Existence of positive periodic solutions for a generalized predator-prey model with diffusion feedback controls

Abstract

By employing the continuation theorem of coincidence degree theory, we derive a sufficient condition for the existence and attractivity of at least a positive periodic solution of the generalized predator-prey model with exploited term.

MSC:92D25, 34C25, 34K15.

1 Introduction

In recent years, the existence of positive periodic solutions of the prey-predator model has been widely studied [1–3]. The qualitative analysis of predator-prey systems is an interesting mathematical problem and has attracted a great attention of many mathematicians and biologists [4, 5]. Recently, Xu and Chen [6] investigated the two-species ratio-dependent predator-prey different model with time delay. Since a realistic model requires the inclusion of the effect of changing environment, recently, Shihua and Feng [7] have considered the following model:

$$\{\begin{array}{c}{x}_1^{\mathrm{\prime }}(t)=x_1(t)(a_1(t)-a_{11}(t)x_1(t)-\frac{a_{13}(t)x_3(t)}{m(t)x_3(t)+x_1(t)})\hfill \\ \phantom{x_1^{\mathrm{\prime }}(t)=}+D_1(t)(x_2(t)-x_1(t))-h_1(t),\hfill \\ x_2^{\mathrm{\prime }}(t)=x_2(t)(a_2(t)-a_{22}(t)x_2(t))+D_2(t)(x_1(t)-x_2(t))-h_2(t),\hfill \\ x_3^{\mathrm{\prime }}(t)=x_3(t)(-a_3(t)-a_4(t)x_3(t)+\frac{a_{31}(t)x_1(t-\tau )}{m(t)x_3(t-\tau )+x_1(t-\tau )})-h_3(t),\hfill \end{array}$$

(1.1)

where $D_i(t)$ ($i=1,2$), $a_i(t)$ ($i=1,2,3$), $a_{11}(t)$, $a_{13}(t)$, $a_{22}(t)$, $a_{31}(t)$ and $m(t)$ are strictly positive continuous w-periodic functions.

In the paper, we will study the following model:

$$\{\begin{array}{c}{x}_1^{\mathrm{\prime }}(t)=x_1(t)(g_1(t,x_1(t))-\frac{a_{13}(t)x_3(t)}{m(t)x_3(t)+x_1(t)})\hfill \\ \phantom{x_1^{\mathrm{\prime }}(t)=}+D_1(t)(x_2(t)-x_1(t))-h_1(t),\hfill \\ x_2^{\mathrm{\prime }}(t)=x_2(t)(g_2(t,x_2(t))+D_2(t)(x_1(t)-x_2(t))-h_2(t),\hfill \\ x_3^{\mathrm{\prime }}(t)=x_3(t)(-a_3(t)-a_4(t)x_3(t)+\frac{a_{31}(t)x_1(t-\tau )}{m(t)x_3(t-\tau )+x_1(t-\tau )})-h_3(t),\hfill \end{array}$$

(1.2)

where $D_i(t)$ ($i=1,2$), $a_{13}(t)$, $a_{31}(t)$, $m(t)$ are the same as in model (1.1). Some assumptions on the above functions on $g_i(t,x)$ ($i=1,2$) will be given in next section.

Our aim in this paper is to establish a sufficient condition for the existence and attractivity of at least a positive w-periodic solution of model (1.2).

2 Main result

To obtain the existence of positive periodic solutions of system (1.2), we summarize some concepts and results from [5] that will be basic for this section.

Let X, Z be Banach spaces, let $L:DomL\subset X\to Y$ be a linear mapping, and let $N:X\to Z$ be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if $dimKerL=codimImL<+\mathrm{\infty }$ and ImL is closed in Z. If L is a Fredholm mapping of index zero, there exist continuous projectors $P:X\to Z$ and $Q:Z\to Z$ such that $ImP=KerL$ and $ImL=KerQ=Im(I-Q)$. It follows that $L/DomL\cap KerP:(I-P)X\to ImL$ is invertible. We denote the inverse of that map by Kp. If Ω is an open-bounded subset of X, the mapping N will be called L-compact on $\overline{\mathrm{\Omega }}$ if $QN(\overline{\mathrm{\Omega }})$ is bounded and $Kp(I-Q)N:\overline{\mathrm{\Omega }}\to X$ is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism $J:ImQ\to KerL$.

In the proof of our existence theorem, we will use the continuation theorem of Gaines and Mawhin [8].

Lemma 2.1 [8]

Let L be a Fredholm mapping of index zero and let N be L-compact on $\overline{\mathrm{\Omega }}$. Suppose the following:

• (i) for each $\lambda \in (0,1)$, every solution x of $Lx=\lambda Nx$ is such that $x\overline{\in }\partial \mathrm{\Omega }$;

• (ii) $QNx\ne 0$ for each $x\in \partial \mathrm{\Omega }\cap KerL$;

• (iii) $deg\{JQN,\mathrm{\Omega }\cap KerL,0\}\ne 0$.

The $Lx=Nx$ has at least one solution in $DomL\cap \overline{\mathrm{\Omega }}$.

For convenience, we introduce the notations

$$\overline{f}=\frac{1}{w}{\int }_0^{w}f(t)dt,f^L=\underset{t\in [0,w]}{min}|f(t)|,f^M=\underset{t\in [0,w]}{max}|f(t)|,$$

where f is a continuous w-periodic function.

Our main result on the global existence of a positive periodic solution of system (1.2) is stated in the following theorem.

Theorem 2.1 Assume that

(${\mathrm{H}}_1$) there exists a constant A such that for $\mathrm{\forall }x\in R$, $t\in R$, when $x\ge A$,

$$g_1(t,e^x)\le 0;$$

(${\mathrm{H}}_2$) there exists a constant B such that for $\mathrm{\forall }x\in R$, when $x\ge B$,

$$g_2(t,e^x)\le 0;$$

(${\mathrm{H}}_3$) there exists a constant C ($C<A$) such that for $\mathrm{\forall }x\in R$, $t\in R$, when $x\le C$,

$$e^x{g}_1(t,e^x)\ge {\left(\frac{a_{13}}{m}\right)}^M{e}^x+h_1^M;$$

(${\mathrm{H}}_4$) there exists a constant D ($D<B$) such that for $\mathrm{\forall }x\in R$, $t\in R$, when $x\le D$,

$$e^x{g}_2(t,e^x)\ge h_2^M;$$

(${\mathrm{H}}_5$) $a_{31}^M{e}^{d_1}>h_3^l{m}^l$.

Then system (1.1) has at least one positive w-periodic solution.

Proof

Consider the system

$$\{\begin{array}{c}{u}_1^{\mathrm{\prime }}(t)=g_1(t,e^{u_1(t)})-\frac{a_{13}(t)e^{u_3(t)}}{m(t)e^{u_3(t)}+e^{u_1(t)}}\hfill \\ \phantom{u_1^{\mathrm{\prime }}(t)=}+D_1(t)(e^{u_2(t)-u_1(t)}-1)-h_1(t)e^{-u_1(t)},\hfill \\ u_2^{\mathrm{\prime }}(t)=g_2(t,e^{u_2(t)})+D_2(t)(e^{u_1(t)-u_2(t)}-1)-h_2(t)e^{-u_2(t)},\hfill \\ u_3^{\mathrm{\prime }}(t)=-a_3(t)-a_4(t)e^{u_3(t)}+\frac{a_3(t)e^{u_1(t-\tau )}}{m(t)e^{u_3(t)+e^{u_1(t-\tau )}}}-h_3(t)e^{-u_3(t)}.\hfill \end{array}$$

(2.1)

Let $x_i(t)=e^{u_i(t)}$, $i=1,2,3$, then system (1.2) changes into system (2.1). Hence it is easy to see that system (2.1) has a w-periodic solution ${(u_1^{\ast }(t),u_2^{\ast }(t),u_3^{\ast }(t))}^T$, then ${(e^{u_1^{\ast }(t)},e^{u_2^{\ast }(t)},e^{u_3^{\ast }(t)})}^T$ is a positive w-periodic solution of system (1.2). Therefore, for (1.2) to have at least one positive w-periodic solution, it is sufficient that (2.1) has at least one w-periodic solution. In order to apply Lemma 2.1 to system (2.1), we take

$$X=Z=\{u(t)={(u_1(t),u_2(t),u_3(t))}^T\in C(R,R^3),u(t+w)=u(t)\}$$

and

$$\parallel u\parallel =\parallel {(u_1(t),u_2(t),u_3(t))}^T\parallel =\sum _{i=1}^3\underset{t\in [0,w]}{max}|u_i(t)|$$

for any $u\in Z$ (or Z). Then X and Z are Banach spaces with the norm $\parallel \cdot \parallel$. Let

$$\begin{array}{c}Nu=\left[\begin{array}{c}{g}_1(t,e^{u_1(t)})-\frac{a_{13}(t)e^{u_3(t)}}{m(t)e^{u_3(t)}+e^{u_1(t)}}+D_1(t)(e^{u_2(t)-u_1(t)}-1)-h_1(t)e^{-u_1(t)}\\ g_2(t,e^{u_2(t)})+D_2(t)(e^{u_1(t)-u_2(t)}-1)-h_2(t)e^{-u_2(t)}\\ -a_3(t)-a_4(t)e^{u_3(t)}+\frac{a_3(t)e^{u_1(t-\tau )}}{m(t)e^{u_3(t)+e^{u_1(t-\tau )}}}-h_3(t)e^{-u_3(t)}\end{array}\right],u\in X,\hfill \\ Lu=u^{\prime }=\frac{du(t)}{dt},pu=\frac{1}{w}{\int }_0^{w}u(t)dt,u\in X;\hfill \\ Qz=\frac{1}{w}{\int }_0^{w}z(t)dt,z\in Z.\hfill \end{array}$$

Then it follows that

$$\begin{array}{c}KerL=R^3,ImL=\{z\in Z:{\int }_0^{w}z(t)dt=0\}\text{ is closed in }Z,\hfill \\ dimKerL=3=codimImL,\hfill \end{array}$$

and P, Q are continuous projectors such that

$$ImP=KerL,KerQ=ImL=Im(I-Q).$$

Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) $Kp:ImL\to KerP\cap DomL$ is given by

$$Kp(z)={\int }_0^{t}z(s)ds-\frac{1}{w}{\int }_0^w{\int }_0^{t}z(s)dsdt.$$

Thus

$$QNu=\left[\begin{array}{c}1/w{\int }_0^w{F}_1(s)ds\\ 1/w{\int }_0^w{F}_2(s)ds\\ 1/w{\int }_0^w{F}_3(s)ds\end{array}\right]$$

and

$$Kp(I-Q)Nu=\left[\begin{array}{c}{\int }_0^t{F}_1(s)ds-1/w{\int }_0^w{\int }_0^t{F}_1(s)dsdt+(1/2-t/w){\int }_0^w{F}_1(s)ds\\ {\int }_0^t{F}_2(s)ds-1/w{\int }_0^w{\int }_0^t{F}_2(s)dsdt+(1/2-t/w){\int }_0^w{F}_2(s)ds\\ {\int }_0^t{F}_3(s)ds-1/w{\int }_0^w{\int }_0^t{F}_3(s)dsdt+(1/2-t/w){\int }_0^w{F}_3(s)ds\end{array}\right],$$

where

$$\begin{array}{c}{F}_1(s)=g_1(s,e^{u_1(s)})-\frac{a_{13}(s)e^{u_3(s)}}{m(s)e^{u_3(s)}+e^{u_1(s)}}+D_1(s)(e^{u_2(s)-u_1(s)}-1)-h_1(s)e^{-u_1(s)},\hfill \\ F_2(s)=g_2(s,e^{u_2(s)})+D_2(s)(e^{u_1(s)-u_2(s)}-1)-h_2(s)e^{-u_2(s)}\hfill \end{array}$$

and

$$F_3(s)=-a_3(s)-a_4(s)e^{u_3(s)}+\frac{a_3(s)e^{u_1(s-\tau )}}{m(s)e^{u_3(s-\tau )}+e^{u_1(s-\tau )}}-h_3(s)e^{-u_3(s)}.$$

Obviously, QN and $Kp(I-Q)N$ are continuous. It is not difficult to show that $Kp(I-Q)N(\overline{\mathrm{\Omega }})$ is compact for any open bounded $\mathrm{\Omega }\subset X$ by using the Arzela-Ascoli theorem. Moreover, $QN(\overline{\mathrm{\Omega }})$ is clearly bounded. Thus, N is L-compact on $\overline{\mathrm{\Omega }}$ with any open bounded set $\mathrm{\Omega }\subset X$.

Now we reach the point where we search for an appropriate open bounded subset Ω for the application of the continuation theorem (Lemma 2.1). Corresponding to the operator equation $Lx=\lambda Nx$, $\lambda \in (0,1)$, we have

$$\{\begin{array}{c}{u}_1^{\mathrm{\prime }}(t)=\lambda [g_1(t,e^{u_1(t)})-\frac{a_{13}(t)e^{u_3(t)}}{m(t)e^{u_3(t)}+e^{u_1(t)}}\hfill \\ \phantom{u_1^{\mathrm{\prime }}(t)=}+D_1(t)(e^{u_2(t)-u_1(t)}-1)-h_1(t)e^{-u_1(t)}],\hfill \\ u_2^{\mathrm{\prime }}(t)=\lambda [g_2(t,e^{u_2(t)})+D_2(t)(e^{u_1(t)-u_2(t)}-1)-h_2(t)e^{-u_2(t)}],\hfill \\ u_3^{\mathrm{\prime }}(t)=\lambda [-a_3(t)-a_4(t)e^{u_3(t)}+\frac{a_{31}(t)e^{u_1(t-\tau )}}{m(t)e^{u_3(t)+e^{u_1(t-\tau )}}}-h_3(t)e^{-u_3(t)}].\hfill \end{array}$$

(2.2)

Assume that $u=u(t)\in X$ is a solution of system (2.2) for a certain $\lambda \in (0,1)$.

Because of ${(u_1(t),u_2(t),u_3(t))}^T\in X$, there exist ${\xi }_i,{\eta }_i\in [0,w]$ such that

$$u_i({\xi }_i)=\underset{t\in [0,w]}{max}{u}_i(t),u_i({\eta }_i)=\underset{t\in [0,w]}{min}{u}_i(t),i=1,2,3.$$

It is clear that

$$u_i^{\mathrm{\prime }}({\xi }_i)=0,u_i^{\mathrm{\prime }}({\eta }_i)=0,i=1,2,3.$$

From this and system (2.2), we obtain

$$g_1({\xi }_1,e^{u_1({\xi }_1)})-\frac{a_{13}({\xi }_1)e^{u_3({\xi }_1)}}{m({\xi }_1)e^{u_3({\xi }_1)}+e^{u_1({\xi }_1)}}+D_1({\xi }_1)(e^{u_2({\xi }_1)-u_1({\xi }_1)}-1)-h_1({\xi }_1)e^{-u_1({\xi }_1)}=0,$$

(2.3)

$$g_2({\xi }_2,e^{u_2({\xi }_2)})+D_2({\xi }_2)(e^{u_1({\xi }_2)-u_2({\xi }_2)}-1)-h_2({\xi }_2)e^{-u_2({\xi }_2)}=0,$$

(2.4)

$$-a_3({\xi }_3)-a_4({\xi }_3)e^{u_3({\xi }_3)}+\frac{a_{31}({\xi }_3)e^{u_1({\xi }_3-\tau )}}{m({\xi }_3)e^{u_3({\xi }_3-\tau )+e^{u_1({\xi }_3-\tau )}}}-h_3({\xi }_3)e^{-u_3({\xi }_3)}=0,$$

(2.5)

$$\begin{array}{c}{g}_1({\eta }_1,e^{u_1({\eta }_1)})-\frac{a_{13}({\eta }_1)e^{u_3({\eta }_1)}}{m({\eta }_1)e^{u_3({\eta }_1)}+e^{u_1({\eta }_1)}}+D_1({\eta }_1)(e^{u_2({\eta }_1)-u_1({\eta }_1)}-1)\\ -h_1({\eta }_1)e^{-u_1({\eta }_1)}=0,\end{array}$$

(2.6)

$$g_2({\eta }_2,e^{u_2({\eta }_2)})+D_2({\eta }_2)(e^{u_1({\eta }_2)-u_2({\eta }_2)}-1)-h_2({\eta }_2)e^{-u_2({\eta }_2)}=0,$$

(2.7)

$$-a_3({\eta }_3)-a_4({\eta }_3)e^{u_3({\eta }_3)}+\frac{a_{31}({\eta }_3)e^{u_1({\eta }_3-\tau )}}{m({\eta }_3)e^{u_3({\eta }_3-\tau )+e^{u_1({\eta }_3-\tau )}}}-h_3({\eta }_3)e^{-u_3({\eta }_3)}=0.$$

(2.8)

There are two cases to consider for (2.3) and (2.4).

Case 1. Assume that $u_1({\xi }_1)\ge u_2({\xi }_2)$, then $u_1({\xi }_1)\ge u_2({\xi }_1)$.

From this and (2.3), we have

$$g_1({\xi }_1,e^{u_1({\xi }_1)})=\frac{a_{13}({\xi }_1)e^{u_3({\xi }_3)}}{m({\xi }_1)e^{u_3({\xi }_1)}+e^{u_1({\xi }_1)}}-D_1({\xi }_1)(e^{u_2({\xi }_1)-u_1({\xi }_1)}-1)+h_1({\xi }_1)e^{-u_1({\xi }_1)}>0,$$

which, together with condition (${\mathrm{H}}_1$) in Theorem 2.1, gives

$$u_1({\xi }_1)<A.$$

(2.9)

Thus

$$u_2({\xi }_2)\le u_1({\xi }_1)<A.$$

(2.10)

Case 2. Assume that $u_1({\xi }_1)\le u_2({\xi }_2)$, then $u_1({\xi }_2)<u_2({\xi }_2)$.

From this and (2.4), we have

$$g_2({\xi }_2,e^{u_2({\xi }_2)})=-D_2({\xi }_2)(e^{u_1({\xi }_2)-u_2({\xi }_2)}-1)+h_2({\xi }_2)e^{-u_2({\xi }_2)}>0,$$

which, together with condition (${\mathrm{H}}_2$) in Theorem 2.1, gives

$$u_2({\xi }_2)<B.$$

(2.11)

Thus

$$u_1({\xi }_1)\le u_2({\xi }_2)<B.$$

(2.12)

From Case 1 and Case 2, we obtain

$$u_1({\xi }_1)<max\{A,B\}\stackrel{\mathrm{def}}{=}{d}_1,$$

(2.13)

$$u_2({\xi }_2)<max\{A,B\}=d_2.$$

(2.14)

From (2.5), we get

$$a_4{e}^{u_3({\xi }_3)}\le a_4({\xi }_3)e^{u_3({\xi }_3)}\le \frac{a_{31}({\xi }_3)e^{u_1({\xi }_3-\tau )}}{m({\xi }_3)e^{u_3({\xi }_3-\tau )+e^{u_1({\xi }_3-\tau )}}}<a_{31}^M.$$

Thus

$$u_3({\xi }_3)\le ln\left(\frac{a_{31}^M}{a_4^l}\right)\stackrel{\mathrm{def}}{=}{d}_3.$$

(2.15)

There are two cases to consider for (2.6) and (2.7).

Case 1. Assume that $u_1({\eta }_1)\le u_2({\eta }_2)$, then $u_1({\eta }_1)<u_2({\eta }_1)$. From this and (2.5), we have

$$\begin{array}{rcl}{g}_1({\eta }_1,e^{u_1({\eta }_1)})& =& \frac{a_{13}({\eta }_1)e^{u_3({\eta }_1)}}{m({\eta }_1)e^{u_3({\eta }_1)}+e^{u_1({\eta }_1)}}-D_1({\eta }_1)(e^{u_2({\eta }_1)-u_1({\eta }_1)}-1)+h_1({\eta }_1)e^{-u_1({\eta }_1)}\\ <& \frac{a_{13}({\eta }_1)e^{u_3({\eta }_1)}}{m({\eta }_1)e^{u_3({\eta }_1)}+e^{u_1({\eta }_1)}}+h_1({\eta }_1)e^{-u_1({\eta }_1)}<{\left(\frac{a_{13}}{m}\right)}^M+h_1^M{e}^{-u_1({\eta }_1)},\end{array}$$

which, together with condition (${\mathrm{H}}_3$) in Theorem 2.1, gives

$$u_1({\eta }_1)>C.$$

(2.16)

Hence

$$u_2({\eta }_2)>u_1({\eta }_1)>C.$$

(2.17)

Case 2. Assume that $u_1({\eta }_1)\ge u_2({\eta }_2)$, then $u_1({\eta }_2)\ge u_2({\eta }_2)$. From this and (2.7), we have

$$g_2({\eta }_2,e^{u_2({\eta }_2)})=-D_2({\eta }_2)(e^{u_1({\eta }_2)-u_2({\eta }_2)}-1)+h_2({\eta }_2)e^{-u_2({\eta }_2)}<h_2^M{e}^{-u_2({\eta }_2)},$$

which, together with condition (${\mathrm{H}}_4$) in Theorem 2.1, gives

$$u_2({\eta }_2)>D.$$

(2.18)

Hence

$$u_1({\eta }_1)>u_2({\eta }_2)>D.$$

(2.19)

From Case 1 and Case 2, we have

$$u_1({\eta }_1)>min\{C,D\}\stackrel{\mathrm{def}}{=}{\rho }_1,$$

(2.20)

$$u_2({\eta }_2)>min\{C,D\}={\rho }_1.$$

(2.21)

From Theorem 2.1(${\mathrm{H}}_5$), we get

$$u_3({\eta }_3)>ln\left(\frac{h_3^l}{a_{31}^M-a_3^l}\right).$$

(2.22)

From (2.11)-(2.22), we obtain, for $\mathrm{\forall }t\in R$,

$$\begin{array}{c}|u_1(t)|\le max\{|d_1|,|{\rho }_1|\}\stackrel{\mathrm{def}}{=}{R}_1,\hfill \\ |u_2(t)|\le max\{|d_1|,|{\rho }_1|\}\stackrel{\mathrm{def}}{=}{R}_2\hfill \end{array}$$

and

$$|u_3(t)|\le max\{|a_3|,|{\rho }_3|\}\stackrel{\mathrm{def}}{=}{R}_3.$$

Clearly, $R_i$ ($i=1,2,3$) are independent of λ. Denote $M={\sum }_{i=1}^3{R}_i+R_0$, here $R_0$ is taken sufficiently large such that each solution ${({\alpha }^{\ast },{\beta }^{\ast },{\gamma }^{\ast })}^T$ of the system

$$\begin{array}{c}{g}_1(t,e^{\alpha })-\frac{{\overline{a}}_{13}{e}^{\gamma }}{m(t_3)e^{\gamma }+e^{\alpha }}+{\overline{D}}_1(e^{\beta -\alpha }-1)-{\overline{h}}_1{e}^{-\alpha }=0,\hfill \\ g_2(t_2,e^{\beta })+{\overline{D}}_2(e^{\alpha -\beta }-1)-{\overline{h}}_2{e}^{-\beta }=0,\hfill \\ -{\overline{a}}_3-a_4{e}^r+\frac{{\overline{a}}_{31}{e}^{\alpha }}{m(t_4)e^{\gamma }+e^{\alpha }}-{\overline{h}}_3{e}^{-\gamma }=0\end{array}$$

(2.23)

satisfies $\parallel {({\alpha }^{\ast },{\beta }^{\ast },{\gamma }^{\ast })}^T\parallel =|{\alpha }^{\ast }|+|{\beta }^{\ast }|+|{\gamma }^{\ast }|<M$, provided that system (2.23) has a solution or a number of solutions, and that

$$max\{|d_1|,|{\rho }_1|\}+max\{|d_1|,|{\rho }_1|\}+max\{|d_3|,|{\rho }_3|\}<M,$$

where $t_i\in (0,w)$ will appear in $QNu$ below.

Now we take $\mathrm{\Omega }=\{u={(u_1(t),u_2(t),u_3(t))}^T\in X:\parallel u\parallel <M\}$. This satisfies condition (i) of Lemma 2.1. When $u\in \partial \mathrm{\Omega }\cap KerL=\partial \mathrm{\Omega }\cap R^3$, u is a constant vector in $R^3$ with ${\sum }_{i=1}^3|u_i|=M$. If system (2.23) has one or more solutions, then

$$QNu=\left[\begin{array}{c}{g}_1(t_1,e^{u_1})-\frac{{\overline{a}}_{13}{e}^{u_3}}{m(t_3)e^{u_3}+e^{u_1}}+{\overline{D}}_1(e^{u_2-u_1}-1)-{\overline{h}}_1{e}^{-u_1}\\ g_2(t_2,e^{u_2})+{\overline{D}}_2(e^{u_1-u_2}-1)-{\overline{h}}_2{e}^{-u_2}\\ -{\overline{a}}_3-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{13}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}-{\overline{h}}_3{e}^{-u_3}\end{array}\right]\ne {(0,0,0)}^T,$$

where $t_i\in (0,w)$ are one constant.

If system (2.23) does not have a solution, then naturally

$$QNu\ne {(0,0,0)}^T.$$

This shows that condition (ii) of Lemma 2.1 is satisfied finally. We will prove that condition (iii) of Lemma 2.1 is satisfied. We only prove that when $u\in \partial \mathrm{\Omega }\cap KerL=\partial \mathrm{\Omega }\cap R^3$, $deg\{JQNu,\partial \mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\ne 0$. When $u\in \partial \mathrm{\Omega }\cap KerL=\partial \mathrm{\Omega }\cap R^3$, u is a constant vector in $R^3$ with ${\sum }_{i=1}^3|u_i|=M$. Our proof will be broken into three steps as follows.

Step 1. We prove

$$\begin{array}{c}deg\{JQNu,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{(g(t_1,e^{u_1}),g(t_2,e^{u_2}),-{\overline{a}}_3-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}-h_3{e}^{-u_3})}^T,\hfill \\ \mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}.\hfill \end{array}$$

To this end, we define the mapping ${\varphi }_1:DomL\times [0,1]\to X$ by

$$\begin{array}{rcl}{\varphi }_1(u_1,u_2,u_3,{\mu }_1)& =& \left[\begin{array}{c}{g}_1(t_1,e^{u_1})\\ g_2(t_2,e^{u_2})\\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}\end{array}\right]\\ +{\mu }_1\left[\begin{array}{c}-\frac{{\overline{a}}_{13}{e}^{u_3}}{m(t_3)e^{u_3}+e^{u_1}}+{\overline{D}}_1(e^{u_2-u_1}-1)-{\overline{h}}_1{e}^{u_1}\\ {\overline{D}}_2(e^{u_1-u_2}-1)-{\overline{h}}_2{e}^{u_2}\\ -{\overline{a}}_3-{\overline{h}}_3{e}^{-u_3}\end{array}\right],\end{array}$$

where ${\mu }_1\in [0,1]$ is a parameter, when $u={(u_1,u_2,u_3)}^T\in \partial \mathrm{\Omega }\cap KerL=\partial \mathrm{\Omega }\cap R^3$, u is a constant vector in $R^3$ with ${\sum }_{i=1}^3|u_i|=M$. We will show that when $u\in \partial \mathrm{\Omega }\cap KerL$, ${\varphi }_1(u_1,u_2,u_3,{\mu }_1)\ne 0$, if the conclusion is not true, i.e., the constant vector u with ${\sum }_{i=1}^3|u_i|=M$ satisfies ${\varphi }_1(u_1,u_2,u_3,{\mu }_1)=0$, then from

$$\begin{array}{c}{g}_1(t_1,e^{u_1})+{\mu }_1(\frac{-{\overline{a}}_{13}{e}^{u_3}}{m(t_3)e^{u_3}+e^{u_1}}+{\overline{D}}_1(e^{u_2-u_1}-1))-{\overline{h}}_1{e}^{u_1}=0,\hfill \\ g_2(t_2,e^{u_2})+{\mu }_1({\overline{D}}_2(e^{u_1-u_2}-1)-{\overline{h}}_2{e}^{u_2})=0,\hfill \\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}-{\mu }_1({\overline{a}}_3+{\overline{h}}_3{e}^{-u_3})=0\hfill \end{array}$$

it follows the arguments of (2.11)-(2.22) that

$$|u_i|<R_i,i=1,2,3.$$

Thus

$$\sum _{i=1}^3|u_i|<\sum _{i=1}^3{R}_i<M,$$

which contradicts the fact that ${\sum }_{i=1}^3|u_i|=M$.

According to topological degree theory, we have

$$\begin{array}{c}deg\left\{(JQN,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T)\right\}\hfill \\ =deg\{{\varphi }_1{(u_1,u_2,u_3,1)}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{\varphi }_1{(u_1,u_2,u_3,0)}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{(g_1(t_1,e^{u_1}),g_2(t_2,e^{u_2}),-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}})}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}.\hfill \end{array}$$

Step 2. We prove

$$\begin{array}{c}deg\{{(g_1(t_1,e^{u_1}),g_2(t_2,e^{u_2}),-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}})}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{({\overline{a}}_1-{\overline{a}}_{11}{e}^{u_1},g(t_2,e^{u_2}),-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}})}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\},\hfill \end{array}$$

where ${\overline{a}}_1$, ${\overline{a}}_{11}$ are two chosen positive constants such that

$$C<ln\frac{{\overline{a}}_1}{{\overline{a}}_{11}}<A.$$

To this end, we define the mapping ${\varphi }_2:DomL\times [0,1]\to X$ by

$$\begin{array}{rcl}{\varphi }_2(u_1,u_2,u_3,{\mu }_2)& =& {\mu }_2\left[\begin{array}{c}{\overline{a}}_1-{\overline{a}}_{11}{e}^{u_1}\\ g_2(t_2,e^{u_2})\\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}\end{array}\right]\\ +(1-{\mu }_2)\left[\begin{array}{c}{g}_1(t_1,e^{u_1})\\ g_2(t_2,e^{u_2})\\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}\end{array}\right]\\ =& \left[\begin{array}{c}{\mu }_2({\overline{a}}_1-{\overline{a}}_{11}{e}^{u_1})+(1-{\mu }_2)g_1(t_1,e^{u_1})\\ g_2(t_2,e^{u_2})\\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}\end{array}\right],\end{array}$$

where ${\mu }_2\in [0,1]$ is a parameter. We will prove that when $u\in \partial \mathrm{\Omega }\cap KerL$, ${\varphi }_2(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$. When $u\in \partial \mathrm{\Omega }\cap KerL=\partial \mathrm{\Omega }\cap R^3$, u is a constant vector in $R^3$ with ${\sum }_{i=1}^3|u_i|=M$. Now we consider two possible cases:

$$(\mathrm{i})u_1\ge A;(\mathrm{ii})u_1<A.$$

• (i) When $u_1\ge A$, from condition (iii) in Theorem 2.1, we have $g(t_1,e^{u_1})\le 0$. Moreover, ${\overline{a}}_1-{\overline{a}}_{11}{e}^{u_1}\le {\overline{a}}_1-{\overline{a}}_{11}{e}^A<0$, thus ${\mu }_2({\overline{a}}_1-{\overline{a}}_{11}{e}^{u_1})+(1-{\mu }_2)g(t_1,e^{u_1})<0$. Therefore, ${\varphi }_1(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$.

• (ii) When $u_1<A$, if $u_1\le C$, from condition (${\mathrm{H}}_3$) in Theorem 2.1, we have $g(t_1,e^{u_1})>0$. However, ${\overline{a}}_1-{\overline{a}}_{11}{e}^{u_1}\ge {\overline{a}}_1-{\overline{a}}_{11}{e}^C>0$. Therefore, ${\varphi }_1(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$. If $u_1>C$, we also consider two possible cases: (a) $u_2\ge B$; (b) $u_2<B$. (a) When $u_2\ge B$, from condition (${\mathrm{H}}_2$) in Theorem 2.1, we have

  $$g_2(t_2,e^{u_2})<0.$$

Therefore ${\varphi }_1(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$. (b) When $u_2<B$, if $u_2\le D$, then from condition (${\mathrm{H}}_4$) in Theorem 2.1, we obtain $g_2(t_2,e^{u_2})>0$. Consequently, ${\varphi }_2(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$. If $u_2>D$, we can claim when $u\in \partial \mathrm{\Omega }\cap KerL=\partial \mathrm{\Omega }\cap R^3$, ${\varphi }_2(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$. Otherwise, from

$$-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}=0$$

we have

$$e^{u_3}<\frac{{\overline{a}}_{31}}{{\overline{a}}_4}$$

and

$$e^{u_3}>\frac{-{\overline{a}}_4{e}^{{\rho }_1}+\sqrt{{(-{\overline{a}}_4{e}^{{\rho }_1})}^2+4-{\overline{a}}_{4}m(t_4){\overline{a}}_{31}{e}^{{\rho }_1}}}{2{\overline{a}}_{4}m(t_4)}>0,$$

i.e.,

$$\begin{array}{c}{u}_3<ln{\overline{a}}_{31}-ln{\overline{a}}_4,\hfill \\ u_3>ln\frac{-{\overline{a}}_4{e}^{{\rho }_1}+\sqrt{{(-{\overline{a}}_4{e}^{{\rho }_1})}^2+4-{\overline{a}}_{4}m(t_4){\overline{a}}_{31}{e}^{{\rho }_1}}}{2{\overline{a}}_{4}m(t_4)}.\hfill \end{array}$$

Thus

$$\begin{array}{c}|u_1|<max\{|d_1|,|{\rho }_1|\},\hfill \\ |u_2|<max\{|d_1|,|{\rho }_1|\}\hfill \end{array}$$

and

$$|u_3|<max\{|d_3|,|{\rho }_3|\}.$$

Therefore

$$\begin{array}{rcl}\sum _{i=1}^3|u_i|& <& max\{|d_1|,|{\rho }_1|\}+max\{|d_1|,|{\rho }_1|\}\\ +max\{|d_3|,|{\rho }_3|\}<M,\end{array}$$

which contradicts the fact that ${\sum }_{i=1}^3|u_i|=M$. Based on the above discussion, for any $u\in \partial \mathrm{\Omega }\cap KerL$, we have ${\varphi }_2(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$. According to topological degree theory, we obtain

$$\begin{array}{c}deg\{{(g_1(t_1,e^{u_1}),g_2(t_2,e^{u_2}),-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}})}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{\varphi }_2{(u_1,u_2,u_3,1)}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{\varphi }_2{(u_1,u_2,u_3,0)}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{(a_1-a_{11}{e}^{u_1},g_2(t_2,e^{u_2}),-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}})}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}.\hfill \end{array}$$

Step 3. We prove

$$\begin{array}{c}deg\{{(a_1-a_{11}{e}^{u_1},g_2(t_2,e^{u_2}),-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}})}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{(a_1-a_{11}{e}^{u_1},a_2-a_{22}{e}^{u_2},-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}})}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}.\hfill \end{array}$$

To this end, we define the mapping ${\varphi }_3:DomL\times [0,1]\to X$ by

$$\begin{array}{rcl}{\varphi }_3(u_1,u_2,u_3,{\mu }_3)& =& {\mu }_3\left[\begin{array}{c}{a}_1-a_{11}{e}^{u_1}\\ a_2-a_{22}{e}^{u_2}\\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}\end{array}\right]\\ +(1-{\mu }_3)\left[\begin{array}{c}{a}_1-a_{11}{e}^{u_1}\\ g_2(t_2,e^{u_2})\\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}\end{array}\right]\\ =& \left[\begin{array}{c}{a}_1-a_{11}{e}^{u_1}\\ {\mu }_3({\overline{a}}_2-{\overline{a}}_{22}{e}^{u_2})+(1-{\mu }_3)g(t_2,e^{u_2})\\ -{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}\end{array}\right],\end{array}$$

where ${\mu }_3\in [0,1]$ is a parameter and $a_2$, $a_{22}$ are two chosen positive constants such that $D<ln\frac{a_2}{a_{22}}<B$. We will prove that when $u\in \partial \mathrm{\Omega }\cap KerL$, ${\varphi }_3(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$. If it is not true, then the constant vector u satisfies ${\varphi }_3(u_1,u_2,u_3,{\mu }_2)\ne {(0,0,0)}^T$ with ${\sum }_{i=1}^3|u_i|=M$. Thus we have

(2.24) implies

$$C<u_1=ln\frac{a_1}{a_{11}}<A.$$

(2.27)

We claim that $u_2<B$; otherwise, if $u_2\ge B$, then from condition (${\mathrm{H}}_2$) in Theorem 2.1, we have

$$(1-u_3)g_2(t_2,e^{u_2})<0.$$

Consequently,

$${\mu }_3(a_2-a_{22}{e}^{u_2})+(1-{\mu }_3)g_2(t_2,e^{u_2})<0,$$

which contradicts (2.23). We also claim that $u_2>D$. If $u_2\le D$, then $g_2(t_2,e^{u_2})>0$. However, $a_2-a_{22}{e}^{u_2}>a_2-a_{22}{e}^D>0$.

Thus

$$u_3(a_2-a_{22}{e}^{u_2})+(1-{\mu }_3)g_2(t_2,e^{u_2})>0,$$

which contradicts (2.24). (2.26) gives

$$-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}=0,$$

that is,

$$\begin{array}{c}{u}_3<ln{\overline{a}}_{31}-ln{\overline{a}}_4,\hfill \\ u_3>ln\frac{-{\overline{a}}_4{e}^{{\rho }_1}+\sqrt{{(-{\overline{a}}_4{e}^{{\rho }_1})}^2+4-{\overline{a}}_{4}m(t_4){\overline{a}}_{31}{e}^{{\rho }_1}}}{2{\overline{a}}_{4}m(t_4)}.\hfill \end{array}$$

Thus

$$\begin{array}{c}|u_1|<max\{|d_1|,|{\rho }_1|\},\hfill \\ |u_2|<max\{|d_1|,|{\rho }_1|\}\hfill \end{array}$$

and

$$|u_3|<max\{|d_3|,|{\rho }_3|\}.$$

Therefore

$$\begin{array}{rcl}\sum _{i=1}^3|u_i|& <& max\{|d_1|,|{\rho }_1|\}+max\{|d_1|,|{\rho }_1|\}\\ +max\{|d_3|,|{\rho }_3|\}<M,\end{array}$$

which leads to a contradiction. Therefore, by means of topological degree theory, we have

$$\begin{array}{c}deg\{(a_1-a_{11}{e}^{u_1},g_2(t_2,e^{u_2}),-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}),\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{\varphi }_3{(u_1,u_2,u_3,1)}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{{\varphi }_3{(u_1,u_2,u_3,0)}^T,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{(a_1-a_{11}{e}^{u_1},a_2-a_{22}{e}^{u_2},-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}),\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}.\hfill \end{array}$$

From the proof of the three steps above, we obtain

$$\begin{array}{c}deg\{JQNu,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =deg\{(a_1-a_{11}{e}^{u_1},a_2-a_{22}{e}^{u_2},-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}),\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}.\hfill \end{array}$$

Because of condition (${\mathrm{H}}_5$) in Theorem 2.1, the system of algebraic equations

$$\{\begin{array}{c}{a}_1-a_{11}x=0,\hfill \\ a_2-a_{22}y=0,\hfill \\ -{\overline{a}}_{4}I+\frac{{\overline{a}}_{31}x}{m(t_4)z+x}=0\hfill \end{array}$$

has a unique solution ${(x^{\ast },y^{\ast },z^{\ast })}^T$ which satisfies

$$x^{\ast }=\frac{a_1}{a_{11}}>0,y^{\ast }=\frac{a_2}{a_{22}}>0,z^{\ast }=\frac{a_4{x}^{\ast }+\sqrt{{(a_4{x}^{\ast })}^2+4a_{4}m(t_4){\overline{a}}_{31}{x}^{\ast }}}{2{\overline{a}}_{4}m(t_4)}>0.$$

Thus

$$\begin{array}{c}deg\{(a_1-a_{11}{e}^{u_1},a_2-a_{22}{e}^{u_2},-{\overline{a}}_4{e}^{u_3}+\frac{{\overline{a}}_{31}{e}^{u_1}}{m(t_4)e^{u_3}+e^{u_1}}),\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}\hfill \\ =sign\left|\begin{array}{ccc}-a_{11}{x}^{\ast }& 0& 0\\ 0& -a_{22}{y}^{\ast }& 0\\ \cdots & 0& -{\overline{a}}_4{z}^{\ast }-\frac{{\overline{a}}_{31}m(t_4)x^{\ast }{z}^{\ast }}{{[m(t_4)z^{\ast }+x^{\ast }]}^2}\end{array}\right|=-1.\hfill \end{array}$$

Therefore, from (2.20), we have

$$deg\{JQNu,\mathrm{\Omega }\cap KerL,{(0,0,0)}^T\}=-1.$$

This completes the proof of Theorem 2.1. □

3 An example

Consider the system

$$\{\begin{array}{c}{x}_1^{\mathrm{\prime }}=x_1(t)(a_1(t)-a_{11}(t)x_1(t)-\frac{a_{13}(t)x_3(t)}{m(t)x_3(t)+x_1(t)})+D_1(t)(x_2(t)-x_1(t)),\hfill \\ x_2^{\mathrm{\prime }}=x_2(t)(a_2(t)-a_{22}(t)x_2(t))+D_2(t)(x_1(t)-x_2(t)),\hfill \\ x_3^{\mathrm{\prime }}=x_3(t)(-a_3(t)+\frac{a_{31}(t)x_1(t-T)}{m(t)x_3(t-T)+x_1(t-T)}),\hfill \end{array}$$

(3.1)

where $\tau >0$ is a positive constant, all the parameters are positive continuous w-periodic functions with periodic $w>0$.

In Theorem 2.1, $g_1(t,e^x)=a_1(t)-a_{11}(t)e^x$, $g_2(t,e^x)=a_2(t)-a_{22}(t)e^x$. It is easily shown that if $x\ge ln(\frac{a_1^m}{a_{11}^l})=A$, then $g_1(t,e^x)\le 0$ and if $x\ge ln(\frac{a_2^M}{a_{22}^l})=B$, then $g_2(t,e^x)\le 0$. We also can show if

$$\begin{array}{c}x\le ln\frac{a_1^M-{(\frac{a_{13}}{m})}^M}{a_{11}^l}=C,\hfill \\ g_1(t,e^x)\ge {\left(\frac{a_{11}}{m}\right)}^M\hfill \end{array}$$

and if $x\le ln\frac{a_2^M}{a_{22}^l}=D$, then $g_2(t,e^x)>0$.

(${\mathrm{H}}_1$) $a_1^m>{(\frac{a_{13}}{m})}^M$,

(${\mathrm{H}}_2$) ${\overline{a}}_{31}>{\overline{a}}_3$.

By Theorem 2.1, we have the following theorem.

Theorem 3.1 If (${\mathrm{H}}_1$) and (${\mathrm{H}}_2$) hold, the system (3.1) has at least one positive w-periodic solution. Consider the system

$$\{\begin{array}{c}{x}_1^{\mathrm{\prime }}=x_1(t)(a_1(t)-a_{11}(t)x_1(t)-\frac{a_{13}(t)x_1(t)x_3(t)}{m(t)x_3^2(t)+x_1^2(t)})+D_1(t)(x_2(t)-x_1(t)),\hfill \\ x_2^{\mathrm{\prime }}=x_2(t)(a_2(t)-a_{22}(t)x_2(t))+D_2(t)(x_1(t)-x_2(t)),\hfill \\ x_3^{\mathrm{\prime }}=x_3(t)(-a_3(t)+\frac{a_{31}(t)x_1^2(t-\tau )}{m(t)x_3^2(t-\tau )+x_1^2(t-\tau )}),\hfill \end{array}$$

(3.2)

where $z>0$ is a positive constant, all the parameters are positive continuous w-periodic functions with periodic $w>0$.

In Theorem 2.1, $g_1(t,e^x)=a_1(t)-a_{11}(t)e^x$, $g_2(t,e^x)=a_2(t)-a_{22}(t)e^x$. It is easily shown that if $x\ge ln\frac{a_1^M}{a_{11}^l}=A$, then $g_1(t,e^x)\le 0$ and if $x\ge ln\frac{a_2^M}{a_{22}^l}=B$, then $g_2(t,e^x)\le 0$. We also can show if $x\le ln\frac{a_1^M-{(\frac{a_{13}}{2\sqrt{m}})}^M}{a_{11}^l}=C$, $g_1(t,e^x)\ge {(\frac{a_{13}}{2\sqrt{m}})}^m$ and if $x\le ln\frac{a_2^M}{a_{22}^l}=D$, then $g_2(t,e^x)\ge 0$.

(${\mathrm{H}}_1^{\mathrm{\prime }}$) $a_1^M>{(\frac{a_{13}}{2\sqrt{m}})}^M$,

(${\mathrm{H}}_2^{\mathrm{\prime }}$) ${\overline{a}}_{31}>{\overline{a}}_3$.

By Theorem 2.1, we have the following theorem.

Theorem 3.2 If (${\mathrm{H}}_1^{\mathrm{\prime }}$) and (${\mathrm{H}}_2^{\mathrm{\prime }}$) hold, system (3.2) has at least one positive w-periodic solution.