We consider that Assumption 1.1 always holds. Taking a biological meaning of the model into account, we discuss the solution of system (2) with the initial condition \((x(0), y(0), z(0))\in \mathbb{R}_{+}^{3}\) in \(\mathbb{R}_{+}^{3} = \{(x,y,z)\in \mathbb{R}^{3}| x>0,y>0,z>0\}\). On the basis of Lemma 2.7, together with constructing a proper Lyapunov function, the sufficient condition of a positive period solution of (2) will be obtained. For convenience, we let

$$\begin{aligned} \lambda =& \frac{1}{\theta } \int _{0}^{\theta } \biggl(-2a_{21}^{u}a_{31}^{u} \biggl[D_{1}(t)-\frac{\alpha ^{2}(t)}{2} \biggr] + a_{11}^{l}a_{31}^{l} \biggl[D_{2}(t)+ \frac{\beta ^{2}(t)}{2} \biggr] \\ &{}+ a_{11}^{l}a_{21}^{l} \biggl[D_{3}(t)+ \frac{\gamma ^{2}(t)}{2} \biggr] \biggr)\,dt. \end{aligned}$$

### Theorem 3.1

*If* \(\lambda >0\) *and*

$$\begin{aligned}& a_{12}^{l} \geq a_{21}^{u}, a_{13}^{l} \geq a_{31}^{u}, a_{22}^{l} \geq a_{32}^{u}, \end{aligned}$$

(5)

$$\begin{aligned}& 2a_{31}^{u}a_{12}^{l}\geq a_{11}^{l}a_{32}^{u}, \end{aligned}$$

(6)

*then for system* (2), *there exists at least one periodic solution with* *θ* *as its period*.

### Proof

(4) holds in that parameters of system (2) are all continuous and bounded periodic functions. Now we will prove that conditions (I) and (II) of Lemma 2.7 hold. The Lyapunov function \(V: [0,+\infty )\times \mathbb{R}_{+}^{3}\rightarrow \mathbb{R}\) on \(\mathbb{C}^{2}\) is defined as

$$ \begin{aligned} V(t,x,y,z) &= M \bigl(2a_{21}^{u}a_{31}^{u} \ln x +a_{11}^{l}a_{31}^{l} \ln y + a_{11}^{l}a_{21}^{l}\ln z \bigr) + \frac{(x+y+z)^{2}}{2} + M \varpi \\ &= V_{1}(x,y,z) + V_{2}(x,y,z) + V_{3}(t), \end{aligned} $$

where \(M = (2/\lambda )\max \{1,\sup_{(x,y,z)\in \mathbb{R}_{+}^{3}}Q(x,y,z) \}\),

$$ \begin{aligned} Q(x,y,z) =& -\frac{1}{2} a_{11}^{l}x^{3}+ \biggl(D_{1}^{u}+ \frac{(\alpha ^{2})^{u}}{2} \biggr)x^{2} +\frac{(\beta ^{2})^{u}}{2}y^{2} + \frac{(\gamma ^{2})^{u}}{2}z^{2} - \bigl(D_{1}^{l}+D_{2}^{l} \bigr)xy \\ & + \bigl(-D_{3}^{l}+D_{1}^{u} \bigr)xz + \bigl(-D_{3}^{l}-D_{2}^{l} \bigr)yz, \end{aligned} $$

\(V_{2}(x,y,z)=\frac{(x+y+z)^{2}}{2}\), \(V_{3}(t) = M\varpi \), obviously \(M\lambda \geq 2\). Let

$$ \begin{aligned}[b] \dot{\varpi } ={}&{-} \frac{1}{\theta } \int _{0}^{\theta } \biggl(-2a_{21}^{u}a_{31}^{u} \biggl[D_{1}(t)-\frac{\alpha ^{2}(t)}{2} \biggr] \\ &{}+ a_{11}^{l}a_{31}^{l} \biggl[D_{2}(t)+\frac{\beta ^{2}(t)}{2} \biggr] + a_{11}^{l}a_{21}^{l} \biggl[D_{3}(t)+\frac{\gamma ^{2}(t)}{2} \biggr] \biggr)\,dt \\ &{} -2a_{21}^{u}a_{31}^{u} \biggl[D_{1}(t)-\frac{\alpha ^{2}(t)}{2} \biggr] + a_{11}^{l}a_{31}^{l} \biggl[D_{2}(t)+\frac{\beta ^{2}(t)}{2} \biggr] \\ &{} + a_{11}^{u}a_{21}^{u} \biggl[D_{3}(t)+\frac{\gamma ^{2}(t)}{2} \biggr] \\ ={}& {-}\lambda + \biggl(-2a_{21}^{u}a_{31}^{u} \biggl[D_{1}(t)- \frac{\alpha ^{2}(t)}{2} \biggr] \\ &{}+ a_{11}^{l}a_{31}^{l} \biggl[D_{2}(t)+\frac{\beta ^{2}(t)}{2} \biggr] + a_{11}^{l}a_{21}^{l} \biggl[D_{3}(t)+\frac{\gamma ^{2}(t)}{2} \biggr] \biggr). \end{aligned} $$

(7)

It is obvious that \(\varpi (t)\) is a periodic function with period *θ*. As a matter of fact, integrating (7) from *t* to \(t+\theta \), we get

$$ \begin{aligned} \varpi (t+\theta ) - \varpi (\theta )={}& \int _{t}^{t+ \theta } \dot{\varpi }(s)\,ds \\ ={}& {-} \int _{0}^{\theta } \biggl(-2a_{21}^{u}a_{31}^{u} \biggl[D_{1}(s)- \frac{\alpha ^{2}(s)}{2} \biggr] + a_{11}^{l}a_{31}^{l} \biggl[D_{2}(s)+ \frac{\beta ^{2}(s)}{2} \biggr] \\ &{}+ a_{11}^{l}a_{21}^{l} \biggl[D_{3}(s)+\frac{\gamma ^{2}(s)}{2} \biggr] \biggr)\,ds + \int _{0}^{\theta } \biggl(-2a_{21}^{u}a_{31}^{u} \biggl[D_{1}(s)- \frac{\alpha ^{2}(s)}{2} \biggr] \\ &{}+ a_{11}^{l}a_{31}^{l} \biggl[D_{2}(s)+\frac{\beta ^{2}(s)}{2} \biggr] + a_{11}^{l}a_{21}^{l} \biggl[D_{3}(s)+\frac{\gamma ^{2}(s)}{2} \biggr] \biggr)\,ds = 0. \end{aligned} $$

Now we will prove that condition (I) of Lemma 2.7 holds. Since quadratic terms of \(V(t,x,y,z)\) are all positive, then

$$ \inf_{(x,y,z)\in \mathbb{R}_{+}^{3} \backslash E_{\kappa }} V(t,x,y,z) \rightarrow \infty , \quad \text{when } \kappa \rightarrow \infty , $$

where \(E_{\kappa } = (\frac{1}{\kappa },\kappa )\times ( \frac{1}{\kappa },\kappa )\times (\frac{1}{\kappa },\kappa )\). Now we will prove that condition (II) of Lemma 2.7 holds. By using Itô’s formula and condition (6), we have

$$\begin{aligned} \mathscr{L}V_{1}(x,y,z) ={}& M \biggl(2a_{21}^{u}a_{31}^{u} \bigl[D_{1}(t) - a_{11}(t)x - a_{12}(t)y - a_{13}(t)z -\delta _{1}(t)S(t) \bigr] \\ &{}+ a_{11}^{l}a_{31}^{l} \bigl[-D_{2}(t) + a_{21}(t)x-a_{22}(t)z - \delta _{2}(t)S(t) \bigr] \\ &{}+ a_{11}^{l}a_{21}^{l} \bigl[-D_{3}(t) +a_{31}(t)x + a_{32}(t)y - \delta _{2}(t)S(t) \bigr] \\ &{}- a_{21}^{l}a_{31}^{l} \frac{\alpha ^{2}(t)}{2} - a_{11}^{l}a_{31}^{l} \frac{\beta ^{2}(t)}{2} + a_{11}^{u}a_{21}^{u} \frac{\gamma ^{2}(t)}{2} \biggr) \\ \leq{} & M \biggl( 2a_{21}^{u}a_{31}^{u} \biggl[D_{1}(t)- \frac{\alpha ^{2}(t)}{2} \biggr] - a_{11}^{l}a_{31}^{l} \biggl[D_{2}(t)+ \frac{\beta ^{2}(t)}{2} \biggr] \\ &{}- a_{11}^{l}a_{21}^{l} \biggl[D_{3}(t)+\frac{\gamma ^{2}(t)}{2} \biggr] \biggr). \end{aligned}$$

(8)

Considering (7) and (8) together, we get

$$ \mathscr{L}(V_{1}+V_{3}) = -M\lambda . $$

(9)

Similarly, we get the following conclusion by using (5):

$$\begin{aligned} \mathscr{L}V_{2}(x,y,z) =& (x+y+z) \biggl( D_{1}(t)x-a_{11}(t)x^{2}-a_{12}(t)xy-a_{13}(t)xz- \delta _{1}(t)S -D_{2}(t)y \\ &{} +a_{21}(t)xy-a_{22}(t)yz-\delta _{2}(t)S -D_{3}(t)z+a_{31}(t)xz+a_{32}(t)yz -\delta _{3}(t)S \\ &{} +\frac{\alpha ^{2}(t)}{2}x^{2} +\frac{\beta ^{2}(t)}{2}y^{2} + \frac{\gamma ^{2}(t)}{2}z^{2} \biggr) \\ \leq& -a_{11}^{l}x^{3}+ \biggl[D_{1}^{u}+\frac{(\alpha ^{2})^{u}}{2} \biggr]x^{2} + \biggl[-D_{2}^{l}+ \frac{(\beta ^{2})^{u}}{2} \biggr]y^{2} + \biggl[-D_{3}^{l}+ \frac{(\beta ^{2})^{u}}{2} \biggr]z^{2} \\ &{} + \bigl(-D_{2}^{l}+D_{2}^{u} \bigr)xy + \bigl(-D_{3}^{l}+D_{1}^{u} \bigr)xz - \bigl(D_{3}^{l} + D_{2}^{l} \bigr)yz. \end{aligned}$$

(10)

Considering (9) and (10), we get

$$ \begin{aligned}[b] \mathscr{L}V(t,x,y,z)\leq{} & {-}M\lambda -a_{11}^{l}x^{3}+ \biggl[D_{1}^{u}+ \frac{(\alpha ^{2})^{u}}{2} \biggr]x^{2} + \biggl[-D_{2}^{l}+ \frac{(\beta ^{2})^{u}}{2} \biggr]y^{2} \\ &{} + \biggl[-D_{3}^{l}+\frac{(\beta ^{2})^{u}}{2} \biggr]z^{2} + \bigl(-D_{2}^{l}+D_{2}^{u} \bigr)xy \\ & {}+ \bigl(-D_{3}^{l}+D_{1}^{u} \bigr)xz - \bigl(D_{3}^{l} + D_{2}^{l} \bigr)yz \\ ={}& {-}M\lambda -\frac{1}{2}a_{11}^{l}x^{3}-D_{2}^{l}y^{2} -D_{3}^{l}z^{2} +Q(x,y,z). \end{aligned} $$

(11)

Now we define a bounded close set

$$ \mathscr{D}= \biggl\{ (x,y,z)\in \mathbb{R}_{+}^{3}: \epsilon \leq x\leq \frac{1}{\epsilon }, \epsilon \leq y\leq \frac{1}{\epsilon }, \epsilon \leq z\leq \frac{1}{\epsilon } \biggr\} , $$

where \(0<\epsilon <1\). We choose *ϵ* small enough such that

$$\begin{aligned}& -M\lambda - \frac{a_{11}^{l}}{2\epsilon ^{3}} + Q_{ \sup } \leq -1, \end{aligned}$$

(12)

$$\begin{aligned}& -M\lambda - \frac{D_{2}^{l}}{\epsilon ^{2}} + Q_{\sup } \leq -1, \end{aligned}$$

(13)

$$\begin{aligned}& -M\lambda - \frac{D_{3}^{l}}{\epsilon ^{2}} + Q_{\sup } \leq -1, \end{aligned}$$

(14)

where \(Q_{\sup } = \sup_{(x,y,z)\in \mathbb{R}_{+}^{3}} Q(x,y,z)\). Let

$$\begin{aligned}& \mathscr{D}_{\epsilon }^{1} = \bigl\{ (x,y,z)\in \mathbb{R}_{+}^{3}: 0< x< \epsilon \bigr\} ,\qquad \mathscr{D}_{\epsilon }^{2} = \bigl\{ (x,y,z)\in \mathbb{R}_{+}^{3}: 0< y< \epsilon \bigr\} , \\& \mathscr{D}_{\epsilon }^{3} = \bigl\{ (x,y,z)\in \mathbb{R}_{+}^{3}: 0< z< \epsilon \bigr\} ,\qquad \mathscr{D}_{\epsilon }^{4} = \biggl\{ (x,y,z)\in \mathbb{R}_{+}^{3}: x> \frac{1}{\epsilon } \biggr\} , \\& \mathscr{D}_{\epsilon }^{5} = \biggl\{ (x,y,z)\in \mathbb{R}_{+}^{3}: y> \frac{1}{\epsilon } \biggr\} ,\qquad \mathscr{D}_{\epsilon }^{6} = \biggl\{ (x,y,z)\in \mathbb{R}_{+}^{3}: z> \frac{1}{\epsilon } \biggr\} . \end{aligned}$$

A complementary set of \(\mathscr{D}\) can be denoted as \(\mathscr{D}^{C} = \mathscr{D}_{\epsilon }^{1} \cup \mathscr{D}_{\epsilon }^{2} \cup \mathscr{D}_{\epsilon }^{3} \cup \mathscr{D}_{\epsilon }^{4} \cup \mathscr{D}_{\epsilon }^{5} \cup \mathscr{D}_{\epsilon }^{6}\). Now we prove that \(\mathscr{L}V(t,x,y,z)\leq -1\) is valid on \([0,+\infty ) \times \mathscr{D}^{C}\).

Case 1. When \((t,x,y,z)\in [0,+\infty )\times (\mathscr{D}_{\epsilon }^{1} \cup \mathscr{D}_{\epsilon }^{2} \cup \mathscr{D}_{\epsilon }^{3})\), we have

$$ \begin{aligned}[b] \mathscr{L}V(t,x,y,z) &\leq -M\lambda +Q(x,y,z)\leq -M \lambda +Q_{\sup } \\ &\leq -M\lambda +\frac{M\lambda }{2} = -\frac{M\lambda }{2} \leq -1. \end{aligned} $$

(15)

Case 2. When \((t,x,y,z)\in [0,+\infty )\times \mathscr{D}_{\epsilon }^{4}\), on the basis of (12), we have

$$ \mathscr{L}V(t,x,y,z) \leq -M\lambda -\frac{a_{11}^{l}}{2}x^{3} + Q(x,y,z) \leq -M\lambda - \frac{a_{11}^{l}}{2\epsilon ^{3}} + Q_{\sup } \leq -1. $$

(16)

Case 3. When \((t,x,y,z)\in [0,+\infty )\times \mathscr{D}_{\epsilon }^{5}\), from (13), we get

$$ \mathscr{L}V(t,x,y,z) \leq -M\lambda - D_{2}^{l}y^{2} + Q(x,y,z)\leq -M \lambda - \frac{D_{2}^{l}}{\epsilon ^{2}} + Q_{\sup } \leq -1. $$

(17)

Case 4. When \((t,x,y,z)\in [0,+\infty )\times \mathscr{D}_{\epsilon }^{6}\), on the basis of (14), we get

$$ \mathscr{L}V(t,x,y,z) \leq -M\lambda - D_{3}^{l}y^{2} + Q(x,y,z)\leq -M \lambda - \frac{D_{3}^{l}}{\epsilon ^{2}} + Q_{\sup } \leq -1. $$

(18)

Thus, from (15)–(18), we get

$$ \mathscr{L}V(t,x,y,z) \leq -1, \quad \forall (t,x,y,z)\in [0,+\infty ) \times \mathscr{D}^{C}. $$

(19)

So Lemma 2.7 (II) is true, and there exists a periodic solution of system (2) with period *θ*. Besides, from Lemma 2.9, there exists a unique positive solution of system (2). Thus there exists at least one periodic solution of system (2) with period *θ*. □

Thanks to Part 3 in [3], we obtain the condition of a periodic solution of system (2), which is an expansion of Theorem 1 in [6].