Leslie[1] introduced the famous Leslie predator-prey system

$ \begin{eqnarray*} \begin{array}{ll} \dot{x}(t)=x(t)\big[a-bx(t)\big]-p(x)y(t), \\ \dot{y}(t)=y(t)\big[e-f\frac{y(t)}{x(t)}\big], \end{array}\end{eqnarray*} $

where $x(t), y(t)$ stand for the population (the density) of the prey and the predator at time $t$, respectively, and $p(x)$ is the so-called predator functional response to prey. The term $f\frac{y}{x}$ of the above equation is called Leslie-Gower term, which measures the loss in the predator population due to rarity (per capita $y/x$) of its favorite food. In case of severe scarcity, $y$ can switch over to other populations but its growth will be limited by the fact that its most favorite food $x$ is not available in abundance. This situation can be taken care of by adding a positive constant $k$ to the denominator, see [2-7] and references cited therein.

It is well known that time delays play important roles in many biological dynamical systems. In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the populations to fluctuate (see [5-7]). Furthermore, the existence of periodic solutions may be changed. Naturally, more realistic and interesting models of population interactions should take into account both the seasonality of changing environment and the effects of time delay.

In recent years, Leslie-Gower model with Holling-type Ⅱ has been extensively studied by many scholars, many excellent results were obtained concerned with the persistent property and positive periodic solution of the system (see [18-23] and the reference therein). Because Holling-type Ⅲ can describe the relationship between the predator and prey clearly. So Zhang et al [7] studied the following system

$ \begin{eqnarray} \left\{\begin{array}{ll} x'(t)=x(t)\bigg[r_{1}-b_{1}x(t-\tau_{1})-\frac{a_{1}x(t)y(t)}{x^{2}(t)+k_{1}}\bigg], \\ y'(t)=y(t)\bigg[r_{2}-\frac{a_{2}y(t-\tau_{2})}{x(t-\tau_{2})+k_{2}}\bigg], \end{array}\right. \end{eqnarray} $

(1.1)

where $x(t)$ and $y(t)$ represent the densities of the prey and predator population, respectively; $\tau_{i}\geq 0$; $r_{1}$, $b_{1}$, $a_{1}$, $k_{1}$, $r_{2}$, $a_{2}$, and $k_{2}$ are positive values. Some sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to the two delays are obtained; however, Zhang did not give sufficient conditions for the existence of positive periodic solutions and permanence. Moreover, We known that coincidence degree theory is an important method to investigate the positive periodic solutions, some excellent results have been obtained concerned with the existence of periodic solutions of the predator-prey system(see [8-14] and the references therein).

Stimulated by the above reasons, In this paper we consider the following system:

$ \begin{eqnarray} \left\{\begin{array}{ll} x'(t)=x(t)\bigg[r_{1}(t)-b(t)x(t-\tau_{1}(t))-\frac{a_{1}(t)x(t)y(t-\sigma(t))}{x^{2}(t)+k_{1}^{2}(t)}\bigg], \\ y'(t)=y(t)\bigg[r_{2}(t)-\frac{a_{2}(t)y(t-\tau_{2}(t))}{x(t-\tau_{2}(t))+k_{2}(t)}\bigg], \end{array}\right. \end{eqnarray} $

(1.2)

where $x(t)$ and $y(t)$ represent the densities of the prey and predator population, respectively; $b(t)$, $a_{1}(t)$, $a_{2}(t)$, $k_{1}(t)$, $k_{2}(t)$, $\sigma(t)$, $\tau_{i}(t)\, \, i=1, 2$ are all positive periodic continuous functions with period $\omega >0$; $r_{i}(t)\in C(R, R)\, \, i=1, 2$ are $\omega-$periodic continuous functions. In addition, we request that $\int_{0}^{\omega}r_{i}dt>0, i=1, 2$, and the growth functions $r_{i}(t), i=1, 2$ are not necessarily positive, because the environment fluctuates randomly. Obviously, where $k^{2}_{1}$, $k_{2}$ are positive constants, system (1.1) is the special case of (1.2).

To our knowledge, no such work has been done on the existence of positive periodic solutions and permanence of (1.2). Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [15], to derive a set of easily verifiable sufficient conditions for the existence of positive solutions. Then by utilizing the comparison obtain sufficient conditions for permanence of system (1.2).

Let $X$, $Z$ be real Banach spaces, $L : \mathit{\rm{Dom}} L\subset X \rightarrow Z$ be a linear mapping, and $N : X\rightarrow Z$ be a continuous mapping. The mapping $L$ is said to be a Fredholm mapping of index zero if dim Ker $L$ = codim Im$L <+\infty$ and Im $L$ is closed in $Z$. If $L$ is a Fredholm mapping of index zero, then there exist continuous projectors $P : X\rightarrow X$ and $Q : Z\rightarrow Z$ such that Im $P$ = Ker $L$, Ker $Q$ = Im$L$ = Im$(I-Q)$. It follows that the restriction $L_{P}$ of $L$ to $\mathit{\rm{Dom}} L\cap \mathit{\rm{Ker}} P : (I-P)X \rightarrow \mathit{\rm{Im}} L$ is invertible. Denote the inverse of $L_{P}$ by $K_{P}$. The mapping $N$ is said to be $L$-compact on $\overline{\Omega}$ if $\Omega$ is an open bounded subset of $X$, $QN(\overline{\Omega})$ is bounded and $K_{P}(I-Q)N : \overline{\Omega}\rightarrow X$ is compact. Since Im$Q$ is isomorphic to Ker$L$, there exist an isomorphic $J : \mathit{\rm{Im}} L\rightarrow \mathit{\rm{Ker}} L$.

Lemma 2.1 (Continuation Theorem [15]) Let $\Omega \subset X$ be an open bounded set, $L$ be a Fredholm mapping of index zero and $N$ be $L$-compact on $\overline{\Omega}$. Suppose that

(ⅰ) for each $\lambda \in(0, 1), x\in \partial\Omega\cap\mathit{\rm{Dom}} L, Lx\neq\lambda Nx$;

(ⅱ) for each $x\in \partial\Omega\cap\mathit{\rm{Ker}} L, QNx\neq0$;

(ⅲ) $\mathit{\rm{deg}}$$\{JQN, \Omega\cap\mathit{\rm{Ker}}L, 0\}\neq0$.

Then $Lx=Nx$ has at least one solution in $\overline{\Omega}\cap\mathit{\rm{Dom}} L$.

Lemma 2.2 [17] Suppose that $g\in PC_{\omega}^{1}=\{x: x\in C^{1}(R, R), x(t+\omega)\equiv x(t)\}$, then

$ \begin{eqnarray*} 0\leq \max\limits_{s\in [0, \omega]}g(s)-\min\limits_{s\in [0, \omega]}g(s)\leq \frac{1}{2}\int_{0}^{\omega}|g'(s)|\mathit{\rm{d}}s. \end{eqnarray*} $

For convenience, we denote

$ \begin{eqnarray*} \bar{f}=\frac{1}{\omega}\int_{0}^{\omega}f(t)\mathit{\rm{d}}t, \quad f^{L}=\min\limits_{t\in [0, \omega]}f(t), \quad f^{U}=\max\limits_{t\in [0, \omega]}f(t), \end{eqnarray*} $

where $f(t)$ is a continuous $\omega$-periodic function.

Theorem 3.1  Assume $\overline{r}_{1}>\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}$ hold, where $H_{2}$ is defined in the proof, Then system (1.2) has at least one positive $\omega$-periodic solution.

Proof  Let $x(t)=e^{x_{1}(t)}$, $y(t)=e^{x_{2}(t)}$, then from (1.2) we have

$ \begin{eqnarray} \left\{\begin{array}{ll} x_{1}'(t)=r_{1}(t)-b(t)\exp\{x_{1}(t-\tau_{1}(t))\}-\frac{a_{1}(t)\exp\{x_{1}(t)+x_{2}(t-\sigma(t))\}}{\exp\{2x_{1}(t)\}+k_{1}^{2}(t)}, \\ x_{2}'(t)=r_{2}(t)-\frac{a_{2}(t)\exp\{x_{2}(t-\tau_{2}(t))\}}{\exp\{x_{1}(t-\tau_{2}(t))\}+k_{2}(t)}. \end{array}\right. \end{eqnarray} $

(3.1)

It is easy to see that if the system (3.1) has one $\omega$-periodic solution $(x_{1}^{\ast}(t), x_{2}^{\ast}(t))^{T}$, then $(x^{\ast}(t), y^{\ast}(t))^{T}=(e^{x_{1}^{\ast}(t)}, e^{x_{2}^{\ast}(t)})^{T}$ is a positive $\omega$-periodic solution of (1.2). Therefore, we only need to prove that (3.1) has at least one $\omega$-periodic solution.

Take $ X=Z=\{x(t)=(x_{1}(t), x_{2}(t))^{T}\in C(R, R^{2}): x(t+\omega)=x(t)\} $ and denote

$ \begin{eqnarray*} \|x\|=\max\limits_{t\in [0, \omega]}\{|x_{1}(t)|+|x_{2}(t)|\}. \end{eqnarray*} $

Then $X$ and $Z$ are Banach spaces when they are endowed with the norms $\|\cdot\|$.

We define operators $L, P$ and $Q$ as follows, respectively,

$ \begin{eqnarray*} L: \mathit{\rm{Dom}}L\cap X\rightarrow Z, \quad Lx=x'; \quad P(x)=\frac{1}{\omega}\int_{0}^{\omega}x(t)\mathit{\rm{d}}t, \, \, x\in X, \end{eqnarray*} $

$ \begin{eqnarray*} Q(x)=\frac{1}{\omega}\int_{0}^{\omega}z(t)\mathit{\rm{d}}t, \, \, z\in Z, \end{eqnarray*} $

where $\mathit{\rm{Dom}}L=\{x\in X: x(t)\in C^{1}(R, R^{2})\}$, and define $N:X\rightarrow Z$ by the form

$ \begin{eqnarray*} Nx=\left[\begin{array}{c}r_{1}(t)-b(t)\exp\{x_{1}(t-\tau_{1}(t))\}-\frac{a_{1}(t)\exp\{x_{1}(t)+x_{2}(t-\sigma(t))\}}{\exp\{2x_{1}(t)\}+k_{1}^{2}(t)}\\ r_{2}(t)-\frac{a_{2}(t)\exp\{x_{2}(t-\tau_{2}(t))\}}{\exp\{x_{1}(t-\tau_{2}(t))\}+k_{2}(t)}\end{array}\right]. \end{eqnarray*} $

Evidently, $\mathit{\rm{Ker}}L=R^{2}$, $\mathit{\rm{Im}}L=\{z|z\in Z, \, \int_{0}^{\omega}z(t)\mathit{\rm{d}}t=0\}$ is closed in $Z$. $\mathit{\rm{dim}}\, \, \mathit{\rm{Ker}}L=\mathit{\rm{codim}}\, \, \mathit{\rm{Im}}L=2$, and $P$, $Q$ are continuous projectors such that

$ \begin{eqnarray*} \mathit{\rm{Im}}P=\mathit{\rm{Ker}}L, \quad \mathit{\rm{Ker}}Q=\mathit{\rm{Im}}L=\mathit{\rm{Im}}(I-Q), \end{eqnarray*} $

thus, $L$ is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to $L$) $K_{P}$: $\mathit{\rm{Im}}L \rightarrow \mathit{\rm{Ker}} P\cap \mathit{\rm{Dom}} L$ has the form

$ \begin{eqnarray*} K_{P}(z)=\int_{0}^{t}z(s)\mathit{\rm{d}}s-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}z(s)\mathit{\rm{d}}s\mathit{\rm{d}}t. \end{eqnarray*} $

Thus

$ \begin{eqnarray*} QNx=\begin{bmatrix}\frac{1}{\omega}\int_{0}^{\omega}\big[r_{1}(t)-b(t)\exp\{x_{1}(t-\tau_{1}(t))\}-\frac{a_{1}(t)\exp\{x_{1}(t)+x_{2}(t-\sigma(t))\}}{\exp\{2x_{1}(t)\}+k_{1}^{2}(t)}\big]\mathit{\rm{d}}t\\ \frac{1}{\omega}\int_{0}^{\omega}\big[r_{2}(t)-\frac{a_{2}(t)\exp\{x_{2}(t-\tau_{2}(t))\}}{\exp\{x_{1}(t-\tau_{2}(t))\}+k_{2}(t)}\big]\mathit{\rm{d}}t\end{bmatrix} \end{eqnarray*} $

and

$ \begin{eqnarray*} &&K_{P}(I-Q)Nx\\ &&=\begin{bmatrix}\int_{0}^{t}\big[r_{1}(s)-b(s)\exp\{x_{1}(s-\tau_{1}(s))\}-\frac{a_{1}(s)\exp\{x_{1}(s)+x_{2}(s-\sigma(s))\}}{\exp\{2x_{1}(s)\}+k_{1}^{2}(s)}\big]\mathit{\rm{d}}s\\ \int_{0}^{t}\big[r_{2}(s)-\frac{a_{2}(s)\exp\{x_{2}(s-\tau_{2}(s))\}}{\exp\{x_{1}(s-\tau_{2}(s))\}+k_{2}(s)}\big]\mathit{\rm{d}}s\end{bmatrix}\\ &&-\begin{bmatrix}\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}[r_{1}(s)-b(s)\exp\{x_{1}(s-\tau_{1}(s))\}-\frac{a_{1}(s)\exp\{x_{1}(s)+x_{2}(s-\sigma(s))\}}{\exp\{2x_{1}(s)\}+k_{1}^{2}(s)}\big]\mathit{\rm{d}}s\mathit{\rm{d}}t\\ \frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}\big[r_{2}(s)-\frac{a_{2}(s)\exp\{x_{2}(s-\tau_{2}(s))\}}{\exp\{x_{1}(s-\tau_{2}(s))\}+k_{2}(s)}\big]\mathit{\rm{d}}s\mathit{\rm{d}}t\end{bmatrix}\\ &&-\begin{bmatrix}(\frac{t}{\omega}-\frac{1}{2})\int_{0}^{\omega}\big[r_{1}(s)-b(s)\exp\{x_{1}(s-\tau_{1}(s))\}-\frac{a_{1}(s)\exp\{x_{1}(s)+x_{2}(s-\sigma(s))\}}{\exp\{2x_{1}(s)\}+k_{1}^{2}(s)}\big]\mathit{\rm{d}}s\\ (\frac{t}{\omega}-\frac{1}{2})\int_{0}^{\omega}\big[r_{2}(s)-\frac{a_{2}(s)\exp\{x_{2}(s-\tau_{2}(s))\}}{\exp\{x_{1}(s-\tau_{2}(s))\}+k_{2}(s)}\big]\mathit{\rm{d}}s\end{bmatrix}. \end{eqnarray*} $

Obviously, $QN$ and $K_{P}(I-Q)N$ are continuous. Moreover, $QN(\overline{\Omega})$ $K_{p}(I-Q)N(\overline{\Omega})$ are relatively compact for any open bounded set $\Omega\subset X$. Hence, $N$ is $L-$compact on $\overline{\Omega}$, here $\Omega$ is any open bounded set in $X$.

Corresponding to the operator equation $Lx=\lambda Nz, \lambda\in (0, 1)$, we have

$ \begin{eqnarray} x_{1}'(t)&=&\lambda\bigg[r_{1}(t)-b(t)\exp\{x_{1}(t-\tau_{1}(t))\}-\frac{a_{1}(t)\exp\{x_{1}(t)+x_{2}(t-\sigma(t))\}}{\exp\{2x_{1}(t)\}+k_{1}^{2}(t)}\bigg], \nonumber\\ x_{2}'(t)&=&\lambda\bigg[r_{2}(t)-\frac{a_{2}(t)\exp\{x_{2}(t-\tau_{2}(t))\}}{\exp\{x_{1}(t-\tau_{2}(t))\}+k_{2}(t)}\bigg]. \end{eqnarray} $

(3.2)

Suppose that $x(t)=(x_{1}(t), x_{2}(t))^{T}\in X$ is an $\omega$-periodic solution of system (3.2) for a certain $\lambda \in (0, 1)$. By integrating (3.2) over the interval $[0, \omega]$, we obtain

$ \int_{0}^{\omega}\bigg[r_{1}(t)-b(t)\exp\{x_{1}(t-\tau_{1}(t))\}-\frac{a_{1}(t)\exp\{x_{1}(t)+x_{2}(t-\sigma(t))\}}{\exp\{2x_{1}(t)\}+k_{1}^{2}(t)}\bigg]\mathit{\rm{d}}t=0, $

(3.3)

$ \int_{0}^{\omega}\bigg[r_{2}(t)-\frac{a_{2}(t)\exp\{x_{2}(t-\tau_{2}(t))\}}{\exp\{x_{1}(t-\tau_{2}(t))\}+k_{2}(t)}\bigg]\mathit{\rm{d}}t=0. $

(3.4)

From (3.2)-(3.4), we obtain

$ \begin{eqnarray} \int_{0}^{\omega}|x'_{1}(t)|\mathit{\rm{d}}t&=&\lambda\int_{0}^{\omega}|r_{1}(t)-b(t)\exp\{x_{1}(t-\tau_{1}(t))\}-\frac{a_{1}(t)\exp\{x_{1}(t)+x_{2}(t-\sigma(t))\}}{\exp\{2x_{1}(t)\}+k_{1}^{2}(t)}|\mathit{\rm{d}}t\nonumber\\ &\leq&(\overline{|r_{1}|}+\bar{r}_{1})\omega \end{eqnarray} $

(3.5)

and

$ \begin{eqnarray} \int_{0}^{\omega}|x'_{2}(t)|\mathit{\rm{d}}t\leq\int_{0}^{\omega}|r_{2}(t)-\frac{a_{2}(t)\exp\{x_{2}(t-\tau_{2}(t))\}}{\exp\{x_{1}(t-\tau_{2}(t))\}+k_{2}(t)}|\mathit{\rm{d}}t\leq(\overline{|r_{2}|}+\bar{r}_{2})\omega. \end{eqnarray} $

(3.6)

Noting that $x=(x_{1}(t), x_{2}(t))^{T}\in X$. Then there exist $\xi_{i}, \eta_{i}\in [0, \omega]$ such that

$ \begin{eqnarray} x_{i}(\xi_{i})=\sup\limits_{t\in [0, \omega]}x_{i}(t), \quad x_{i}(\eta_{i})=\inf\limits_{t\in [0, \omega]}x_{i}(t), \quad i=1, 2. \end{eqnarray} $

(3.7)

It follows from (3.3) and (3.7) that

$ \begin{eqnarray*} \omega\overline{r}_{1}\geq\int_{0}^{\omega}b(t)\exp\{x_{1}(t-\tau_{1})\}\mathit{\rm{d}}t\geq \omega\overline{b}e^{x_{1}(\eta_{1})}, \end{eqnarray*} $

which implies that

$ \begin{eqnarray} x_{1}(\eta_{1})\leq\ln \frac{\overline{r}_{1}}{\overline{b}}. \end{eqnarray} $

(3.8)

It follows from (3.5), (3.8) and Lemma 2.2 that, for any $t\in[0, \omega]$,

$ \begin{eqnarray} x_{1}(t)\leq x_{1}(\eta_{1})+\frac{1}{2}\int_{0}^{\omega}|x'_{1}(t)|\mathit{\rm{d}}t\leq \ln \frac{\overline{r}_{1}}{\overline{b}}+\frac{1}{2}\omega(\overline{|r_{1}|}+\bar{r}_{1})\triangleq H_{1}. \end{eqnarray} $

(3.9)

From (3.7), (3.9) and (3.4) that

$ \begin{eqnarray*} \frac{\omega \overline{a}_{2}e^{x_{2}(\eta_{2})}}{e^{H_{1}}+k_{2}^{U}}\leq \int_{0}^{\omega}\frac{a_{2}e^{x_{2}(t)}}{e^{x_{1}(t)}+k_{2}(t)}\mathit{\rm{d}}t=\omega \overline{r}_{2}, \end{eqnarray*} $

i.e.,

$ \begin{eqnarray*} x_{2}(\eta_{2})\leq \ln \frac{\overline{r}_{2}(e^{H_{1}}+k_{2}^{U})}{\overline{a}_{2}} \end{eqnarray*} $

which together with (3.6) and Lemma 2.2 imply

$ \begin{eqnarray} x_{2}(t)\leq x_{2}(\eta_{2})+\frac{1}{2}\int_{0}^{\omega}|x'_{2}(t)|\mathit{\rm{d}}t\leq \ln \frac{\overline{r}_{2}(e^{H_{1}}+k_{2}^{U})}{\overline{a}_{2}}+\frac{1}{2}\omega(\overline{|r_{2}|}+\overline{r}_{2})\triangleq H_{2} \end{eqnarray} $

(3.10)

In addition, from (3.3) and (3.7), we get

$ \begin{array}{l} \omega \bar b{e^{{x_1}({\xi _1})}} \ge \omega {{\bar r}_1} - \int_0^\omega {\frac{{{a_1}(t)\exp \{ {x_1}(t) + {x_2}(t - \sigma (t))\} }}{{\exp \{ 2{x_1}(t)\} + k_1^2(t)}}} \\ \;\;\;\;\;\;\;\;\;\;\;\; \ge \omega {{\bar r}_1} - \omega \overline {(\frac{{{a_1}}}{{2{k_1}}})} {e^{{H_2}}}, \end{array} $

which implies that

$ \begin{eqnarray*} x_{1}(\xi_{1})\geq\ln\frac{\overline{r}_{1}-\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}}{\bar{b}}, \end{eqnarray*} $

Then together with (3.5) and Lemma 2.2 imply

$ \begin{eqnarray} x_{1}(t)\geq x_{1}(\xi_{1})-\frac{1}{2}\int_{0}^{\omega}|x'_{1}(t)|\mathit{\rm{d}}t\geq\ln\frac{\overline{r}_{1}-\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}}{\bar{b}}-\frac{1}{2}\omega(\overline{|r_{1}|}+\bar{r}_{1})\triangleq H_{3}. \end{eqnarray} $

(3.11)

From (3.9), (3.7) and (3.4), we have

$ \begin{eqnarray*} \frac{\omega \bar{a}_{2}e^{x_{2}(\xi_{2})}}{e^{H_{3}}+k_{2}^{L}}\geq\int_{0}^{\omega}\frac{a_{2}(t)\exp\{x_{2}(t)\}}{\exp\{x_{1}(t)\}+k_{2}(t)}\mathit{\rm{d}}t=\omega\bar{r}_{2}, \end{eqnarray*} $

i.e.,

$ \begin{eqnarray*} x_{2}(\xi_{2})\geq\ln\frac{\bar{r}_{2}(\exp\{H_{3}\}+k_{2}^{L})}{\bar{a}_{2}}, \end{eqnarray*} $

which, together with (3.6) and Lemma 2.2 imply

$ \begin{eqnarray} x_{2}(t)\geq x_{2}(\xi_{2})-\frac{1}{2}\int_{0}^{\omega}|x'_{2}(t)|\mathit{\rm{d}}t\geq\ln\frac{\bar{r}_{2}(\exp\{H_{3}\}+k_{2}^{L})}{\bar{a}_{2}}-\frac{1}{2}\omega(\overline{|r_{2}|}+\overline{r}_{2})\triangleq H_{4}, \end{eqnarray} $

(3.12)

It follows from (3, 9)-(3.12) that

$ \begin{eqnarray} \|x\|\leq |H_{1}|+|H_{2}|+|H_{3}|+|H_{4}|\triangleq H_{0}. \end{eqnarray} $

(3.13)

Obviously, $H_{0}$ is independent of $\lambda$.

Considering the following algebraic equations

$ \begin{eqnarray} \left\{\begin{array}{ll} \bar{r}_{1}-\bar{b}\exp\{x_{1}\}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{1}(t)\exp\{x_{1}+x_{2}\}}{\exp\{2x_{1}\}+k^{2}(t)}\mathit{\rm{d}}t=0, \\ \bar{r}_{2}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{2}(t)\exp\{x_{2}\}}{\exp\{x_{1}\}+k_{2}(t)}=0 \end{array}\right. \end{eqnarray} $

(3.14)

If the system (3.14) has a solution or a number of solutions $x^{\ast}=(x_{1}^{\ast}, x_{2}^{\ast})^{T}$, then similar arguments as those of (3.9)-(3.12) show that

$ x_{1}^{\ast} \leq \ln\frac{\bar{r}_{1}}{\bar{b}}\leq H_{1}, x_{2}^{\ast} \leq \ln\frac{\bar{r}_{2}(e^{H_{1}}+k_{2}^{U})}{\bar{a}_{2}}\leq H_{2}, \\ x_{1}^{\ast} \geq \ln\frac{\bar{r}_{1}-\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}}{\bar{b}}\geq H_{3}, x_{2}^{\ast} \geq \ln\frac{\bar{r}_{2}(e^{H_{3}}+k_{2}^{L})}{\bar{a}_{2}}\geq H_{4}. $

Hence

$ \begin{eqnarray} \|x^{\ast}\|=\|(x_{1}^{\ast}, x_{2}^{\ast})^{T}\|=\max\{|x_{1}^{\ast}|+|x_{2}^{\ast}|\}<H_{0}. \end{eqnarray} $

(3.15)

Set $\Omega=\{x=(x_{1}, x_{2})^{T}\in X: \|x\| <H_{0}\}$. Then, $Lx\neq \lambda Nx$ for $x\in\partial\Omega$ and $\lambda\in (0, 1)$, that is $\Omega$ satisfies condition (ⅰ) in Lemma 2.1.

Suppose $x\in \partial\Omega\cap {\rm{Ker}} L$ with $\|x\|=H_{0}$. If the (3.14) has at least one solution, we obtain from (3.15) that

$ \begin{eqnarray*} QNx=\begin{bmatrix}\bar{r}_{1}-\bar{b}\exp\{x_{1}\}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{1}(t)\exp\{x_{1}+x_{2}\}}{\exp\{2x_{1}\}+k_{1}^{2}(t)}\mathit{\rm{d}}t\\ \bar{r}_{2}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{2}(t)\exp\{x_{2}\}}{\exp\{x_{1}\}+k_{2}(t)}\mathit{\rm{d}}t\end{bmatrix}\neq 0. \end{eqnarray*} $

If the system (3.14) does not have a solution, then

$ \begin{eqnarray*} QNx=\begin{bmatrix}\bar{r}_{1}-\bar{b}\exp\{x_{1}\}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{1}(t)\exp\{x_{1}+x_{2}\}}{\exp\{2x_{1}\}+k_{1}^{2}(t)}\mathit{\rm{d}}t\\ \bar{r}_{2}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{2}(t)\exp\{x_{2}\}}{\exp\{x_{1}\}+k_{2}(t)}\mathit{\rm{d}}t\end{bmatrix}\neq 0. \end{eqnarray*} $

Thus, the condition $(ⅱ)$ in Lemma 2.1 is satisfied.

Finally in order to prove (ⅲ) in Lemma 2.1 we define homomorphism mapping

$ \begin{eqnarray*} J:\mathit{\rm{Im}}Q\rightarrow \mathit{\rm{Ker}}L, \quad x\rightarrow x, \end{eqnarray*} $

and

$\begin{eqnarray*} &&H:\quad \mathit{\rm{Dom}}X\times [0, 1], \\ &&H(x_{1}, x_{2}, \mu)=\begin{bmatrix}\bar{r}_{1}-\bar{b}e^{x_{1}}-\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}\\ \bar{r}_{2}-\frac{\bar{a}_{2}e^{x_{2}}}{e^{H_{3}}+k^{L}_{2}}\end{bmatrix} +\mu\begin{bmatrix}\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{1}e^{x_{1}+x_{2}}}{e^{2x_{1}}+k_{1}^{2}(t)}\mathit{\rm{d}}t\\ \frac{\bar{a}_{2}e^{x_{2}}}{e^{H_{3}}+k^{L}_{2}}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{2}(t)\exp\{x_{2}\}}{\exp\{x_{1}\}+k_{2}(t)}\mathit{\rm{d}}t\end{bmatrix}, \end{eqnarray*} $

where $\mu\in[0, 1]$ is a parameter. We will show that if $x=(x_{1}, x_{2})^{T}\in \partial\Omega\cap {\rm{Ker}} L$, $x=(x_{1}, x_{2})^{T}$ is a constant vector in $R^{2}$ with $\max\{|x_{1}|, |x_{2}|\}=H_{0}$, then $H(x_{1}, x_{2}, \mu)\neq 0$. Otherwise, suppose that $x=(x_{1}, x_{2})^{T}\in R^{2}$ with $\max\{|x_{1}|, |x_{2}|\}=H_{0}$: satisfying $H(x_{1}, x_{2}, \mu)=0$, that is,

$ \begin{eqnarray*} &&\bar{r}_{1}-\bar{b}e^{x_{1}}-\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}+\mu[\overline{(\frac{a_{1}}{2k_{1}})}e^{H_{2}}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{1}e^{x_{1}+x_{2}}}{e^{2x_{1}}+k_{1}^{2}(t)}\mathit{\rm{d}}t]=0, \\ &&\bar{r}_{2}-\frac{\bar{a}_{2}e^{x_{2}}}{e^{H_{3}}+k^{L}_{2}}+\mu[\frac{\bar{a}_{2}e^{x_{2}}}{e^{H_{3}}+k^{L}_{2}}-\frac{1}{\omega}\int_{0}^{\omega}\frac{a_{2}(t)\exp\{x_{2}\}}{\exp\{x_{1}\}+k_{2}(t)}\mathit{\rm{d}}t]=0. \end{eqnarray*} $

Similar argument as those of (3.14), (3.15) show that

$ \begin{eqnarray*} \|x\|=\max\{|x_{1}|+|x_{2}|\}\leq H_{0}, \end{eqnarray*} $

which is a contradiction.

Hence by a direct calculation, we have

$ \begin{array}{l} {\rm{deg}}\{ JQN,\Omega \cap {\rm{Ker}}L,0\} {\rm{ = deg}}\{ H({x_1},{x_2},1),\Omega \cap {\rm{Ker}}L,0\} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ = deg}}\{ H({x_1},{x_2},0),\Omega \cap {\rm{Ker}}L,0\} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ne 0. \end{array} $

(3.16)

So, (ⅲ) in Lemma 2.2 is satisfied. By applying Lemma 2.1, we conclude that system (1.2) has at least one positive $\omega$-periodic solution. The proof is complete.

Remark 3.1 It is notable that our result only need $b(t)$, $a_{1}(t)$, $a_{2}(t)$, $k_{1}(t)$, $k_{2}(t)$, $\tau_{i}(t)\, \, i=1, 2$, $\sigma(t)$ are all positive $\omega$-periodic continuous functions; but $r_{i}(t)\in C(R, R)\, \, i=1, 2$ are $\omega-$periodic continuous functions, $\int_{0}^{\omega}r_{i}(t)dt>0, i=1, 2$, and the growth functions $r_{i}(t), i=1, 2$ are not necessarily positive. It is reasonable on the biology. In addition, one can easily find that time delays $\tau_{i}(t)\, \, i=1, 2$, $\sigma(t)$ do not necessarily remain nonnegative. Moreover, Theorem 3.1 will remain valid for systems (1.2) if the delayed terms are replaced by the term with discrete time delays, state-dependent delays, or deviating argument. Hence, time delays of any type or the deviating argument have no effect on the existence of positive solutions.

If the time delayed term $\sigma (t)$ vanishes, $\tau_{1}(t)\equiv \tau_{1}$, $\tau_{2}\equiv\tau_{2}$ and $k^{2}_{1}(t)\equiv k_{1}$, $k_{2}(t)\equiv k_{2}$, then the system (1.2) is reduced to the system (1.1) which has been studied by Zhang et al in [7]. Thus from Theorem 3.1, we have the following result.

Corollary 3.1 Assume ${\overline{r}_{1}}>\frac{\overline{a}_{1}}{2k_{1}}e^{H^{\ast}_{2}}$ hold, where

$ H^{\ast}_{2}=\ln \frac{\overline{r}^{\ast}_{2}(e^{H^{\ast}_{1}}+k_{2})}{\overline{a}_{2}}+\frac{1}{2}\omega(\overline{|r_{2}|}+\overline{r}_{2}),H^{\ast}_{1}= \ln \frac{\overline{r}_{1}}{\overline{b}}+\frac{1}{2}\omega(\overline{|r_{1}|}+\bar{r}_{1}). $

Then system (1.1) has at least one positive $\omega$-periodic solution.

Remark 3.2 In [7], Zhang et al suppose $r_{i}(t), i=1, 2$ are positive. From Corollary 3.1. It is easy to known that $r_{i}(t)\in C(R, R)$, $\int_{0}^{\omega}r_{i}(t)dt>0$, so $r_{i}(t), i=1, 2$ are not necessarily positive. We improve the result of [7].

Definition 4.1 System (1.2) is said to be permanent if there exist positive constants $T$, $M_{i}$, $m_{i}, i=1, 2, $ such that any solution $(x(t), y(t))^{T}$ of (1.2) satisfies $m_{1}\leq x(t)\leq M_{1}, m_{2}\leq y(t)\leq M_{2}$ for $t\geq T.$

Lemma 4.1[16] If $a>0$, $b>0$, $\tau(t)\geq 0$, then

(1) If $y'(t)\leq y(t)[b-ay(t-\tau(t))]$, then there exists a constant $T>0$ such that $y(t)\leq \frac{b}{a}\exp\{b\tau^{U}\}$ for $t>T; $

(2) If $y'(t)\geq y(t)[b-ay(t-\tau(t))]$, then there exists a constant $T$ and $M$ such that $y(t) <M$ for $t>T, $ then for any small constant $\varepsilon>0$ there exists a constant $T^{\ast}>T$ such that $y(t)\geq \min\{\frac{b}{a}\exp\{(b-aM)\tau^{U}\}, \frac{b}{a}-\varepsilon\}$ for $t\geq T^{\ast}.$

Lemma 4.2  There exists positive constant $T_{0}$ such that the solution $(x(t), y(t))$ of (1.2) satisfies

$ \begin{eqnarray*} 0<x(t)\leq M_{1}, \quad \mathit{\rm{and}}\quad 0<y(t)\leq M_{2}, \quad \mathit{\rm{for}} \;t\geq T_{0}, \end{eqnarray*} $

where

$ M_{1}=\frac{r_{1}^{U}}{b_{1}^{L}}\exp\{r_{1}^{U}\tau_{1}^{U}\}$, $M_{2}=\frac{(M_{1}+k_{2}^{U})r_{2}^{U}}{a_{2}^{L}}\exp\{r_{2}^{U}\tau_{2}^{U}\}. $

Proof   If follows from system (1.2) that

$ \begin{eqnarray*} x'(t)\leq x(t)[r_{1}^{U}-b^{L}x(t-\tau_{1}(t))]. \end{eqnarray*} $

From Lemma 4.1 yield that there exists a positive constant $T_{1}$ such that $x(t)\leq M_{1}, \quad \mathit{\rm{for}} \quad t\geq T_{1}.$ Then we get

$ \begin{eqnarray*} y'(t)\leq y(t)[r_{2}^{U}-\frac{a_{2}^{L}y(t-\tau_{2}(t))}{M_{1}+k_{2}^{U}}], \quad \mathit{\rm{for}} \quad t\geq T_{1}. \end{eqnarray*} $

So, there exists a positive $T_{0}\geq T_{1}$ such that $ y(t)\leq M_{2}, \quad \mathit{\rm{for}} \quad t\geq T_{0}. $

Lemma 4.3 If $\Delta_{1}>0$ then there exists a positive constant $T^{\ast}$ such that the solution $(x(t), y(t))$ of system (1.2) satisfies

$ \begin{eqnarray*} x(t)\geq m_{1}, \quad \mathit{\rm{and}}\quad y(t)\geq m_{2}, \quad \mathit{\rm{for}} \quad t\geq T^{\ast}, \end{eqnarray*} $

where $\varepsilon$ is a small enough positive constant, and

$ \begin{eqnarray*} &&\Delta_{1}=\big[r_{1}-\frac{a_{1}M_{2}}{2k_{1}}\big]^{L}, m_{1}=\min\bigg\{\frac{\Delta_{1}}{b^{U}}\exp\big\{(\Delta_{1}-b^{U}M_{1})\tau_{1}^{U}\big\}, \frac{\Delta_{1}}{b^{U}}-\varepsilon\bigg\}, \\ && m_{2}=\min\bigg\{\frac{r_{2}^{L}k_{2}^{L}}{a_{2}^{U}}\exp\big\{(r_{2}^{L}-\frac{a_{2}^{U}M_{2}}{k_{2}^{L}})\tau_{2}^{U}\big\}, \frac{r_{2}^{L}k_{2}^{L}}{a_{2}^{U}}-\varepsilon\bigg\}. \end{eqnarray*} $

Proof   If follows from Lemma 4.2 and system (1.2) that, for $t\geq T_{0}$,

$ \begin{eqnarray} \left\{\begin{array}{ll} x'(t)\geq x(t)[\Delta_{1}-b^{U}x(t-\tau_{1}(t))], \\ y'(t)\geq y(t)[r_{2}^{L}-\frac{a_{2}^{U}y(t-\tau_{2}(t))}{k_{2}^{L}}]. \end{array}\right. \end{eqnarray} $

(4.1)

which, together with Lemma 4.1 and 4.2, implies that there exists a positive constant $T^{\ast}\geq T_{0}$ such that $x(t)\geq m_{1}$ and $y(t)\geq m_{2}$ for $t\geq T^{\ast}$.

From Lemma 4.2 and 4.3, we can get the following result on the permanence of System (1.1).

Theorem 4.1  If $\Delta_{1}>0$, then system (1.2) is permanent.

Similar to the proofs of Lemma 4.2 and Lemma 4.3, we have

Corollary 4.1  If $\Delta_{1}>0$, then system (1.1) is permanent.

Example 1  Consider the following equation

$ \left\{ {\begin{array}{*{20}{l}} {x'(t) = x(t)[{r_1}(t) - b(t)x(t - {\tau _1}) - \frac{{{a_1}(t)x(t)y(t)}}{{{x^2}(t) + k_1^2}}],}\\ {y'(t) = y(t)[{r_2}(t) - \frac{{{a_2}(t)y(t - {\tau _2})}}{{x(t - {\tau _2}) + {k_2}}}].} \end{array}} \right. $

(4.2)

where $r_{1}(t)=3+2\sin(12\pi t)$, $b(t)=1-0.1\sin(12\pi t)$, $a_{1}(t)=0.5+0.1\sin(12\pi t)$, $k_{1}^{2}=9$, $r_{2}(t)=0.8+0.2\sin(12\pi t)$, $\tau_{1}=\frac{1}{10}$, $\tau_{2}=0$, $a_{2}(t)=0.3-0.1\sin(12\pi t)$, and $k_{2}=1$, It is easy to calculation, and all the conditions in Theorem 3.1, 3.2 and 4.1 hold. So we know the system (4.3) has at least one positive periodic solution and permanent (see Figures 1, 2. We take $x(0)=1, y(0)=5$ and $x(0))=4, y(0)=5$).

Example 2   If $r_{1}(t)=8+2\sin(2\pi t)$, $b(t)=2-0.1\sin(2\pi t)$, $a_{1}(t)=0.5+0.1\sin(2\pi t)$, $k_{1}^{2}(t)=9$, $r_{2}(t)=0.8+0.2\sin(2\pi t)$, $\tau_{1}(t)=1$, $\tau_{2}(t)=0.5$, $\sigma(t)=0$, $a_{2}(t)=0.3-0.1\sin(2\pi t)$, and $k_{2}(t)=1$, It is easy to calculation, and all the conditions in Theorem 3.1, 3.2 and 4.1 hold. So we know the system (4.2) has at least one positive periodic solution and permanent (see Figure 3).