1}} (t-\tau )- \omega_{u_{2}} (t-\tau )<\frac{\varepsilon }{2}, $$
and
$$\omega_{u_{2}}(t)-\omega_{u_{2}-\delta _{1}}(t)\le \omega_{u_{2}} (t-\tau )-\phi _{u_{2}-\delta _{1}} (t-\tau )<\frac{\varepsilon }{2}. $$
Thus,
$$\omega_{u_{2}+\delta _{1}}(t)-\omega_{u_{2}-\delta _{1}}(t)<\varepsilon . $$
That is, the inequality \eqref{eqb14} is true for $t\in [t_{0},(m+1)T ]$. An induction argument proves that \eqref{eqb14} is true for $t\in [kT,(k+1)T ]$ with $k\ge m+1$, and hence \eqref{eqb11} and \eqref{eqb14} are true. This proves that $\omega_{u_{2}}(t)$ is stable. Therefore, it is globally asymptotically stable.

\textbf{$(2)$} It follows from Lemma 2.9(2) that, model \eqref{eqa1}-\eqref{eqa2} has unique $T$-periodic, denoted by 
$\phi_{u^{*} }(t)$. Thus, from Lemma 2.3, model \eqref{eq7}-\eqref{eq8} has unique $T$-periodic solution. Denote the unique $T$-periodic solution by $\omega_{u^{*} }(t) $. We first show that $\omega_{u^{*} }(t) $ to \eqref{eq7}-\eqref{eq8} is stable. For any $\varepsilon >0$, we only need to prove that there exists $\delta =\delta (\varepsilon ,t_{0})>0$ such that for $\phi \in C([t_{0} -\tau ,t_{0}],(0,\frac{a-\mu }{a\eta }))$ and $\left\lvert \phi (t)-\omega_{u^{*} }(t) \right\rvert<\delta$,
\begin{equation}\label{eqb15}
\left\lvert \omega(t;t_{0},\phi )-\omega_{u^{*} }(t) \right\rvert<\varepsilon .
\end{equation}
From Lemma 2.9(2), $\phi_{u^{*} }(t)$ is stable for \eqref{eqa1}-\eqref{eqa2}. Then, for above $\epsilon>0$, there exists $\delta _{1}>0$, such that $\left\lvert \phi_{u}(t)-\phi_{u^{*}}(t) \right\rvert< \frac{\varepsilon  }{2}$  for $\left\lvert u-u^{*} \right\rvert< \delta _{1}$ and $t\ge 0$.
Choose
$$\delta =\min\left \{ \min_{t\in [0,t_{0}]}\left\lvert \phi _{u^{*}+\delta _{1} }(t)-\phi_{u^{*}}(t)\right\rvert, \min_{t\in [0,t_{0}]}\left\lvert \phi_{u^{*}}(t)-\phi _{u^{*}-\delta _{1} }(t)\right\rvert  \right \}. $$
Then, for any $\phi \in C([t_{0} -\tau ,t_{0}],(0,\frac{a-\mu }{a\eta }))$ and $\left\lvert \phi(t)-\phi_{u^{*}}(t) \right\rvert < \delta $ for $t\in [t_{0} -\tau ,t_{0}]$, we have
$$\phi _{u^{*}-\delta _{1} }(t)\le \phi (t)\le \phi _{u^{*}+\delta _{1} }(t),~t\in [t_{0} -\tau ,t_{0}].$$
Which leads to
$$\omega _{u^{*}-\delta _{1} }(t)\le \omega(t;t_{0},\phi  )\le \omega_{u^{*}+\delta _{1} }(t),~t\ge t_{0} .$$
The proof of
\begin{equation}\label{eqb16}
\omega_{u^{*}+\delta _{1} }(t)-\omega_{u^{*}-\delta _{1} }(t)<\varepsilon ,~t\ge t_{0} ,
\end{equation}
is the same as the proof of \eqref{eqb14} with $u_{2} $ replaced by $u^{*}$. Hence, the inequality \eqref{eqb16} holds, which completes the proof. That is, $\omega_{u^{*} }(t) $ of \eqref{eq7}-\eqref{eq8} is stable.

We next show $\omega_{u^{*}}(t)$ to \eqref{eq7}-\eqref{eq8} is globally attractive. That is, for any initial condition $\phi \in C([t_{0} -\tau ,t_{0}],(0,\frac{a-\mu}{a\eta } ))$, we have
$$\lim_{t \to \infty}[\omega(t;t_{0},\phi)-\omega_{u^{*}}(t)]=0. $$
Again, following the lines as above to prove the attractivity of $\omega_{u_{2} }(t)$ of \eqref{eq7}-\eqref{eq8}, we can prove that $\omega(t;t_{0},\phi)$ is sandwiched between two special solutions $\omega_{\bar{u} } (t)$ and $\omega_{\underline{u} } (t)$, with $\bar{u}$ and $\underline{u}$ defined the same as in $(3.10)$, which satisfy
$$\lim_{t \to \infty} \omega_{\bar{u} } (t)=\lim_{t \to \infty} \omega_{\underline{u} } (t)=\omega_{u^{*}}(t).$$
Therefore, $\omega_{u^{*} }(t) $ is globally attractive.
This completes the proof Theorem 3.1. $\hfill\square$
\vspace{10pt}

\textbf{Theorem 3.2.}
Assume that the release amount satisfies $c\ge c^{*}$. The following two statements are true.

$(1)$ The origin $\omega _{0} $ of \eqref{eq7}-\eqref{eq8} is globally asymptotically stable if and only if $T\le T^{*}$.

$(2)$ Model \eqref{eq7}-\eqref{eq8} has a unique globally asymptotically stable $T$-periodic solution if $T> T^{*} $.
\vspace{10pt}

\textbf{Proof.} \textbf{$(1)$} It follows from Lemma 2.6 (i) that, if $c\ge c^{*} $ and $T\le T^{*}$, then $\phi_{0} $ of \eqref{eqa1}-\eqref{eqa2} is globally asymptotically stable. Which implies that $\phi _{u}(T)<u $, for all $u>0$. Hence, for any $\varepsilon >0$, there exists $\delta _{1}\in (0, \varepsilon )$ such that, for any $u\in (0, \delta _{1})$,
\begin{equation}\label{eqb17}
\phi _{u}(t)<\varepsilon.
\end{equation}
Denote $\delta =\frac{\mu }{a+\mu }\bar{ h}(h_{m}(\delta _{1} ) )$. In order to show that $\omega_{0} $ of \eqref{eq7}-\eqref{eq8} is stable. We need to prove  that if $\phi \in C([t_{0}-\tau , t_{0}],(0,\delta ))$, then
\begin{equation}\label{eqb18}
\omega(t;t_{0},\phi  )<\varepsilon ,t\ge t_{0} .
\end{equation}
From \eqref{eq7}-\eqref{eq8}, for $t\in [t_{0},(m+1)T ]$, we have
$${\omega}' (t;t_{0},\phi  ):={\omega}' (t)\le a\omega(t-\tau )-\mu \omega(t)<a\delta-\mu \omega(t).$$
Or equivalently,
\begin{equation}\label{eqb19}
\omega(t;t_{0},\phi)<\omega(t_{0})e^{-\mu(t-t_{0})}+ a\delta \frac{1}{\mu}(1-e^{-\mu(t-t_{0})} )<\frac{a+\mu }{\mu }\delta =\bar{h }(h_{m}(\delta _{1} ) )<\varepsilon .
\end{equation}
Hence, the inequality \eqref{eqb18} is true for $t\in (t_{0},(m+1)T ]$.

For $t\in ((m+1)T,(m+2)T ]$, it is obviously from Lemma 2.1 that
$$\phi (t)<\frac{\mu }{a+\mu} \bar{h} (h_{m}(\delta _{1}))<\bar{h} (h_{m}(\delta _{1}))\le \phi _{\delta _{1}}(t)<\phi _{\varepsilon }(t),t\in [t_{0}-\tau ,(m+1)T ].$$
Then it follows from \eqref{eqb17},  Lemma 2.1 and Lemma 2.5 that
$$\omega(t;t_{0},\phi)<\omega_{\varepsilon }(t)\le \omega_{\varepsilon }(t-\tau ) =\phi _{\varepsilon }(t-\tau )<\varepsilon ,~t\in ((m+1)T,(m+2)T].$$
Hence,  the inequality  \eqref{eqb18} is true for $t\in ((m+1)T,(m+2)T]$. Which, together with \eqref{eqb19} implies that \eqref{eqb18} is true for $t\in (t_{0} ,(m+2)T]$.
Repeatedly using Lemma 2.5 and Lemma 2.1, we can prove that \eqref{eqb18} holds in each $T$-periodic interval $[kT,(k+1)T]$ for all $k\ge m+2 $. Therefore, $\omega_{0} $ of \eqref{eq7}-\eqref{eq8} is stable.

Next we will show that $\omega_{0} $ of \eqref{eq7}-\eqref{eq8} is globally attractive. That is, for any $\phi \in C([t_{0}-\tau ,t_{0}],(0,\frac{a-\mu }{a\eta }))$, we need to show that
\begin{equation}\label{eqb20}
\lim_{t \to \infty}\omega(t;t_{0},\phi)=0.
\end{equation}
From the proof of Theorem 3.1, if $\bar{u}>h^{-1} _{m} (\bar{h}^{-1}(\eta _{2})) $, then
$$\phi (t)<\phi _{\bar{u} }(t),~t\in [t_{0}-\tau ,t_{0}]. $$
Hence, from Lemma 2.5 we yields
$$\omega_{\bar{u} }(t)\ge \omega(t;t_{0},\phi ),t\ge t_{0}.$$
To prove $(3.20)$, we only need to show
\begin{equation}\label{eqb21}
\lim_{t \to \infty} \omega_{\bar{u} }(t)=0.
\end{equation}
Notice $\phi _{\bar{u} }(T)<\bar{u} $, then the sequence $\left \{ \omega_{\bar{u} }(t+n\tau ) \right \} $ is monotonically decreasing. Therefore, from Lemma 2.4, the function
$\psi _{\bar{u}}(t)=\lim_{n \to \infty}  \omega_{\bar{u} }(t+n\tau )$ is a $T$-periodic solution to \eqref{eqa1}-\eqref{eqa2}. Hence, $\psi _{\bar{u}}(t)\equiv 0$ owing to the global asymptotic stability of $\omega_{0} $, thus \eqref{eqb21} holds.

On the other hand, assume that $\omega _{0} $ is globally asymptotically stable. We will prove $T\le T^{*} $. For any $u>0$, from \eqref{a}, we have
$${h}' (0)=e^{(a-\mu )(T-T^{*} ) }.$$
Thus, to show $T\le T^{*} $, it suffices to verify that ${h}' (0)\le 1$.

Suppose ${h}' (0)>1$, then there exists $\delta _{*}>0 $ small enough such that $h(u)>u$, for $u\in (0,\delta _{*})$.  Which together with Lemma 2.1 implies that $\left \{ h_{n}(u)  \right \} $ is strictly increasing. Thus, $\lim_{n \to \infty}  h_{n}(u)\ne 0 $ for $u\in (0,\delta _{*})$, a contradiction. Hence $T\le T^{*}$.

\textbf{$(2)$} It follows from Lemma2.6(ii) that, if $c\ge  c^{*} $ and $T>T^{*}$, then \eqref{eqa1}-\eqref{eqa2} has a unique $T$-periodic solution $\phi _{u^{*} } (t)$, which is globally asymptotically stable. Hence, according to Lemma 2.3, the existence and uniqueness of the $T$-periodic solution $\omega  _{u^{*} } (t)$ of \eqref{eq7}-\eqref{eq8} can be guaranteed. The proof of the global asymptotic stability of $\omega_{u^{*} }(t) $ is the same as Theorem 3.1. So we omit it here. $\hfill\square$
\vspace{10pt}

\textbf{Theorem 3.3.}
Assume that the release amount satisfies $g^{*}<c<c^{*}  $. Then the following two statements are true.

$(1)$ The origin $\omega_{0}$ of \eqref{eq7}-\eqref{eq8} is asymptotically stable if and only if $T<T^{*}$.

$(2)$ System \eqref{eq7}-\eqref{eq8} has a unique globally asymptotically stable $T$-periodic solution if and only if $T\ge  T^{*} $.
\vspace{10pt}

\textbf{Proof.}
\textbf{(1)} Assume $\omega_{0}$ is asymptotically stable. We need to show $T<T^{*}$. From \eqref{a}, we have ${h}' (0)=e^{(a-\mu)(T-T^{*} )}$. It suffices to verify ${h}' (0)<1$. Assume by contradiction that ${h}' (0)\ge 1$. For case ${h}' (0)>1$, there exists $\delta>0$ small enough such that $h(u)>u$ for $u\in (0,\delta)$. From Lemma 2.1, $\left \{ h_{n}(u)  \right \} $ is strictly increasing for $u\in (0,\delta )$. Thus, $\lim_{n \to \infty}  h_{n}(u)\ne 0$, a contradiction to the asymptotical stability of $\phi _{0} $, it also contradicts to the asymptotical stability of $\omega_{0}$. Similarly, for each ${h}' (0)=1$, we obtain $T=T^{*} $. Which yields the existence of at least one $T$-periodic solution, which contradicts to the asymptotical stability of $\omega_{0}$. Thus ${h}' (0)<1$, that is, $T<T^{*}$.

Hereafter, if $T<T^{*}$, we need to prove that $\omega_{0}$ is asymptotically stable. From Theorem 3.3(2) in \cite{ref6}, it showed that $\phi_{0}$ of \eqref{eqa1}-\eqref{eqa2} is asymptotically stable. Therefore, the proof process for the asymptotic stability of $\omega _{0}$ of \eqref{eq7}-\eqref{eq8} is the same as Theorem 3.2(1) and it is omitted here.

\textbf{(2)} It follows from \cite{ref6} (Theorem 3.3(1) and Theorem 3.5) that, there is a unique globally asymptotically stable $T$-periodic solution $\phi _{u^{*} } (t)$ for \eqref{eqa1}-\eqref{eqa2} if $T\ge T^{*} $ and $g^{*}<c<c^{*}  $, then the proof process for the global asymptotic stability of $T$-periodic solution for \eqref{eq7}-\eqref{eq8} is the same as Theorem $3.1$. Here, we not prove it.

Next, if \eqref{eq7}-\eqref{eq8} has a unique globally asymptotically stable $T$-periodic solution, then we will show $T\ge  T^{*} $. In fact, if $T<T^{*} $, then, by the conclusion in Theorem 3.3(1) shown above, the origin $\omega_{0}$ is asymptotically stable, again a contradiction to the assumption. Thus there is a unique globally asymptotically stable $T$-periodic solution if and only if $T\ge  T^{*} $. So we finish the proof. $\hfill\square$
\vspace{10pt}

\section{Numerical Simulations}

Concerning the natural mortality $\mu$ of adult mosquitoes, we assume that the number of surviving mosquitoes obeys the differential equation ${\omega}' (t)=-\mu \omega (t)$, whose general solution is $\omega (t)=\omega (0)e^{-\mu t}  $. Thus, $\mu =\frac{\ln 2}{h_{f} } $, where $h_{f}$ is the half-life of adult mosquitoes. While the lifespan of $Aedes~albopictus$ is influenced by various external factors like seasonal changes, temperature fluctuations, and food availability, it follows from \cite{ref24} that the half-life of $Aedes~albopictus$ ranges from 5 to 30 days \cite{ref5, ref25}, leading to
$$\mu\in [\frac{\ln 2}{30}, \frac{\ln 2}{5}]\approx [0.0231,0.14].$$
Due to a female mosquito laying between 28.36 and 224.5 eggs per batch \cite{ref15}. Meanwhile, the average probability of egg survival through larval, pupal stages and subsequent emergence as an adult is from $\frac{4}{1000}$ \cite{ref16} to $\frac{(5+312)}{2000} $ \cite{ref25}. This leads to the estimation of $a$ as $a\in [0.11344,19.7]$. It is widely accepted that the maturation period of wild mosquitoes ranges from 4 days to a month \cite{ref5}, thus $\tau \in [4,30]$. We choose:
\begin{equation}\label{c}
a=0.19,\mu=0.11,\tau =20,\bar{T} =2.5.
\end{equation}
Assuming an initial population of 1000 individuals. The density-dependent mortality rate is relatively difficult to define. Recall that without intervention of Wolbachia-infected male mosquitoes, wild mosquitoes stabilize at $\bar{v} =\frac{a-\mu }{a\eta } $, which could be interpreted as the carrying capacity of wild mosquitoes. we take $\eta \approx 0.000421$ such that
$\frac{a-\mu }{a\eta } \approx 1000$. Thus, by calculation we have $c^{*}\approx 1727.489,~g^{*}\approx 181.841,~T^{*}\approx 5.9375$.

\subsection{Example 4.1.}
Take parameters $c=120$, $m=4$ and $T=5<T^{*}$, it follows from Theorem 3.1 (1) that there exists two $T$-periodic solutions, $w_{0} $ and the bigger T-periodic solution is asymptotically stable as shown in the left of Fig. 1. If $m=2,~T=10>T^{*}$, it follows from Theorem 3.1 (2) that there exists a unique globally asymptotically stable $T$-periodic solution as shown in the right of Fig. 1.


% 图片并排放置
\begin{figure}[ht]
  \centering
  % 第一张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{311}
    \label{fig:image1}
  \end{minipage}
  % 第二张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{312}
    \label{fig:image2}
  \end{minipage}
  \hfill
  \caption{The solution graphs of \eqref{eq7}-\eqref{eq8} for different values of $c$ and $T$ satisfying conditions in Theorem 3.1.}
\end{figure}

\subsection{Example 4.2.}
For fixed parameters in \eqref{c}, we let $c=3000,~m=2,~T=10$ then $c>c^{*},T>T^{*}$, Fig.2(right) manifests that \eqref{eq7}-\eqref{eq8} has a unique periodic solution, which is globally asymptotically stable. Taking $m=4$, $T=5$, we get $c>c^{*},T<T^{*}$. Hence, it follows from Theorem 3.2(1) that the origin of \eqref{eq7}-\eqref{eq8} is globally asymptotically stable.


\begin{figure}[ht]
  \centering
  % 第一张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{321}
    \label{fig:image1}
  \end{minipage}
  % 第二张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{322}
    \label{fig:image2}
  \end{minipage}
  \hfill
  \caption{The solution graphs of \eqref{eq7}-\eqref{eq8} for different values of $c$ and $T$ satisfying conditions in Theorem 3.2.}
\end{figure}

\subsection{Example 4.3.}
For given parameters in \eqref{c}, we let $c=1200,~m=2,~T=10$ then $g^{*}<c<c^{*},T>T^{*}$, the first graph of Fig.3 supports our theoretical results in Theorem 3.3(1). Taking $m=4$ and $T=5<T^{*}$, we obtain the last image in Fig.3.

\begin{figure}[ht]
  \centering
  % 第一张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{331}
    \label{fig:image1}
  \end{minipage}
  % 第二张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{332}
    \label{fig:image2}
  \end{minipage}
  \hfill
  \caption{The solution graphs of \eqref{eq7}-\eqref{eq8} for different values of $c$ and $T$ satisfying conditions in Theorem 3.3.}
\end{figure}

\subsection{Example 4.4.}
If $r$ is much greater than 2, the system will exhibit complex dynamic behaviors. Let us consider $a = 2.43$ and $\mu = 0.03$, where $a$ is 81 times larger than $\mu $. In this case, the corresponding values are $c^{*}=80808.08,~g^{*}=19952.613,~T^{*}=2.53125$. Take $\tau =12$ and keep other parameter values remain unchanged. When $c = 80900,~m = 4$, and $T = 3$, it follows that $c>c^{*},T>T^{*}$, as shown in the left figure of Figure 4. However, when the parameter $T$ is adjusted to 2.8, as depicted in the right figure, it becomes challenging to observe the periodic solution. This suggests that the dynamics of the model have undergone complex transformations.

\begin{figure}[ht]
  \centering
  % 第一张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{341}
    \label{fig:image1}
  \end{minipage}
  % 第二张图
  \begin{minipage}[b]{0.4\textwidth}
    \includegraphics[width=\textwidth]{342}
    \label{fig:image2}
  \end{minipage}
  \hfill
  \caption{If $r$ is much larger than 2 then the switching system exhibits complex dynamical behavior. }
\end{figure}

\section{Conclusions and discussions}

In this paper,we investigate the existence, uniqueness and stability of $T$-periodic solutions to model \eqref{eq7}-\eqref{eq8} by using $Poincar\acute{e} $ map method ,  comparison principle, etc. These results may be helpful for optimal control of mosquito vector or mosquito-borne diseases. This paper defines two thresholds, $g^{*} $ and $c^{*}$, based on which we conduct a relatively complete analysis of the dynamics of models \eqref{eq7}-\eqref{eq8}.

For the case of $c\in (0,g^{*}] $,  we describe the local asymptotic stability of the origin $\omega _{0}$ and the stability of two periodic solutions when $T<T^{*} $, where the larger periodic solution is stable and the smaller one is unstable. When $T\ge T^{*} $, the unique $T$-periodic solution is globally asymptotically stable.For the case of $c\ge c^{*} $, the global asymptotic stability of the origin $\omega _{0} $ and the existence of globally asymptotically stable $T$-periodic solutions of equations \eqref{eq7}-\eqref{eq8} depend on whether the release waiting period is less than $T^{*}$ or greater than $T^{*}$ in Theorem 3.2. For the case of $g^{*}< c< c^{*} $ , we obtained the necessary and sufficient conditions for asymptotic stability of $\omega _{0} $ and the unique $T$-periodic solution. In a biological sense, the global asymptotic stability of the origin indicates that mosquitoes in the wild will extinct over time. To maintain a certain level of mosquito population in the wild, it is mathematically necessary to find a suitable periodic solution.

This paper adopts a release strategy of releasing a fixed number of sterile mosquitoes at a fixed time point. Future work could consider releasing different numbers of sterile male mosquitoes based on the number of mosquitoes in different seasonal stages, that is, how to study the switching systems between sterile male mosquitoes and wild mosquitoes when the threshold number is higher.

\section*{Acknowledgment}

The authors thank the editors and the anonymous reviewers for their resourceful and valuable comments and constructive suggestions. This work was supported by the National Natural Science Foundation of China (62076104), the Science Foundation for Distinguished Youth Scholars of Fujian Province (2021J06025), and the Science and Technology Bureau of Quanzhou (2022NS070).


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