ABSTRACT

In this paper, a hibernation plankton-nutrient chemostat model with delayed response in growth is considered. By using the stroboscopic map and the theorem of impulsive delay differential equation, a plankton-extinction boundary periodic solution is obtained. The sufficient conditions on the permanence and globally attractive of the chemostat system are also obtained. Our main results reveal that the delayed response in growth plays an important role on the dynamical behaviors of system.

Keywords:

Chemostat Model, Hibernation, Delayed Response in Growth, Permanence, Extinction

1. Introduction

The chemostat is an important experimental instrument used to provide a controlled environment. Under this condition, the experimenter can adjust the parameters of system and get the final outcome. This chemostat model had been discussed by Smith and Waltman in [1] . In fact, the taken nutrient will not immediately absorbed by microorganism. In other words, nutrients with transformation from the substrate to microorganism have a lag time. Many scholars [2] [3] [4] [5] make discussion about chemostat model with discrete time delay. However, the system will have some changes because of the influence of climate; these perturbations break the continuity of the system. So the impulsive differential equations are considered into the system in [6] [7] [8] [9] . It is important for us to know more about ecology.

In recent years, some authors pay more attention to the hibernation of the plankton. The hibernation has an important sense of adaptation in ecology. Due to unfavorable environmental conditions, the plankton enters a hibernation state in advance. In order to save energy, plankton must maintain the weak life period overcoming the difficulties, such as drought stress, cold climate and temperature. The pressures elimination will restore growth. By hibernation, animals can reduce energy requirement and survive a few months in [10] . Some scholars also proposed that hibernation can make animals through hardship on cold environments and limited availability of food in [11] . However, there are many factors of plankton movements in the lakes, such as currents and river diffusion. These researches are seen in Levin and Segel [12] and Okubo [13] . Ruan discussed Turing instability and the existence of travelling wave solutions in [14] .

Furthermore, it is necessary to study a chemostat model with hibernation and impulsive diffusion on nutrients. In [15] , the author considered the dynamics of a plankton-nutrient chemostat model with hibernation and it was described by impulsive switched systems as follows

$\{\begin{array}{l}\begin{array}{r}\hfill {\stackrel{\dot{}}{S}}_1\left(t\right)=D\left(S_1^0-S_1\left(t\right)\right),\\ \hfill {\stackrel{\dot{}}{S}}_2\left(t\right)=-D{S}_2\left(t\right)-\frac{m{S}_2\left(t\right)x\left(t\right)}{K_1\left(K+S_2\left(t\right)\right)},\\ \hfill \stackrel{\dot{}}{x}\left(t\right)=-Dx\left(t\right)+\frac{m{S}_2\left(t\right)x\left(t\right)}{K+S_2\left(t\right)},\end{array}\}t\in \left(n\tau ,\left(n+L\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=S_1\left(t\right)-d{S}_1\left(t\right),\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+d{S}_1\left(t\right),\\ \hfill x\left(t^{+}\right)=x\left(t\right),\end{array}\}t=\left(n+L\right)\tau ,\hfill \\ \begin{array}{r}\hfill {\stackrel{\dot{}}{S}}_1\left(t\right)=D\left(S_1^0-S_1\left(t\right)\right),\\ \hfill {\stackrel{\dot{}}{S}}_2\left(t\right)=-D{S}_2\left(t\right),\\ \hfill \stackrel{\dot{}}{x}\left(t\right)=-d_{1}x\left(t\right),\end{array}\}t\in \left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=S_1\left(t\right)+{\mu }_1,\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+{\mu }_2,\\ \hfill x\left(t^{+}\right)=x\left(t\right).\end{array}\}t=\left(n+1\right)\tau .\hfill \end{array}$ (1.1)

where $\tau \in R_{+}=\left[0,\infty \right)$ , $n\in Z_{+}$ , $Z_{+}$ is the set of all positive integers; $S_1\left(t\right)$ and $S_2\left(t\right)$ represent the concentration of the nutrient in the river and reservoir at time $t$ respectively. $x\left(t\right)$ is the concentration of the plankton in the reservoir at time $t$ . $S_1^0$ is the input nutrient concentration in the river. $D$ is the dilution rate. $K_1$ is the yield of plankton $x\left(t\right)$ per unit mass of substrate. $d_1$ is the death rate of the plankton in the intervals of hibernation. $S_1\left(\left(n+L\right){\tau }^{+}\right)$ and $S_2\left(\left(n+L\right){\tau }^{+}\right)$ are the concentration of the nutrient in the river and reservoir immediately after the $n$ th diffusion pulse at time $t=n\tau$ respectively, while $S_1\left(\left(n+L\right)\tau \right)$ and $S_2\left(\left(n+L\right)\tau \right)$ are the concentration of the nutrient in the river and reservoir before the $n$ th diffusion pulse at time $t=\left(n+L\right)\tau$ separately. Due to the effect climate, the period of system is divided into two sections. That is normal seasons and drought seasons. In the normal seasons, the plankton grow regularly. The plankton is in hibernation in the drought seasons. $t=\left(n+L\right)\tau$ are moments of torrential rain, the nutrient is diffusing between rivers and reservoir in moments of torrential rain. $t=\left(n+1\right)\tau$ are moments of rainy season. ${\mu }_1$ and ${\mu }_2$ are the amount of nutrients coming from surrounding soil in moments of rainy season.

Based on the above discussion, we consider the following a hibernation plankton-nutrient chemostat model with delayed response in growth

$$\{\begin{array}{l}\begin{array}{r}\hfill {\dot{S}}_1\left(t\right)=D\left(S_0-S_1\left(t\right)\right),\\ \hfill {\dot{S}}_2\left(t\right)=-D{S}_2\left(t\right)-\frac{m{S}_2\left(t\right)x\left(t\right)}{\delta \left(K+S_2\left(t\right)\right)},\\ \hfill \dot{x}\left(t\right)=-Dx\left(t\right)+{\text{e}}^{-D{\tau }_1}\frac{m{S}_2\left(t-{\tau }_1\right)x\left(t-{\tau }_1\right)}{K+S_2\left(t-{\tau }_1\right)},\end{array}\}t\in \left(n\tau ,\left(n+L\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=\left(1-d\right)S_1\left(t\right),\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+d{S}_1\left(t\right),\\ \hfill x\left(t^{+}\right)=x\left(t\right),\end{array}\}t=\left(n+L\right)\tau ,\hfill \\ \begin{array}{r}\hfill {\dot{S}}_1\left(t\right)=D\left(S_0-S_1\left(t\right)\right),\\ \hfill {\dot{S}}_2\left(t\right)=-D{S}_2\left(t\right),\\ \hfill \dot{x}\left(t\right)=-d_{1}x\left(t\right),\end{array}\}t\in \left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=S_1\left(t\right)+{\mu }_1,\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+{\mu }_2,\\ \hfill x\left(t^{+}\right)=x\left(t\right).\end{array}\}t=\left(n+1\right)\tau .\hfill \end{array}$$ (1.2)

Suppose system (1.2) is connected by impulsive diffusion spread between rivers and reservoirs. There is no nutrients input in reservoir. Nutrient input is thought to come from the upper stream. Where constant ${\tau }_1\ge 0$ represents the time delay involved in the conversion of nutrient to plankton. Due to the chemostat outflow, ${\text{e}}^{-D{\tau }_1}$ is the positive constant, since it is assumed that the current change in biomass depends on the amount of nutrient consumed ${\tau }_1$ units of time before time $t$ and that survive in the chemostat the ${\tau }_1$ units of time assumed necessary to complete the nutrient conversion process. Other parameters are the same as system (1.1). $x\left(t\right)$ is continuous on $\left(n\tau ,\left(n+L\right)\tau \right]$

and $\left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right]$ , there exists $x\left(\left(n+L\right){\tau }^{+}\right)={\lim }_{t\to \left[\left(n+L\right){\tau }^{+}\right]}x\left(t\right)$ and $x\left(\left(n+1\right){\tau }^{+}\right)={\lim }_{t\to \left[\left(n+1\right){\tau }^{+}\right]}x\left(t\right)$ .

For system (1.2), we will discuss the sufficient and necessary conditions for the permanence and extinction. This paper can be summarized as follows. In Section 2, we present some preliminary results about system (1.2). Our results about extinction are stated and proven in Section 3. In Section 4, we study the permanence of system (1.2). Finally, we give a brief discussion and numerical analysis.

2. Preliminary Results

In this part, we will give some lemmas which will be useful for our main results.

Lemma 1. [16] Consider the following impulsive differential system

$\{\begin{array}{ll}\stackrel{\dot{}}{w}\left(t\right)\le (\ge )p\left(t\right)w\left(t\right)+q\left(t\right),\hfill & t\ne t_k,k\in N\hfill \\ w\left(t_k^{+}\right)\le (\ge )d_{k}w\left(t_k\right)+b_k,\hfill & t=t_k,k\in N.\hfill \end{array}$

where $p\left(t\right),q\left(t\right)\in c\left(R_{+},R\right),d_k\ge 0$ , and $b_k$ is the constant. Assume $\left(A_1\right)$ the sequence $t_k$ satisfies $0\le t_0<t_1<t_2<\cdots$ , with ${\lim }_{t\to \infty }{t}_k^{+}=\infty$ ; $\left(A_2\right)$ $w\in P{C}^{\prime }\left(R_{+},R\right)$ and w(t) is left-continuous at $t_k,k\in N$ . Then

$\begin{array}{r}\hfill \begin{array}{l}w\left(t\right)\le (\ge )w\left(t_0\right){\displaystyle \underset{t_0<t_k<t}{\prod }}{d}_k\exp \left({\displaystyle {\int }_{t_0}^t}p\left(s\right)\text{d}s\right)+{\displaystyle \underset{t_0<t_k<t}{\sum }}\left({\displaystyle \underset{t_k<t_j<t}{\prod }}{d}_j\exp \left({\displaystyle {\int }_{t_k}^t}p\left(s\right)\text{d}s\right)\right)b_k\\ +{\displaystyle {\int }_{t_0}^t}{\displaystyle \underset{s<t_k<t}{\prod }}{d}_k\exp \left({\displaystyle {\int }_s^t}p\left(\theta \right)\text{d}\theta \right)q\left(s\right)\text{d}s,t\ge t_0.\end{array}\end{array}$

Lemma 2. [17] Consider the following delay differential equation:

$\stackrel{\dot{}}{x}\left(t\right)=r_{1}x\left(t-{\tau }_1\right)-r_{2}x\left(t\right)\mathrm{,}$

where $\begin{array}{l}a,b\hfill \end{array}$ and ${\tau }_1$ are all positive constants and $x\left(t\right)>0$ for $t\in \left[-{\tau }_1,0\right]$ .

a) If $r_1<r_2$ , then ${\lim }_{t\to \infty }x\left(t\right)=0$ ;

b) If $r_1>r_2$ , then ${\lim }_{t\to \infty }x\left(t\right)=\infty$ .

Lemma 3. For any positive solution $\left(S_1\left(t\right),\mathrm{<mi>\; </mi>}{S}_2\left(t\right),\mathrm{<mi>\; </mi>}x\left(t\right)\right)$ of system (1.2) satisfy $S_1\left(0\right)>,S_2\left(0\right)>0,x\left(0\right)>0$ , there exists a constant $M>0$ , such that

$\underset{t\to \infty }{{\displaystyle \lim \sup }}{S}_1\left(t\right)\le M,\underset{t\to \infty }{{\displaystyle \lim \sup }}{S}_2\left(t\right)\le M,\underset{t\to \infty }{{\displaystyle \lim \sup }}x\left(t\right)\le M\mathrm{,}$

where $M=\frac{D{S}_0}{\lambda }+\frac{\left({\mu }_1+{\mu }_2\right){\text{e}}^{\lambda \tau }}{{\text{e}}^{\lambda \tau }-1},\lambda =\min \left\{D,d_1\right\}$ .

The proof of Lemma 3 is simple so we omit it here.

3. Extinction

The solution of system (1.2) corresponding to $x\left(t\right)=0$ is called plankton-ex- tinction periodic solution. For system (1.2), if we select $x\left(t\right)\equiv 0$ , then system (1.2) becomes the following model

$\{\begin{array}{l}\begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{S}}_1\left(t\right)=D\left(S_0-S_1\left(t\right)\right),\\ {\stackrel{\dot{}}{S}}_2\left(t\right)=-D{S}_2\left(t\right)\end{array}\end{array}\}t\in \left(n\tau ,\left(n+L\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=S_1\left(t\right)-d{S}_1\left(t\right),\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+d{S}_1\left(t\right),\end{array}\}t=\left(n+L\right)\tau ,\hfill \\ \begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{S}}_1\left(t\right)=D\left(S_0-S_1\left(t\right)\right),\\ {\stackrel{\dot{}}{S}}_2\left(t\right)=-D{S}_2\left(t\right),\end{array}\end{array}\}t\in \left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=S_1\left(t\right)+{\mu }_1,\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+{\mu }_2.\end{array}\}t=\left(n+1\right)\tau .\hfill \end{array}$ (3.1)

Integrating and solving the system (3.1) equations between pulses, we have

Consider the stroboscopic map of system (3.1), from the third, fourth, seventh and eighth equations of system (3.1) we have:

$$\{\begin{array}{ll}{S}_1\left(\left(n+1\right){\tau }^{+}\right)=\left(1-d\right){\text{e}}^{-D\tau }{S}_1\left(n{\tau }^{+}\right)+A,\hfill & n\tau <t\le \left(n+L\right)\tau ,\hfill \\ S_2\left(\left(n+1\right){\tau }^{+}\right)=d{\text{e}}^{-D\tau }{S}_1\left(n{\tau }^{+}\right)+{\text{e}}^{-D\tau }{S}_2\left(n{\tau }^{+}\right)+B,\hfill & n\tau <t\le \left(n+L\right)\tau .\hfill \end{array}$$ (3.3)

where $A=\left[1-{\text{e}}^{-D\left(1-L\right)\tau }+\left(1-d\right){\text{e}}^{-D\left(1-L\right)\tau }-\left(1-d\right){\text{e}}^{\left(-D\tau \right)}\right]S_0+{\mu }_1$ ,

$B=d{S}_0\left({\text{e}}^{-D\left(1-L\right)\tau }-{\text{e}}^{-D\tau }\right)+{\mu }_2$ . Equation (3.3) is difference equations. The dynamical behaviors of system (3.3) with equation (3.2) have been decided to the dynamical behaviors of system (3.1). So we focus on discussing System (3.3). System (3.3) has the following unique solution

$\{\begin{array}{l}{S}_1^{*}=\frac{A}{1-\left(1-d\right){\text{e}}^{-D\tau }}>0,\hfill \\ S_2^{*}=\frac{B\left[1-\left(1-d\right){\text{e}}^{-D\tau }\right]+Ad{\text{e}}^{-D\tau }}{\left(1-{\text{e}}^{-D\tau }\right)\left[1-\left(1-d\right){\text{e}}^{-D\tau }\right]}>0.\hfill \end{array}$ (3.4)

To change System (3.3) to a map, we define the map $F:R_{+}^2\to R_{+}^2:$

$\{\begin{array}{l}{F}_1\left(x\right)=\left(1-d\right){\text{e}}^{-D\tau }{x}_1+A,\hfill \\ F_2\left(x\right)=d{\text{e}}^{-D\tau }{x}_1+{\text{e}}^{-D\tau }{x}_2+B.\hfill \end{array}$ (3.5)

$F\left(x\right)$ is the map calculate at the point $x=\left(x_1,x_2\right)\in R_{+}^2$ . According to the lemma 3.2 and 3.4 of Refer [15] , we obtain

Hence, system (1.2) has a positive plankton-extinction periodic solution $\left(\widetilde{S_1\left(t\right)}\mathrm{,\; <mi>\; </mi>}\widetilde{S_2\left(t\right)}\mathrm{,\; <mi>\; </mi>0}\right)$ . In what follows, we will study the globally attractive of the plankton-extinction periodic solution $\left(\widetilde{S_1\left(t\right)}\mathrm{,\; <mi>\; </mi>}\widetilde{S_2\left(t\right)}\mathrm{,\; <mi>\; </mi>0}\right)$ of system (1.2).

Theorem 1. The periodic solution $\left(\widetilde{S_1\left(t\right)}\mathrm{,\; <mi>\; </mi>}\widetilde{S_2\left(t\right)}\mathrm{,\; <mi>\; </mi>0}\right)$ of system (1.2) is globally attractive, if

${\text{e}}^{-D{\tau }_1}\frac{m{\hat{S}}_2}{K+{\hat{S}}_2}<D,$ (3.7)

where ${\hat{S}}_2=S_2^{*}\left({\text{e}}^{-D\tau }+{\text{e}}^{-DL\tau }\right)+d{S}_1^{*}{\text{e}}^{-D\tau }+d{S}_0{\text{e}}^{-D\tau }\left({\text{e}}^{DL\tau }-1\right)$ and parametric $S_1^{*}$ and $S_2^{*}$ are given in (3.4).

Proof. Suppose $\left(S_1\left(t\right),\mathrm{<mi>\; </mi>}{S}_2\left(t\right),\mathrm{<mi>\; </mi>}x\left(t\right)\right)$ is any positive solution of system (1.2) with $S_1\left(0\right)>0,S_2\left(0\right)>0,x\left(0\right)>0$ . Based on the condition (3.7), we set

$P\left(z\right)=\frac{m{\text{e}}^{-D{\tau }_1}z}{K+z}$ , then $P\left(z\right)$ is strictly increasing function for arbitrary $z\ge 0$ . We may select sufficiently small $\epsilon >0$ , such that

$\frac{m{\text{e}}^{-D{\tau }_1}\eta }{K+\eta }<D,$ (3.8)

where $\eta ={\hat{S}}_2+\epsilon$ . From the second equation of (1.2) we have $\stackrel{\dot{}}{S}\left(t\right)\le -DS\left(t\right)$ .

Consider the following equations with pulse

$\{\begin{array}{l}\begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{Z}}_1\left(t\right)=D\left(S_0-Z_1\left(t\right)\right)\\ {\stackrel{\dot{}}{Z}}_2\left(t\right)=-D{Z}_2\left(t\right)\end{array}\end{array}\}t\in \left(n\tau ,\left(n+L\right)\tau \right],\hfill \\ \begin{array}{r}\hfill Z_1\left(t^{+}\right)=Z_1\left(t\right)-d{Z}_1\left(t\right),\\ \hfill Z_2\left(t^{+}\right)=Z_2\left(t\right)+d{Z}_1\left(t\right),\end{array}\}t=\left(n+L\right)\tau ,\hfill \\ \begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{Z}}_1\left(t\right)=D\left(S_0-Z_1\left(t\right)\right),\\ {\stackrel{\dot{}}{Z}}_2\left(t\right)=-D{Z}_2\left(t\right),\end{array}\end{array}\}t\in \left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right],\hfill \\ \begin{array}{r}\hfill Z_1\left(t^{+}\right)=Z_1\left(t\right)+{\mu }_1,\\ \hfill Z_2\left(t^{+}\right)=Z_2\left(t\right)+{\mu }_2.\end{array}\}t=\left(n+1\right)\tau .\hfill \end{array}$ (3.9)

From(3.6) and (3.9) we have that $\widetilde{S_1\left(t\right)}\le Z_1\left(t\right),\widetilde{S_2\left(t\right)}\le Z_2\left(t\right)$ and $Z_1\left(t\right)\to \widetilde{S_1\left(t\right)}\mathrm{,}{Z}_2\left(t\right)\to \widetilde{S_2\left(t\right)}$ as $t\to \infty$ . Therefore, there exist a integer $n_1$ and an arbitrary positive parameter $\epsilon$ , such that

$S_2\left(t\right)\le {\hat{S}}_2+\epsilon =:\eta$ (3.10)

for all $t\ge n_1\tau$ , where ${\hat{S}}_2=S_2^{*}\left({\text{e}}^{-D\tau }+{\text{e}}^{-DL\tau }\right)+d{S}_1^{*}{\text{e}}^{-D\tau }+d{S}_0{\text{e}}^{-D\tau }\left({\text{e}}^{DL\tau }-1\right)$ . For $t>n_1\tau +{\tau }_1$ , from (3.10) and the second equation of (1.1), we have

$\stackrel{\dot{}}{x}\left(t\right)\le \frac{m{\text{e}}^{-D{\tau }_1}\eta }{K+\eta }x\left(t-{\tau }_1\right)-Dx\left(t\right)\mathrm{.}$

Consider the following impulsive differential equation

$\stackrel{\dot{}}{y}\left(t\right)=\frac{m{\text{e}}^{-D{\tau }_1}\eta }{K+\eta }y\left(t-{\tau }_1\right)-Dy\left(t\right)\mathrm{.}$

According to lemma 2 and condition (3.8), we obtain that ${\lim }_{t\to \infty }y\left(t\right)=0$ . Since $X\left(s\right)=y\left(s\right)>0$ when $s\in \left[-{\tau }_1,0\right]$ , by the impulsive delay differential equation and the nonnegative of the solutions, we obtain $x\left(t\right)\to 0$ as $t\to \infty$ . Without loss of generality, for all $t\ge 0$ , we may suppose that $$0<x\left(t\right)<\epsilon$$ . By

the second equation of the system (1.2), we have

${\stackrel{\dot{}}{S}}_2\left(t\right)\ge -\left(D+\frac{m\epsilon }{\delta K}\right)S_2\left(t\right)$

Consider the following comparison system with pulse

$\{\begin{array}{l}\begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{Z}}_3\left(t\right)=D\left(S_0-Z_3\left(t\right)\right),\\ {\stackrel{\dot{}}{Z}}_4\left(t\right)=-\left(D+\frac{m\epsilon }{\delta K}\right)Z_4\left(t\right)\end{array}\end{array}\}t\in \left(n\tau ,\left(n+L\right)\tau \right],\hfill \\ \begin{array}{r}\hfill Z_3\left(t^{+}\right)=Z_3\left(t\right)-d{Z}_3\left(t\right),\\ \hfill Z_4\left(t^{+}\right)=Z_4\left(t\right)+d{Z}_3\left(t\right),\end{array}\}t=\left(n+L\right)\tau ,\hfill \\ \begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{Z}}_3\left(t\right)=D\left(S_0-Z_3\left(t\right)\right),\\ {\stackrel{\dot{}}{Z}}_4\left(t\right)=-D{Z}_4\left(t\right),\end{array}\end{array}\}t\in \left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right],\hfill \\ \begin{array}{r}\hfill Z_3\left(t^{+}\right)=Z_3\left(t\right)+{\mu }_1,\\ \hfill Z_4\left(t^{+}\right)=Z_4\left(t\right)+{\mu }_2.\end{array}\}t=\left(n+1\right)\tau .\hfill \end{array}$ (3.11)

The system (3.11) has a positive solution $\left(\widetilde{Z_3\left(t\right)},\widetilde{Z_4\left(t\right)}\right)$ , where $\left(\widetilde{Z_3\left(t\right)},\widetilde{Z_4\left(t\right)}\right)$ are expressed as follows

In which

$Z_4^{*}=\frac{B\left[1-\left(1-d\right){\text{e}}^{-D\tau }\right]+Ad{\text{e}}^{-D\tau }}{\left(1-{\text{e}}^{-\left(D+\frac{m\epsilon L}{\delta K}\right)\tau }\right)\left[1-\left(1-d\right){\text{e}}^{-D\tau }\right]}>0$ (3.13)

and $A=\left[1-{\text{e}}^{-D\left(1-L\right)\tau }+\left(1-d\right){\text{e}}^{-D\left(1-L\right)\tau }-\left(1-d\right){\text{e}}^{\left(-D\tau \right)}\right]S_0+{\mu }_1$ , $B=d{S}_0\left({\text{e}}^{-D\left(1-L\right)\tau }-{\text{e}}^{-D\tau }\right)+{\mu }_2$ . For arbitrary $\epsilon >0$ , there exists a constant $T_1>0$ , such that, for all $T>T_1$ ,

$\widetilde{Z_3\left(t\right)}-{\epsilon }_1<S_1\left(t\right)<\widetilde{S_1\left(t\right)}+{\epsilon }_1,\widetilde{Z_4\left(t\right)}-{\epsilon }_1<S_2\left(t\right)<\widetilde{S_2\left(t\right)}+{\epsilon }_1\mathrm{.}$

Let $\epsilon \to 0$ , we obtain

$\widetilde{S_1\left(t\right)}-{\epsilon }_1<S_1\left(t\right)<\widetilde{S_1\left(t\right)}+{\epsilon }_1,\widetilde{S_2\left(t\right)}-{\epsilon }_1<S_2\left(t\right)<\widetilde{S_2\left(t\right)}+{\epsilon }_1$ (3.14)

For $t\to \infty$ and ${\epsilon }_1\to 0$ , $S_1\left(t\right)\to \widetilde{S_1\left(t\right)}$ and $S_2\left(t\right)\to \widetilde{S_2\left(t\right)}$ . The proof of Theorem 1 is completed.

4. Permanence

In this section we shall study the permanence of system (1.2).

Theorem 2. System (1.2) is permanent, if

${\text{e}}^{-D{\tau }_1}\frac{m{\stackrel{⌣}{S}}_2}{K+{\stackrel{⌣}{S}}_2}>D.$ (4.1)

where ${\stackrel{⌣}{S}}_2=Z_6^{*}\left({\text{e}}^{-\left(D+\frac{m{m}_1^{*}}{\delta K}\right)L\tau }+{\text{e}}^{-\left(D+\frac{m{m}_1^{*}L}{\delta K}\right)\tau }\right)+d{S}_1^{*}{\text{e}}^{-D\tau }+d{S}_0{\text{e}}^{-D\tau }\left({\text{e}}^{DL\tau }+1\right)$ , $S_1^{*}$ and $Z_6^{*}$ are given in (3.4) and (4.8) respectively.

Proof. Suppose $\left(S_1\left(t\right),\mathrm{<mi>\; </mi>}{S}_2\left(t\right),\mathrm{<mi>\; </mi>}x\left(t\right)\right)$ is any positive solution of system (1.2) with $S_1\left(0\right)>0,S_2\left(0\right)>0,x\left(0\right)>0$ . We may rewrite the second equation of the system (1.2) as follows

$$\begin{array}{l}\dot{x}\left(t\right)=\left(\frac{m{\text{e}}^{-D{\tau }_1}{S}_2\left(t\right)}{K+S_2\left(t\right)}-D\right)x\left(t\right)-m{\text{e}}^{-D{\tau }_1}\frac{\text{d}}{\text{d}t}{\displaystyle {\int }_{t-{\tau }_1}^t}\frac{S_2\left(\theta \right)X\left(\theta \right)}{K+S_2\left(\theta \right)}\text{d}\theta .\hfill \end{array}$$ (4.2)

Define

$V\left(t\right)=x\left(t\right)+m{\text{e}}^{-D{\tau }_1}{\displaystyle {\int }_{t-{\tau }_1}^t}\frac{S_2\left(\theta \right)X\left(\theta \right)}{K+S_2\left(\theta \right)}\text{d}\theta .$

Derive $V\left(t\right)$ along with solution of system (1.2), we obtain

$\begin{array}{l}\stackrel{\dot{}}{V}\left(t\right)=D\left[\frac{m{\text{e}}^{-D{\tau }_1}{S}_2\left(t\right)}{D\left(k+S_2\left(t\right)\right)}-1\right]x\left(t\right).\hfill \end{array}$ (4.3)

Set

$m_1^{*}=\frac{\delta K}{m}\left[\frac{1}{\tau }{\ln }^{\frac{S_2^{*}\left(m{\text{e}}^{-D{\tau }_1}-D\right)}{K}}-D\right].$

From (4.4) we obtain $m_1^{*}>0$ . We may select a positive integer ${\epsilon }_1$ small enough, such that

$\frac{m{\text{e}}^{-D{\tau }_1}\beta }{D\left(K+\beta \right)}>1,$ (4.4)

where $\beta =Z_6^{*}\left({\text{e}}^{-\left(D+\frac{m{m}_1^{*}}{\delta K}\right)L\tau }+{\text{e}}^{-\left(D+\frac{m{m}_1^{*}L}{\delta K}\right)\tau }\right)+d{S}_1^{*}{\text{e}}^{-D\tau }+d{S}_0{\text{e}}^{-D\tau }\left({\text{e}}^{DL\tau }+1\right)-{\epsilon }_1.$ For

any nonnegative integer $t_0$ , we claim that inequality $x\left(t\right)<m_1^{*}$ is not hold for all $t\ge t_0$ . Otherwise, there exists a positive parameter $T_1$ , such that $x\left(t\right)<m_1^{*}$ for all $t>T_1$ . From the system (1.2) we obtain

$\{\begin{array}{l}\begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{S}}_1\left(t\right)=D\left(S_0-S_1\left(t\right)\right),\\ {\stackrel{\dot{}}{S}}_2\left(t\right)\ge -\left(D+\frac{m{m}_1^{*}}{\delta K}\right)S_2\left(t\right)\end{array}\end{array}\}t\in \left(n\tau ,\left(n+L\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=S_1\left(t\right)-d{S}_1\left(t\right),\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+d{S}_1\left(t\right),\end{array}\}t=\left(n+L\right)\tau ,\hfill \\ \begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{S}}_1\left(t\right)=D\left(S_0-S_1\left(t\right)\right),\\ {\stackrel{\dot{}}{S}}_2\left(t\right)=-D{S}_2\left(t\right),\end{array}\end{array}\}t\in \left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right],\hfill \\ \begin{array}{r}\hfill S_1\left(t^{+}\right)=S_1\left(t\right)+{\mu }_1,\\ \hfill S_2\left(t^{+}\right)=S_2\left(t\right)+{\mu }_2.\end{array}\}t=\left(n+1\right)\tau .\hfill \end{array}$ (4.5)

Consider the following impulsive differential equation

$\{\begin{array}{l}\begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{Z}}_5\left(t\right)=D\left(S_0-Z_5\left(t\right)\right),\\ {\stackrel{\dot{}}{Z}}_6\left(t\right)=-\left(D+\frac{m{m}_1^{*}}{\delta K}\right)Z_6\left(t\right)\end{array}\end{array}\}t\in \left(n\tau ,\left(n+L\right)\tau \right],\hfill \\ \begin{array}{r}\hfill Z_5\left(t^{+}\right)=Z_5\left(t\right)-d{Z}_5\left(t\right),\\ \hfill Z_6\left(t^{+}\right)=Z_6\left(t\right)+d{Z}_5\left(t\right),\end{array}\}t=\left(n+L\right)\tau ,\hfill \\ \begin{array}{r}\hfill \begin{array}{l}{\stackrel{\dot{}}{Z}}_5\left(t\right)=D\left(S_0-Z_5\left(t\right)\right),\\ {\stackrel{\dot{}}{Z}}_6\left(t\right)=-D{Z}_6\left(t\right),\end{array}\end{array}\}t\in \left(\left(n+L\right)\tau ,\left(n+1\right)\tau \right],\hfill \\ \begin{array}{r}\hfill Z_5\left(t^{+}\right)=Z_5\left(t\right)+{\mu }_1,\\ \hfill Z_6\left(t^{+}\right)=Z_6\left(t\right)+{\mu }_2.\end{array}\}t=\left(n+1\right)\tau .\hfill \end{array}$ (4.6)

The system (4.6) has a unique globally asymptotically stable positive solution as follows

where

$Z_6^{*}=\frac{B\left[1-\left(1-d\right){\text{e}}^{-D\tau }\right]+Ad{\text{e}}^{-D\tau }}{\left(1-{\text{e}}^{-\left(D+\frac{m{m}_1^{*}L}{\delta K}\right)\tau }\right)\left[1-\left(1-d\right){\text{e}}^{-D\tau }\right]}>0.$ (4.8)

There exists $T_2\ge t_0+{\tau }_1$ , such that

$S_2\left(t\right)>{\stackrel{⌣}{S}}_2-{\epsilon }_1=:\beta \text{for}\text{all}t>T_2,$ (4.9)

where ${\stackrel{⌣}{S}}_2=Z_6^{*}\left({\text{e}}^{-\left(D+\frac{m{m}_1^{*}}{\delta K}\right)L\tau }+{\text{e}}^{-\left(D+\frac{m{m}_1^{*}L}{\delta K}\right)\tau }\right)+d{S}_1^{*}{\text{e}}^{-D\tau }+d{S}_0{\text{e}}^{-D\tau }\left({\text{e}}^{DL\tau }+1\right)$ .

Take $t^{*}=\min \left\{T_1,T_2\right\}$ . For any $t\ge t^{*}$ , from (4.3) and (4.9) we obtain

$\stackrel{\dot{}}{V}\left(t\right)>D\left[\frac{m{\text{e}}^{-D{\tau }_1}\beta }{D\left(K+\beta \right)}-1\right]x\left(t\right).$ (4.10)

Let

$\tilde{x}=\underset{t\in \left[t^{*},t^{*}+{\tau }_1\right]}{\min }x\left(t\right).$

In what follows, we prove that $x\left(t\right)<\tilde{x}$ for all $t\ge t^{*}$ . Otherwise, there exists a positive constant $T_3$ , for any $t\in \left[t^{*},t^{*}+{\tau }_1+T_3\right]$ , we have $x\left(t\right)\ge \tilde{x}$ , $x\left(t^{*}+{\tau }_1+T_3\right)=\tilde{x}$ and $x^{\prime }\left(t^{*}+{\tau }_1+T_3\right)\le 0$ . Therefore, according to the third equation of (1.2) and (4.10), we further obtain

$x^{\prime }\left(t^{*}+{\tau }_1+T_3\right)>\left[\frac{m{\text{e}}^{-D{\tau }_1}\beta }{\left(K+\beta \right)}-D\right]x\left(t\right)=D\left[\frac{m{\text{e}}^{-D{\tau }_1}\beta }{D\left(K+\beta \right)}-1\right]x\left(t\right)>0$