Here, we firstly introduce the following useful lemma.
Lemma 8
Assume that model (1) has the solution \((x_{1}(t),x_{2}(t),x_{3}(t),x_{4}(t))\). If there is an \(i\in \{1,2,3\}\) to satisfy \(\lim_{t\to \infty }\langle x_{i}(t)\rangle =0\) a.s., then, for all \(j>i\), \(\lim_{t\to \infty }x_{j}(t)=0\) a.s. holds.
Proof
We first use the Itô formula, then
$$\begin{aligned}& \ln x_{1}(t) = b_{1}t-a_{11} \int _{0}^{t}x_{1}(s) \, \mathrm{d}s-a_{12} \int _{0}^{t}x_{2}(s)\,\mathrm{d}s + \phi _{1}(t), \end{aligned}$$
(21)
$$\begin{aligned}& \begin{aligned} \ln x_{i}(t) ={}&b_{i}t+a_{ii-1} \int _{0}^{t}x_{i-1}(s)\,\mathrm{d}s -a_{ii} \int _{0}^{t}x_{i}(s)\,\mathrm{d}s \\ &{}-a_{ii+1} \int _{0}^{t}x_{i+1}(s)\,\mathrm{d}s+ \phi _{i}(t),\quad i=2,3, \end{aligned} \end{aligned}$$
(22)
and
$$ \ln x_{4}(t) =b_{4}t+a_{43} \int _{0}^{t}x_{3}(s) \, \mathrm{d}s-a_{44} \int _{0}^{t}x_{4}(s) \,\mathrm{d}s+ \phi _{4}(t), $$
(23)
where
$$\begin{aligned}& \begin{aligned} \phi _{1}(t)={}&\sigma _{1}B_{1}(t)+\ln x_{1}(0)+a_{12} \int _{-\tau _{12}}^{0} \int _{t+\theta }^{t}x_{2}(s)\,\mathrm{d}s \,\mathrm{d}\mu _{12}(\theta ) \\ &{}-a_{12} \int _{-\tau _{12}}^{0} \int _{\theta }^{0}x_{2}(s)\,\mathrm{d}s \,\mathrm{d}\mu _{12}( \theta ), \end{aligned} \\& \begin{aligned} \phi _{i}(t)={}&\sigma _{i}B_{i}(t)+\ln x_{i}(0) +a_{ii-1} \int _{-\tau _{ii-1}}^{0} \int _{\theta }^{0}x_{i-1}(s)\,\mathrm{d}s \,\mathrm{d}\mu _{ii-1}(\theta ) \\ &{}-a_{ii-1} \int _{-\tau _{ii-1}}^{0} \int _{t+\theta }^{t}x_{i-1}(s) \, \mathrm{d}s \,\mathrm{d}\mu _{ii-1}(\theta ) +a_{ii+1} \int _{-\tau _{ii+1}}^{0} \int _{t+\theta }^{t}x_{i+1}(s)\,\mathrm{d}s \,\mathrm{d}\mu _{ii+1}(\theta ) \\ &{}-a_{ii+1} \int _{-\tau _{ii+1}}^{0} \int _{\theta }^{0}x_{i+1}(s) \,\mathrm{d}s \,\mathrm{d}\mu _{ii+1}(\theta ),\quad i=2,3, \end{aligned} \\& \begin{aligned} \phi _{4}(t)={}&\sigma _{4}B_{4}(t)+\ln x_{4}(0)+a_{43} \int _{-\tau _{43}}^{0} \int _{\theta }^{0}x_{3}(s)\,\mathrm{d}s \,\mathrm{d}\mu _{43}(\theta ) \\ &{}-a_{43} \int _{-\tau _{43}}^{0} \int _{t+\theta }^{t}x_{3}(s)\,\mathrm{d}s \,\mathrm{d} \mu _{43}(\theta ). \end{aligned} \end{aligned}$$
Obviously, \(\lim_{t\to \infty }\frac{\phi _{i}(t)}{t}=0\) a.s. is obtained for \(i=1,2,3,4\) by Lemma 7. Assume \(\lim_{t\to \infty }\langle x_{i}(t)\rangle =0\) a.s. Then for any constant \(\varepsilon >0\) with \(b_{i+1}+a_{i+1i}\varepsilon <0\) there exists a \(T>0\) to satisfy \(\int _{0}^{t}x_{i}(s)\,\mathrm{d}s<\varepsilon t\) for any \(t\geq T\). Therefore, for \(t\geq T\), by (22) and (23), the following inequality is found:
$$ \ln x_{i+1}(t)\leq b_{i+1}t+a_{i+1i}\varepsilon t -a_{i+1i+1} \int _{0}^{t}x_{i+1}(s) \, \mathrm{d}s+ \phi _{i+1}(t). $$
Thus, by Lemma 5 we derive \(\lim_{t\to \infty }x_{i+1}(t)=0\) a.s. Consequently, \(\lim_{t\to \infty }x_{j}(t)=0\) a.s. for any \(j>i\). □
Remark 3
It is easy for us to find that Lemma 8 also seemingly can be extended to the general n-species stochastic food-chain system with distributed delay and harvesting.
In the following theorem, we state and prove a screening criterion as a main result in this paper on the extinction and persistence in mean of global positive solutions for model (1).
Theorem 1
Suppose that \((x_{1}(t),x_{2}(t),x_{3}(t),x_{4}(t))\) is any positive global solution of model (1). Then we derive:
-
(1)
If \(\Delta _{11}<0\), then \(\lim_{t\to \infty }x_{j}(t)=0\) a.s. for \(j=1,2,3,4\).
-
(2)
If \(\Delta _{11}=0\), then \(\lim_{t\to \infty }\langle x_{1}(t)\rangle =0\) and \(\lim_{t\to \infty }x_{j}(t)=0\) a.s. for \(j=2,3,4\).
-
(3)
If \(\Delta _{11}>0\) and \(\Delta _{22}<0\), then \(\lim_{t\to \infty }\langle x_{1}(t)\rangle = \frac{\Delta _{11}}{H_{1}}\) and \(\lim_{t\to \infty }x_{j}(t)=0\) a.s. for \(j=2,3,4\).
-
(4)
If \(\Delta _{22}=0\), then \(\lim_{t\to \infty }\langle x_{1}(t)\rangle = \frac{\Delta _{11}}{H_{1}}\), \(\lim_{t\to \infty }\langle x_{2}(t)\rangle =0\) and \(\lim_{t\to \infty }x_{j}(t)=0\) a.s. for \(j=3,4\).
-
(5)
If \(\Delta _{22}>0\) and \(\Delta _{33}<0\), then \(\lim_{t\to \infty }\langle x_{j}(t)\rangle = \frac{\Delta _{2j}}{H_{2}}\), \(j=1,2\), and \(\lim_{t\to \infty }x_{j}(t)=0\) a.s. for \(j=3,4\).
-
(6)
If \(\Delta _{33}=0\) and the condition
$$ a_{33}a_{22}H_{2}-a_{12}a_{21}a_{23}a_{32}>0 $$
(24)
holds, then \(\lim_{t\to \infty }\langle x_{j}(t)\rangle = \frac{\Delta _{2j}}{H_{2}}\), \(j=1,2\), \(\lim_{t\to \infty }\langle x_{3}(t)\rangle =0 \) and \(\lim_{t\to \infty }x_{4}(t)=0\) a.s.
-
(7)
If \(\Delta _{33}>0\), \(\Delta _{44}<0\) and the condition (24) holds, then \(\lim_{t\to \infty }\langle x_{j}(t)\rangle = \frac{\Delta _{3j}}{H_{3}}\), \(j=1, 2,3\), \(\lim_{t\to \infty }x_{4}(t)=0\) a.s.
-
(8)
If \(\Delta _{44}=0\) and the condition
$$ (a_{22}a_{33}H_{2}-a_{12}a_{21}a_{23}a_{32})a_{44}H_{3} -a_{23}a_{32}a_{34}a_{43}H_{2}^{2}>0 $$
(25)
holds, then \(\lim_{t\to \infty }\langle x_{j}(t)\rangle = \frac{\Delta _{3j}}{H_{3}}\), \(j=1,2,3\) and \(\lim_{t\to \infty }\langle x_{4}(t)\rangle =0\) a.s.
-
(9)
If \(\Delta _{44}>0\) and the condition (25) holds, then \(\lim_{t\to \infty }\langle x_{j}(t)\rangle = \frac{\Delta _{4j}}{H_{4}}\), \(j=1,2,3,4\), a.s.
Proof
For model (1), \((x_{1}(t),x_{2}(t),x_{3}(t),x_{4}(t))\) can be the positive global solution. Let \(V_{2}(t)=a_{21}\ln x_{1}(t)+a_{11}\ln x_{2}(t)\), \(V_{3}(t)=a_{32}V_{2}(t)+H_{2}\ln x_{3}(t)\) and \(V_{4}(t)=a_{43}V_{3}(t)+H_{3}\ln x_{4}(t)\). From (21)–(23), we obtain
$$ V_{4}(t)=\Delta _{44}t-H_{4} \int _{0}^{t}x_{4}(s)\,\mathrm{d}s+ \phi _{5}(t), $$
(26)
where \(\phi _{5}(t)=a_{21}a_{32}a_{43}\phi _{1}(t)+a_{43}a_{11}a_{32}\phi _{2}(t)+a_{43}H_{2} \phi _{3}(t)+H_{3}\phi _{4}(t)\). we apply the similar method that used for \(\phi _{1}(t)\), \(\lim_{t\to \infty }\frac{\phi _{5}(t)}{t}=0\) a.s. is obtained.
If \(\Delta _{44}>0\), then, by Lemma 7, and for any \(\varepsilon >0\) with \(\Delta _{44}-3\varepsilon >0\), there exists a constant \(T>0\) satisfies \(\ln x_{1}(t)<\frac{\varepsilon }{a_{43}a_{32}a_{21}+1}t\), \(\ln x_{2}(t)<\frac{\varepsilon }{a_{43}a_{32}a_{11}+1}t\) and \(\ln x_{3}(t)<\frac{\varepsilon }{a_{43}H_{2}+1}t\) for all \(t\geq T\). Then, from (26) we obtain
$$ H_{3}\ln x_{4}(t)>(\Delta _{44}-3 \varepsilon )t-H_{4} \int _{0}^{t}x_{4}(s) \,\mathrm{d}s+ \phi _{5}(t) $$
for all \(t\geq T\). Thus, from the arbitrary ε and Lemma 4
$$ \liminf_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \geq \frac{\Delta _{44}}{H_{4}} $$
(27)
is obtained.
If \(\Delta _{44}\leq 0\), then since \(\liminf_{t\to \infty }\langle x_{4}(t)\rangle \geq 0\), we also have \(\liminf_{t\to \infty }\langle x_{4}(t)\rangle \geq \frac{\Delta _{44}}{H_{4}}\). Let \(U_{2}(t)=a_{22}\ln x_{1}(t)-a_{12}\ln x_{2}(t)\) and \(U_{4}(t)=a_{43}U_{2}(t)-a_{12}a_{23}\ln x_{4}(t)\). By (21), (22) and (23), we compute
$$ U_{4}(t)=(a_{43} \Delta _{21}-b_{4}a_{12}a_{23})t+a_{12}a_{23}a_{44} \int _{0}^{t}x_{4}(s) \, \mathrm{d}s-H_{2}a_{43} \int _{0}^{t}x_{1}(s) \, \mathrm{d}s+ \phi _{6}(t), $$
(28)
where \(\phi _{6}(t)=a_{22}a_{43}\phi _{1}(t)-a_{12}a_{43}\phi _{2}(t)-a_{12}a_{23} \phi _{4}(t)\). we apply the similar method that used for \(\phi _{1}(t)\), \(\lim_{t\to \infty }\frac{\phi _{6}(t)}{t}=0\) a.s. is derived. For all \(\varepsilon >0\), there exists a constant \(T>0\) such that \(\ln x_{2}(t)<\frac{\varepsilon }{a_{43}a_{12}+1}t\), \(\ln x_{3}(t)<\frac{\varepsilon }{a_{12}a_{23}+1}t\) and
$$ \int _{0}^{t}x_{4}(s)\,\mathrm{d}s \leq \Bigl( \limsup_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle + \varepsilon \Bigr)t $$
for any \(t\geq T\).
Thus, from (28), we obtain
$$ \begin{aligned} a_{43}a_{22} \ln x_{1}(t)\leq {}& \Bigl(a_{43}\Delta _{21}-b_{4}a_{12}a_{23}+2 \varepsilon +a_{12}a_{23}a_{44} \Bigl(\limsup _{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle + \varepsilon \Bigr) \Bigr)t \\ &{}-H_{2}a_{43} \int _{0}^{t}x_{1}(s)\,\mathrm{d}s+ \phi _{6}(t) \end{aligned} $$
(29)
for any \(t\geq T\). Thus, from the arbitrary ε and Lemma 4
$$ \limsup_{t\to \infty } \bigl\langle x_{1}(t) \bigr\rangle \leq \frac{(a_{43}\Delta _{21}-b_{4}a_{12}a_{23} +a_{12}a_{23}a_{44}\limsup_{t\to \infty }\langle x_{4}(t)\rangle )}{H_{2}a_{43}} $$
(30)
is furthermore obtained.
From (21) and (22), we have
$$ V_{3}(t)=\Delta _{33}t-H_{3} \int _{0}^{t}x_{3}(s)\,\mathrm{d}s+ \phi _{7}(t)-a_{34}H_{2} \int _{0}^{t}x_{4}(s)\,\mathrm{d}s, $$
(31)
where \(\phi _{7}(t)=a_{21}a_{32}\phi _{1}(t)+a_{11}a_{32}\phi _{2}(t)+H_{2} \phi _{3}(t)\). we apply the similar method that used for \(\phi _{1}(t)\), \(\lim_{t\to \infty }\frac{\phi _{7}(t)}{t}=0\) a.s. is obtained. By Lemma 7, and for all \(\varepsilon >0\), there exists a constant \(T>0\) for any \(t\geq T\) such that \(\ln x_{1}(t)<\frac{\varepsilon }{a_{32}a_{21}+1}t\), \(\ln x_{2}(t)<\frac{\varepsilon }{a_{32}a_{11}+1}t\) and \(\int _{0}^{t}x_{4}(s)\,\mathrm{d}s\leq (\limsup_{t\to \infty }\langle x_{4}(t) \rangle +\varepsilon )t\). Thus, for all \(t\geq T\) and by (31)
$$ H_{2}\ln x_{3}(t)>(\Delta _{33}-2 \varepsilon )t-a_{34}H_{2} \Bigl(\limsup _{t \to \infty } \bigl\langle x_{4}(t) \bigr\rangle + \varepsilon \Bigr)t-H_{3} \int _{0}^{t}x_{3}(s) \, \mathrm{d}s+ \phi _{7}(t) $$
is furthermore obtained.
If \(\Delta _{33}-a_{34}H_{2}\limsup_{t\to \infty }\langle x_{4}(t) \rangle >0\), then by Lemma 5 and the arbitrary ε we furthermore have
$$ \liminf_{t\to \infty } \bigl\langle x_{3}(t) \bigr\rangle \geq \frac{\Delta _{33}}{H_{3}}- \frac{a_{34}H_{2}}{H_{3}} \Bigl(\limsup_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \Bigr). $$
(32)
If \(\Delta _{33}-a_{34}H_{2}\limsup_{t\to \infty }\langle x_{4}(t) \rangle \leq 0\), then since \(\liminf_{t\to \infty }\langle x_{3}(t)\rangle \geq 0\), we also have
$$ \liminf_{t\to \infty } \bigl\langle x_{3}(t) \bigr\rangle \geq \frac{\Delta _{33}}{H_{3}}-\frac{a_{34}H_{2}}{H_{3}} \Bigl(\limsup _{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \Bigr). $$
Provided that, for any \(\varepsilon >0\), there is a constant \(T>0\) satisfying for \(t\geq T\)
$$ \int _{0}^{t}x_{1}(s)\,\mathrm{d}s \leq \frac{a_{43}\Delta _{21}-b_{4}a_{12}a_{23}+a_{12}a_{23}a_{44}(\limsup_{t\to \infty }\langle x_{4}(t)\rangle +\varepsilon )}{H_{2}a_{43}} $$
and
$$ \int _{0}^{t}x_{3}(s)\,\mathrm{d}s \geq \frac{\Delta _{33}}{H_{3}}- \frac{a_{34}H_{2}}{H_{3}} \Bigl(\limsup _{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle - \varepsilon \Bigr). $$
Combining with (22), for all \(t\geq T\),
$$ \begin{aligned} \ln x_{2}(t) \leq {}& \biggl[b_{2}+a_{21} \frac{a_{43}\Delta _{21}-b_{4}a_{12}a_{23}+a_{12}a_{23}a_{44} (\limsup_{t\to \infty }\langle x_{4}(t)\rangle +\varepsilon )}{H_{2}a_{43}} \\ &{}-a_{23} \biggl(\frac{\Delta _{33}}{H_{3}}-\frac{a_{34}H_{2}}{H_{3}} \Bigl( \limsup_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle - \varepsilon \Bigr) \biggr) \biggr]t+\phi _{2}(t)-a_{22} \int _{0}^{t}x_{2}(s)\,\mathrm{d}s \end{aligned} $$
(33)
is furthermore obtained.
We have \(\lim_{t\to \infty }\frac{\phi _{2}(t)}{t}=0\) a.s. by Lemma 7. We denote
$$ \begin{aligned} M_{1}={}& b_{2}+a_{21} \frac{(a_{43}\Delta _{21}-b_{4}a_{12}a_{23}+a_{12}a_{23}a_{44}\limsup_{t\to \infty } \langle x_{4}(t)\rangle )}{H_{2}a_{43}} \\ &{}-a_{23} \biggl(\frac{\Delta _{33}}{H_{3}}-\frac{a_{34}H_{2}}{H_{3}}\limsup _{t \to \infty } \bigl\langle x_{4}(t) \bigr\rangle \biggr). \end{aligned} $$
If \(M_{1}\geq 0\), then we can obtain
$$ \begin{aligned} \limsup_{t\to \infty } \bigl\langle x_{2}(t) \bigr\rangle \leq {}&\frac{1}{a_{22}} \biggl[b_{2}+a_{21} \frac{a_{43}\Delta _{21}-b_{4}a_{12}a_{23}}{H_{2}a_{43}}-a_{23} \frac{\Delta _{33}}{H_{3}} \\ &{}+ \biggl(\frac{a_{12}a_{21}a_{23}a_{44}}{a_{43}H_{2}}+ \frac{a_{23}a_{34}H_{2}}{H_{3}} \biggr)\limsup _{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \biggr] = \frac{M_{1}}{a_{22}}. \end{aligned} $$
(34)
If \(M_{1}<0\), then \(\lim_{t\to \infty }x_{2}(t)=0\) is directly obtained. From this and Lemma 8, \(\lim_{t\to \infty }x_{j}(t)=0\), \(j=3,4\), is furthermore derived.
Let \(M_{1}\geq 0\), for all \(\varepsilon >0\), there is a constant \(T>0\) such that
$$ \int _{0}^{t}x_{4}(s)\,\mathrm{d}s \geq \biggl( \frac{\Delta _{44}}{H_{4}}-\varepsilon \biggr)t,\qquad \int _{0}^{t}x_{2}(s)\,\mathrm{d}s \leq \biggl( \frac{M_{1}}{a_{22}}+\varepsilon \biggr)t $$
for any \(t\geq T\). From (22), (27) and (34), we derive for any \(t\geq T\)
$$ \ln x_{3}(t)\leq \biggl(b_{3}+a_{32} \biggl(\frac{M_{1}}{a_{22}}+ \varepsilon \biggr)-a_{34} \biggl( \frac{\Delta _{44}}{H_{4}}-\varepsilon \biggr) \biggr)t-a_{33} \int _{0}^{t}x_{3}(s) \, \mathrm{d}s+ \phi _{4}(t). $$
(35)
We have \(\lim_{t\to \infty }\frac{\phi _{3}(t)}{t}=0\) a.s. by Lemma 7. We denote
$$ M_{2}=b_{3}+\frac{a_{32}}{a_{22}}M_{1}-a_{34} \frac{\Delta _{44}}{H_{4}}. $$
If \(M_{2}\geq 0\), then from the arbitrary ε and Lemma 4 we furthermore have
$$ \limsup_{t\to \infty } \bigl\langle x_{3}(t) \bigr\rangle \leq \frac{1}{a_{33}} \biggl[b_{3}+ \frac{a_{32}M_{1}}{a_{22}}-\frac{a_{34}\Delta _{44}}{H_{4}} \biggr]. $$
(36)
If \(M_{2}<0\), then we obtain \(\lim_{t\to \infty }x_{3}(t)=0\). From this and Lemma 8, \(\lim_{t\to \infty }x_{4}(t)=0\) is furthermore obtained.
Let \(M_{2}\geq 0\). From (36) and for any \(\varepsilon >0\), there exists a constant \(T>0\), we obtain
$$ \int _{0}^{t}x_{3}(s)\,\mathrm{d}s \leq \frac{1}{a_{33}} \biggl[b_{3}+ \frac{a_{32}M_{1}}{a_{22}}- \frac{a_{34}\Delta _{44}}{H_{4}}+ \varepsilon \biggr]t $$
for \(t\geq T\). From (23), we derive
$$ \ln x_{4}(t)\leq \biggl[b_{4}+\frac{a_{43}}{a_{33}} \biggl(b_{3}+ \frac{a_{32}M_{1}}{a_{22}}-\frac{a_{34}\Delta _{44}}{H_{4}}+ \varepsilon \biggr) \biggr]t -a_{44} \int _{0}^{t}x_{4}(s)\,\mathrm{d}s+ \phi _{4}(t) $$
(37)
for any \(t\geq T\). We have \(\lim_{t\to \infty }\frac{\phi _{4}(t)}{t}=0\) a.s. by Lemma 7. We denote
$$ M_{3}=b_{4}+\frac{a_{43}}{a_{33}} \biggl(b_{3}+ \frac{a_{32}M_{1}}{a_{22}}- \frac{a_{34}\Delta _{44}}{H_{4}} \biggr). $$
If \(M_{3}\geq 0\), then from the arbitrary ε and Lemma 4
$$ \begin{aligned} \limsup_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \leq {}&\frac{1}{a_{44}} \biggl[b_{4}+ \frac{a_{43}}{a_{33}} \biggl[b_{3}+ \frac{a_{32}}{a_{22}}M_{1}-a_{34} \frac{\Delta _{44}}{H_{4}} \biggr] \biggr] \\ ={}&\frac{1}{a_{44}} \biggl[b_{4}+\frac{a_{43}}{a_{33}} \biggl[b_{3}+ \frac{a_{32}}{a_{22}} \biggl[b_{2}+a_{21} \frac{a_{43}\Delta _{21}-b_{4}a_{12}a_{23}}{H_{2}a_{43}} \\ &{}-a_{23}\frac{\Delta _{33}}{H_{3}} \biggr]-a_{34} \frac{\Delta _{44}}{H_{4}} \biggr] \biggr] \\ &{}+ \frac{a_{12}a_{21}a_{23}a_{32}a_{44}H_{3}+a_{23}a_{32}a_{34}a_{43}H_{2}^{2}}{a_{22}a_{33}a_{44}H_{2}H_{3}} \limsup_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \end{aligned} $$
(38)
is furthermore obtained. By a detailed calculation we can obtain
$$ \begin{aligned} &\frac{1}{a_{44}} \biggl[b_{4}+ \frac{a_{43}}{a_{33}} \biggl[b_{3}+ \frac{a_{32}}{a_{22}} \biggl[b_{2}+a_{21} \frac{a_{43}\Delta _{21}-b_{4}a_{12}a_{23}}{H_{2}a_{43}}-a_{23} \frac{\Delta _{33}}{H_{3}} \biggr]-a_{34}\frac{\Delta _{44}}{H_{4}} \biggr] \biggr] \\ &\quad =\frac{1}{a_{22}a_{33}a_{44}} \bigl[a_{22}a_{33}a_{44}H_{2}H_{3}-a_{12}a_{21}a_{23}a_{32}a_{44}H_{3}-a_{23}a_{32}a_{34}a_{43}H_{2}^{2} \bigr] \frac{\Delta _{44}}{H_{4}}. \end{aligned} $$
Thus, we furthermore find that (38) is equal with the inequality as follows:
$$ \begin{aligned} & \bigl[a_{22}a_{33}a_{44}H_{2}H_{3}-a_{12}a_{21}a_{23}a_{32}a_{44}H_{3}-a_{23}a_{32}a_{34}a_{43}H_{2}^{2} \bigr] \limsup_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \\ &\quad \leq \bigl[a_{22}a_{33}a_{44}H_{2}H_{3}-a_{12}a_{21}a_{23}a_{32}a_{44}H_{3}-a_{23}a_{32}a_{34}a_{43}H_{2}^{2} \bigr] \frac{\Delta _{44}}{H_{4}}. \end{aligned} $$
(39)
If \(M_{3}<0\), then from (37) and Lemma 5 we directly have \(\lim_{t\to \infty }x_{4}(t)=0\).
Assume \(\Delta _{44}>0\), then we can obtain
$$\begin{aligned}& \begin{aligned} M_{1}\geq {}& b_{2}+a_{21} \frac{a_{43}\Delta _{21}-b_{4}a_{12}a_{23}+a_{12}a_{23}a_{44}\frac{\Delta _{44}}{H_{4}}}{H_{2}a_{43}} \\ &{}-a_{23}\frac{\Delta _{33}}{H_{3}}+ \frac{a_{23}a_{34}H_{2}\Delta _{44}}{H_{3}H_{4}}=a_{22} \frac{\Delta _{41}}{H_{4}}>0, \end{aligned} \\& M_{2}\geq b_{3}+\frac{a_{32}\Delta _{41}}{H_{4}}- \frac{a_{34}\Delta _{44}}{H_{4}}=a_{33}\frac{\Delta _{43}}{H_{4}}>0, \\& M_{3}\geq b_{4}+b_{3} \frac{a_{43}}{a_{33}}- \frac{a_{32}a_{43}\Delta _{41}}{a_{33}H_{4}}- \frac{a_{43}\Delta _{44}}{a_{33}a_{34}H_{4}} =a_{44} \frac{\Delta _{44}}{H_{4}}> 0. \end{aligned}$$
Hence, from (39) and condition (25), \(\limsup_{t\to \infty }\langle x_{4}(t)\rangle \leq \frac{\Delta _{44}}{H_{4}}\) is obtained. Hence, \(\lim_{t\to \infty }\langle x_{4}(t)\rangle = \frac{\Delta _{44}}{H_{4}}\) is directly derived.
From the above conclusion and (30), we have
$$ \begin{aligned} \limsup_{t\to \infty } \bigl\langle x_{1}(t) \bigr\rangle \leq{} &\frac{(a_{43}\Delta _{21}-b_{4}a_{12}a_{23}+a_{12}a_{23}a_{44}\frac{\Delta _{44}}{H_{4}})}{H_{2}a_{43}} \\ ={}&\frac{b_{1}(a_{22}a_{34}a_{43}+a_{22}a_{33}a_{44}+a_{23}a_{32}a_{44})-b_{2}a_{12}(a_{33}a_{44}+a_{34}a_{43})}{H_{4}} \\ &{}+\frac{-b_{4}a_{12}a_{23}a_{34}+b_{3}a_{12}a_{23}a_{44}}{H_{4}}= \frac{\Delta _{41}}{H_{4}}. \end{aligned} $$
(40)
Then, from (32) we furthermore obtain
$$ \liminf_{t\to \infty } \bigl\langle x_{3}(t) \bigr\rangle \geq \frac{\Delta _{33}}{H_{3}}- \frac{a_{34}H_{2}}{H_{3}} \frac{\Delta _{44}}{H_{4}}=\frac{\Delta _{43}}{H_{4}}. $$
(41)
Similarly, by (33)
$$ \limsup_{t\to \infty } \bigl\langle x_{2}(t) \bigr\rangle \leq \frac{b_{2}H_{4}+a_{21}\Delta _{41}-a_{23}\Delta _{43}}{a_{22}H_{4}}= \frac{\Delta _{42}}{H_{4}} $$
(42)
is also obtained. For all \(\varepsilon >0\), there is a \(T>0\) for all \(t\geq T\) such that \(\int _{0}^{t}x_{2}(s)\,\mathrm{d}s<(\frac{\Delta _{42}}{H_{4}}+\varepsilon )t\) and \(\int _{0}^{t}x_{4}(s)\,\mathrm{d}s>(\frac{\Delta _{44}}{H_{4}}-\varepsilon )t\). From (22), we compute
$$ \ln x_{3}(t) \leq b_{3}t+a_{32} \biggl(\frac{\Delta _{42}}{H_{4}}+ \varepsilon \biggr)-a_{33} \int _{0}^{t}x_{3}(s)\,\mathrm{d}s -a_{34} \biggl( \frac{\Delta _{44}}{H_{4}}-\varepsilon \biggr)+\phi _{3}(t). $$
(43)
We have \(\lim_{t\to \infty }\phi _{3}(t)=0\). Thus, from the arbitrariness of ε and Lemma 5
$$ \limsup_{t\to \infty } \bigl\langle x_{3}(t) \bigr\rangle \leq \frac{b_{3}H_{4}+a_{32}\Delta _{42}-a_{34}\Delta _{44}}{a_{33}H_{4}}= \frac{\Delta _{43}}{H_{4}} $$
(44)
is furthermore derived. Hence, \(\lim_{t\to \infty }\langle x_{3}(t)\rangle = \frac{\Delta _{43}}{H_{4}}\) is obtained.
From (21) and (22), we compute
$$ V_{2}(t)=\Delta _{22}t-H_{2} \int _{0}^{t}x_{2}(s) \, \mathrm{d}s-a_{11}a_{23} \int _{0}^{t}x_{3}(s)\,\mathrm{d}s+ \phi _{8}(t), $$
(45)
where \(\phi _{8}(t)=a_{21}\phi _{1}(t)+a_{11}\phi _{2}(t)\). By Lemma 7, \(\lim_{t\to \infty }\frac{\phi _{8}(t)}{t}=0\) a.s. is obtained. From Lemma 7 and for \(\varepsilon >0\), there exists a \(T>0\) satisfying \(\int _{0}^{t}x_{3}(s)\,\mathrm{d}s<(\frac{\Delta _{43}}{H_{4}}+\varepsilon )t\) and \(\ln x_{1}(t)<\frac{\varepsilon }{a_{21}+1}t\) for \(t>T\). Thus
$$ a_{11}\ln x_{2}(t)\geq \biggl(\Delta _{22}-a_{11}a_{23} \biggl( \frac{\Delta _{43}}{H_{4}}+\varepsilon \biggr)-\varepsilon \biggr) t-H_{2} \int _{0}^{t}x_{2}(s) \,\mathrm{d}s+ \phi _{8}(t) $$
(46)
is obtained. Therefore, from the arbitrariness of ε and Lemma 5
$$ \liminf_{t\to \infty } \bigl\langle x_{2}(t) \bigr\rangle \geq \frac{H_{4}\Delta _{22}-a_{11}a_{23}\Delta _{43}}{H_{2}H_{4}}= \frac{\Delta _{42}}{H_{4}} $$
(47)
is furthermore obtained. Then, we obtain \(\lim_{t\to \infty }\langle x_{2}(t)\rangle = \frac{\Delta _{42}}{H_{4}}\).
For any \(\varepsilon >0\), there is a \(T>0\) such that \(\int _{0}^{t} x_{2}(s)\,\mathrm{d}s<(\frac{\Delta _{42}}{H_{4}}+ \varepsilon )\) for any \(t>T\). Hence, from (21), we have
$$ \ln x_{1}(t)\geq \biggl(b_{1}-a_{12} \biggl(\frac{\Delta _{42}}{H_{4}}+ \varepsilon \biggr) \biggr)t-a_{11} \int _{0}^{t}x_{1}(s)\,\mathrm{d}s+ \phi _{1}(t). $$
(48)
Hence, by Lemma 5 and the arbitrariness of ε we furthermore have
$$ \liminf_{t\to \infty } \bigl\langle x_{1}(t) \bigr\rangle \geq \frac{b_{1}H_{4}-a_{12}\Delta _{42}}{a_{11}H_{4}}= \frac{\Delta _{41}}{H_{4}}. $$
(49)
Then, we also have \(\lim_{t\to \infty }\langle x_{1}(t)\rangle = \frac{\Delta _{41}}{H_{4}}\). Therefore, conclusion (9) in Theorem 1 is proved.
Assume \(\Delta _{44}=0\). If there an \(i\in \{1,2,3\}\) such that \(M_{i}<0\), then we furthermore have \(\lim_{t\to \infty }x_{i+1}(t)=0\) from the above discussions. Hence, by Lemma 8, \(\lim_{t\to \infty }x_{4}(t)=0\). Otherwise, we have \(M_{i}\geq 0\), \(i=1,2,3\). Then, from the above discussions we also have
$$ \begin{aligned} & \bigl[a_{22}a_{33}a_{44}H_{2}H_{3}-a_{12}a_{21}a_{23}a_{32}a_{44}H_{3}-a_{23}a_{32}a_{34}a_{43}H_{2}^{2} \bigr] \limsup_{t\to \infty } \bigl\langle x_{4}(t) \bigr\rangle \\ &\quad \leq \bigl[a_{22}a_{33}a_{44}H_{2}H_{3}-a_{12}a_{21}a_{23}a_{32}a_{44}H_{3}-a_{23}a_{32}a_{34}a_{43}H_{2}^{2} \bigr] \frac{\Delta _{44}}{H_{4}}=0. \end{aligned} $$
Therefore, from condition (25) we have \(\lim_{t\to \infty }\langle {x_{4}}\rangle =0\) a.s.
Assume \(\Delta _{44}<0\). Then from (26) we obtain
$$ V_{4}(t)\leq \Delta _{44}t+\phi _{5}(t). $$
Hence,
$$ \limsup_{t\to \infty }\frac{1}{t} \bigl[ \bigl[a_{32} \bigl(a_{21}\ln x_{1}(t)+a_{11} \ln x_{2}(t) \bigr)+H_{2}\ln x_{3}(t) \bigr]a_{43}+H_{3}\ln x_{4}(t) \bigr]\leq \Delta _{44}< 0. $$
Thus, we have
$$ \lim_{t\to \infty } \bigl[ \bigl(x_{1}(t) \bigr)^{a_{43}a_{32}a_{21}} \bigl(x_{2}(t) \bigr)^{a_{43}a_{32}a_{11}} \bigl(x_{3}(t) \bigr)^{a_{43}H_{2}} \bigl(x_{4}(t) \bigr)^{H_{3}} \bigr]=0. $$
This shows that there exists a \(j\in \{1,2,3,4\}\) that satisfies \(\lim_{t\to \infty }x_{j}(t)=0\). Consequently, by Lemma 8, \(\lim_{t\to \infty }x_{4}(t)=0\).
In conclusion, when \(\Delta _{44}\leq 0\) we always obtain \(\lim_{t\to \infty }\langle x_{4}(t)\rangle =0\) or \(\lim_{t\to \infty }x_{4}(t)=0\). Thus, applying the similar arguments used in the proving process of Theorem 1 listed in [28], the remaining conclusions in Theorem 1 can be proved. □
Remark 4
Observe the proving process of the above, the criterion (25) only used to obtain \(\limsup_{t\to \infty }\langle x_{4}(t)\rangle \leq \frac{\Delta _{44}}{H_{4}}\) from the inequality (39). This shows that conditions (24) and (25) appear to be the supererogatory and pure mathematical conditions.
Remark 5
An important and interesting open problem is how to extend Theorem 1 to the general n-species stochastic food-chain system with distributed delay and harvesting.
In the following theorem, we mainly investigate that, for all positive global solutions of model (1), the conclusion about global attractiveness in the expectation.
Theorem 2
For initial conditions \(\phi ,\phi ^{*}\in C([-r,0],R_{+}^{4})\), assume that model (1) has two solutions \((x_{1}(t;\phi ),x_{2}(t;\phi ),x_{3}(t;\phi ),x_{4}(t;\phi ))\) and \((y_{1}(t;\phi ^{*}),y_{2}(t;\phi ^{*}),y_{3}(t;\phi ^{*}), y_{4}(t;\phi ^{*}))\). If there are positive constants \(w_{i}\) (\(i=1,2,3,4\)) such that
$$\begin{aligned}& w_{1}a_{11}-w_{2}a_{21}>0,\qquad w_{i}a_{ii}-w_{i-1}a_{i-1i}-w_{i+1}a_{i+1i}>0\quad (i=2,3), \\& w_{4}a_{44}-w_{3}a_{34}>0. \end{aligned}$$
Then
$$ \lim_{t\to \infty }E \Biggl(\sum_{i=1}^{4} \bigl\vert x_{i}(t,\phi )-y_{i} \bigl(t,\phi ^{*} \bigr) \bigr\vert ^{2} \Biggr)^{ \frac{1}{2}}=0. $$
The proof of Theorem 2 is similar to Theorem 2 from [28]. Hence it is omitted here. Now let \(\mathcal{P}([-r,0],R_{+}^{4})\) represent the whole measurable probability space on \(C([-r,0],R_{+}^{4})\). For \(\mathcal{P}_{1},\mathcal{P}_{2}\in \mathcal{P}([-r,0],R_{+}^{4})\), set the metric as follows:
$$ d_{L}(\mathcal{P}_{1},\mathcal{P}_{2})= \sup_{f\in {L}} \biggl\vert \int _{R_{+}^{4}}f(u) \mathcal{P}_{1}(\mathrm{d}u)- \int _{R_{+}^{4}}f(u)\mathcal{P}_{2}(\mathrm{d}u) \biggr\vert , $$
where
$$ L= \bigl\{ f:\mathcal{C} \bigl([-r,0],R_{+}^{4} \bigr) \rightarrow R: \bigl\vert f(u_{1})-f(u_{2}) \bigr\vert \leq \Vert u_{1}-u_{2} \Vert , \bigl\vert f(\cdot ) \bigr\vert \leq 1 \bigr\} . $$
Let \(p(t,\phi ,\mathrm{d}x)\) represents the transition probability of process \(x(t)=(x_{1}(t),x_{2}(t),x_{3}(t),x_{4}(t))\). In the following theorem, we consider the condition of asymptotically stability. The results as follows are obtained.
Theorem 3
Suppose that positive constants \(q_{i}\) (\(i=1,2,3,4\)) satisfies
$$\begin{aligned}& q_{1}a_{11}-q_{2}a_{21}>0,\qquad q_{i}a_{ii}-q_{i-1}a_{i-1i}-q_{i+1}a_{i+1i}>0\quad (i=2,3),\\& q_{4}a_{44}-q_{3}a_{34}>0. \end{aligned}$$
Then model (1) is asymptotically stable in distribution, i.e., for all initial value \(\phi \in C([-\gamma ,0],R_{+}^{4})\), a unique probability measure \(v(\cdot )\) satisfies the transition probability \(p(t,\phi ,\cdot )\) of solution \((x_{1}(t,\phi ),x_{2}(t,\phi ),x_{3}(t,\phi ),x_{4}(t,\phi ))\) such that
$$ \lim_{t\to \infty }d_{BL} \bigl(p(t,\phi ,\cdot ),v( \cdot ) \bigr)=0. $$
Remark 6
Obviously, Theorems 2 and 3 also seemingly can be extended to the general n-species stochastic food-chain system with distributed delay and harvesting.