Theorem 3.1
If the following condition holds:
$$[H _{1}]\quad \hat{g}_{i} = r_{i}^{l} - \sum_{k = 1,k \ne i}^{n} b_{ik} ^{u} ( t ) M_{k} - \sum_{k = 1}^{m} \frac{c_{ik}^{u}M_{i}N _{k}}{f_{ik}^{l}} > 0, $$
then system (1.2) is permanent, that is, there exists
\(T>0\), for
\(t > T > 0\), the solution
\(( x ( t ) ,y ( t ) ) ^{T}\)
of (1.2) satisfies
\(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n _{j} \le y_{j} ( t ) \le N_{j}\), where
$$\begin{aligned}& m_{i} = \frac{ \hat{g}_{i}}{b_{ii}^{u}}\exp \bigl( \bigl( \hat{g}_{i} - b_{ii}^{u}M _{i} \bigr) \tau \bigr) , \quad\quad M_{i} = \frac{r_{i}^{u}}{b_{ii}^{l}e^{ - r_{i}^{u}\tau }}, \\& n_{j} = \biggl( \frac{\sum_{k = 1}^{n} q_{kj}^{l}m_{k}^{2}}{2 ( \sum_{k = 1} ^{n} ( M_{k}^{2} + f_{kj}^{u} ) ) ( r_{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} ) } \biggr) ^{\frac{1}{1 - \alpha }}, \quad \quad N_{j} = \biggl( \frac{3\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{2r _{j}^{l}} \biggr) ^{\frac{1}{1 - \alpha }} \end{aligned}$$
for
\(i = 1,2,\ldots,n\); \(j = 1,2,\ldots,m\). In this article, the values of
i, j
are no longer repeated.
Proof
By the first equation of (1.2), we get
$$ x'_{i} ( t ) \le x_{i} ( t ) r_{i} ( t ) . $$
(3.1)
Integrating (3.1), we have \(x_{i} ( t ) \le x_{i} ( t - \tau ) \exp ( r_{i}^{u}\tau ) \), \(t > \tau \), that is,
$$ x_{i} ( t - \tau ) \ge x_{i} ( t ) \exp \bigl( - r _{i}^{u}\tau \bigr) ,\quad t > \tau . $$
(3.2)
Combining (3.2) and the first equation of (1.2), we have
$$ x'_{i} ( t ) \le x_{i} ( t ) \bigl[ r_{i}^{u} - b _{ii}^{l}x_{i} ( t ) \exp \bigl( - r_{i}^{u}\tau \bigr) \bigr] ,\quad t > \tau . $$
(3.3)
By applying Lemma 2.4 to (3.3), we obtain
$$ \lim_{t \to + \infty } \sup x_{i} ( t ) \le \frac{r_{i}^{u}}{b _{ii}^{l}e^{ - r_{i}^{u}\tau }} \equiv M_{i}. $$
(3.4)
By (3.4), there exists \(T_{1} > \tau \), when \(t \ge T_{1}\) and \(T_{1} \to \infty \), then
$$ x_{i} ( t ) \le M_{i}. $$
(3.5)
By (3.5), there also exists \(T_{2} = T_{1} + \tau \), when \(t \ge T _{2}\), then
$$ x_{i} ( t - \tau ) \le M_{i}. $$
(3.6)
Combining (3.6) and the second equation of (1.2), we have
$$ \begin{aligned}[b] y'_{j} ( t ) & \le y_{j} ( t ) \Biggl[ \sum_{k = 1} ^{n} q_{kj}^{u} ( t ) y_{j}^{\alpha - 1} ( t ) - r _{j}^{l} ( t ) \Biggr] \\ &\le y_{j}^{\alpha } ( t ) \Biggl[ \sum _{k = 1}^{n} q_{kj} ^{u} ( t ) - r_{j}^{l} ( t ) y^{1 - \alpha } ( t ) \Biggr] ,\quad t \ge T_{2}. \end{aligned} $$
(3.7)
Using Lemma 2.5 to (3.7), then
$$ y_{j} ( t ) \le \biggl[ \frac{\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{r_{j}^{l} ( t ) } + \biggl( y^{1 - \alpha } ( 0 ) - \frac{\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{r _{j}^{l} ( t ) } \biggr) e^{ - r_{j}^{l} ( t ) ( 1 - \alpha ) t} \biggr] ^{\frac{1}{1 - \alpha }},\quad \forall t \ge 0. $$
(3.8)
Therefore, there exists \(T_{3} > 0\) such that
$$ y_{j} ( t ) \le \biggl( \frac{3\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{2r_{j}^{l}} \biggr) ^{\frac{1}{1 - \alpha }} \equiv N_{j},\quad t > T_{3}. $$
(3.9)
Combining (3.5), (3.6), (3.9) and the first equation of (1.2), we get
$$ x_{i}^{\prime } ( t ) \ge x_{i} ( t ) \Biggl[ r _{i}^{l} - \sum_{k = 1,k \ne i}^{n} b_{ik}^{u} ( t ) M_{k} - b_{ii}^{u}x_{i} \bigl( t - \tau_{i} ( t ) \bigr) - \sum_{k = 1}^{m} \frac{c_{ik}^{u}M_{i}N_{k}^{\alpha }}{f_{ik}^{l}} \Biggr] . $$
(3.10)
Suppose \(x_{i} ( \tilde{t} ) \) is any local minimal value of \(x_{i} ( t ) \), then we have
$$ 0 = x_{i}^{\prime } ( \tilde{t} ) \ge x_{i} ( \tilde{t} ) \Biggl[ r_{i}^{l} - \sum _{k = 1,k \ne i}^{n} b_{ik} ^{u} ( t ) M_{k} - b_{ii}^{u}x_{i} \bigl( \tilde{t} - \tau _{i} ( \tilde{t} ) \bigr) - \sum _{k = 1}^{m} \frac{c_{ik} ^{u}M_{i}N_{k}^{\alpha }}{f_{ik}^{l}} \Biggr] . $$
(3.11)
Let
$$ \hat{g}_{i} = r_{i}^{l} - \sum _{k = 1,k \ne i}^{n} b_{ik}^{u} ( t ) M_{k} - \sum_{k = 1}^{m} \frac{c_{ik}^{u}M_{i}N_{k}^{\alpha }}{f _{ik}^{l}}. $$
(3.12)
From (3.11) and (3.12), we have
$$ x_{i} \bigl( \tilde{t} - \tau_{i} ( \tilde{t} ) \bigr) \ge \frac{\hat{g}_{i}}{b_{ii}^{u}}. $$
(3.13)
Integrating (3.10) on \([ \tilde{t} - \tau_{i} ( \tilde{t} ) , \tilde{t} ] \) and noticing that \(\hat{g}_{i} - b_{ii}^{u}x_{i} ( \tilde{t} - \tau_{i} ( \tilde{t} ) ) \le 0\), we obtain
$$ \ln \biggl( \frac{x_{i} ( \tilde{t} ) }{x_{i} ( \tilde{t} - \tau_{i} ( \tilde{t} ) ) } \biggr) \ge \int_{\tilde{t} - \tau_{i} ( \tilde{t} ) }^{\tilde{t}} \bigl( \hat{g}_{i} - b_{ii}^{u}x_{i} \bigl( \tilde{t} - \tau_{i} ( \tilde{t} ) \bigr) \bigr) \,dt \ge \bigl( \hat{g}_{i} - b_{ii} ^{u}M_{i} \bigr) \tau . $$
(3.14)
From (3.13) and (3.14), then
$$ x_{i} ( \tilde{t} ) \ge \frac{\hat{g}_{i}}{b_{ii}^{u}}\exp \bigl( \bigl( \hat{g}_{i} - b_{ii}^{u}M_{i} \bigr) \tau \bigr) . $$
(3.15)
Hence, for \(T_{4} > 0\) and \(t > T_{4}\), we have
$$ x_{i} ( t ) \ge x_{i} ( \tilde{t} ) \ge \frac{ \hat{g}_{i}}{b_{ii}^{u}} \exp \bigl( \bigl( \hat{g}_{i} - b_{ii}^{u}M _{i} \bigr) \tau \bigr) \equiv m_{i}. $$
(3.16)
Combining (3.9), (3.16) and the second equation of (1.2), when \(T_{5} \ge \max \{ T_{3},T_{4} \} > 0\), for \(t > T_{5}\), we get
$$ \begin{aligned} y'_{j} ( t ) &\ge y_{j} ( t ) \Biggl[ - r_{j}^{u} - \sum_{k = 1}^{m} p_{jk}^{u}N_{k} + \sum _{k = 1}^{n} \frac{q_{kj}^{l}m _{k}^{2}}{M_{k}^{2} + f_{kj}^{u}}y_{j}^{\alpha - 1} ( t ) \Biggr] \\ &= y_{j}^{\alpha } ( t ) \Biggl[ \sum _{k = 1}^{n} \frac{q_{kj} ^{l}m_{k}^{2}}{M_{k}^{2} + f_{kj}^{u}} - \Biggl( r_{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} \Biggr) y_{j}^{1 - \alpha } ( t ) \Biggr] . \end{aligned} $$
It follows from Lemma 2.5 that there exists \(T_{6} > 0\) such that
$$ \begin{aligned} y_{j} ( t ) &\ge \biggl( \frac{\sum_{k = 1}^{n} \frac{q_{kj} ^{l}m_{k}^{2}}{M_{k}^{2} + f_{kj}^{u}}}{r_{j}^{u} + \sum_{k = 1}^{m} p _{jk}^{u}N_{k}} + \biggl( y_{j}^{1 - \alpha } ( 0 ) - \frac{ \sum_{k = 1}^{n} \frac{q_{kj}^{l}m_{k}^{2}}{M_{k}^{2} + f_{kj}^{u}}}{r _{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k}} \biggr) e^{ - ( r _{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} ) ( 1 - \alpha ) t} \biggr) ^{\frac{1}{1 - \alpha }} \\ &\ge \biggl( \frac{\sum_{k = 1}^{n} q_{kj}^{l}m_{k}^{2}}{2 ( \sum_{k = 1}^{n} ( M_{k}^{2} + f_{kj}^{u} ) ) ( r _{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} ) } \biggr) ^{\frac{1}{1 - \alpha }} \equiv n_{j}. \end{aligned} $$
Make \(T \ge \max \{ T_{2},T_{5},T_{6} \} > 0\), for \(t > T\), we get \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n_{j} \le y_{j} ( t ) \le N_{j}\).
Therefore system (1.2) is permanent.
Next, we prove that system (1.2) has at least one bounded positive solution for \(t \ge 0\). Define \(\Omega = \{ ( x ( t ) ,y ( t ) ) ^{T} = ( x_{1} ( t ) ,x_{2} ( t ) ,\ldots,x_{n} ( t ) ,y _{1} ( t ) ,y_{2} ( t ) ,\ldots,y_{m} ( t ) ) ^{T}\in R^{n + m}| ( x ( t ) ,y ( t ) ) ^{T}\text{ is the solution of system (1.2), satisfying }m_{i} \le x_{i} ( t ) \le M_{i},n_{j} \le y_{j} ( t ) \le N_{j},t \in R \}\). □
Theorem 3.2
For system (1.2), the set
\(\Omega \ne \emptyset\).
Proof
According to the characteristics of an almost periodic function, for a sequence of \(\{ t_{\gamma } \} \), \(t_{\gamma } \to \infty \) as \(\gamma \to \infty \), then \(r_{i} ( t + t_{\gamma } ) \to r_{i} ( t ) \), \(r_{j} ( t + t_{\gamma } ) \to r_{j} ( t ) \), \(b_{il} ( t + t_{\gamma } ) \to b_{il} ( t ) \), \(p_{jk} ( t + t_{\gamma } ) \to p_{jk} ( t ) \), \(c_{ik} ( t + t_{\gamma } ) \to c_{ik} ( t ) \), \(q_{lj} ( t + t_{\gamma } ) \to q_{lj} ( t ) \), \(\tau_{i} ( t + t_{\gamma } ) \to \tau_{i} ( t ) \), \(f_{ij} ( t + t_{\gamma } ) \to f_{ij} ( t ) \) (\(i,l = 1,2,\ldots,n\); \(j,k = 1,2,\ldots,m\)) uniformly on R as \(\gamma \to \infty \). By Lemma 2.3, system (1.2) has at least one solution \(z ( t ) = ( x ( t ) ,y ( t ) ) ^{T}\) satisfying \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n_{j} \le y_{j} ( t ) \le N_{j}\) when \(t > T\).
Obviously, the sequence \(z ( t + t_{\gamma } ) \) is uniformly bounded and equi-continuous on any bounded subset of R. By the Ascoli theorem, we know there exists a subsequence \(z ( t + t_{\lambda } ) \) which converges to a continuous function
$$ g ( t ) = \bigl( g_{1} ( t ) ,g_{2} ( t ) \bigr) ^{T} = \bigl( g_{11} ( t ) ,g_{21} ( t ) ,\ldots,g _{n1} ( t ) ,g_{12} ( t ) ,g_{22} ( t ) ,\ldots,g _{m2} ( t ) \bigr) ^{T} $$
as \(\lambda \to \infty \) uniformly on any bounded subset of R.
Make \(T_{7} \in R\), suppose \(T_{7} + t_{\lambda } \ge T\) for all λ. When \(t \ge 0\), we obtain
$$\begin{aligned}& \begin{aligned}[b] &x_{i} ( t + t_{\lambda } + T_{7} ) - x_{i} ( t_{\lambda } + T_{7} ) \\ &\quad = \int_{T_{7}}^{t + T_{7}} x_{i} ( s + t_{ \lambda } ) \Biggl( r_{i} ( s + t_{\lambda } ) - \sum _{k = 1}^{n} b_{ik} ( s + t_{\lambda } ) x_{k} \bigl( ( s + t_{\lambda } ) - \tau_{k} ( s + t_{\lambda } ) \bigr) \\ &\quad\quad{} - \sum_{k = 1}^{m} \frac{c_{ik} ( s + t_{\lambda } ) x_{i} ( s + t_{\lambda } ) }{x_{i}^{2} ( s + t_{\lambda } ) + f_{ik} ( s + t_{\lambda } ) }y_{k}^{\alpha } ( s + t_{\lambda } ) \Biggr) \,ds, \end{aligned} \end{aligned}$$
(3.17)
$$\begin{aligned}& \begin{aligned}[b] & y_{j} ( t + t_{\lambda } + T_{7} ) - y_{j} ( t_{\lambda } + T_{7} ) \\ &\quad = \int_{T_{7}}^{t + T_{7}} y_{j} ( t_{\lambda } + s ) \Biggl( - r_{j} ( s + t_{\lambda } ) - \sum _{k = 1}^{m} p_{jk} ( s + t_{\lambda } ) y_{k} ( s + t_{\lambda } ) \\ &\quad\quad{} + \sum_{k = 1}^{n} \frac{q_{kj} ( s + t_{\lambda } ) x_{k} ^{2} ( s + t_{\lambda } ) }{x_{k}^{2} ( s + t_{\lambda } ) + f_{kj} ( s + t_{\lambda } ) }y_{j}^{\alpha - 1} ( s + t_{\lambda } ) \Biggr) \,ds. \end{aligned} \end{aligned}$$
(3.18)
Letting \(\lambda \to \infty \) in (3.17) and (3.18), for \(\forall t \ge 0\), by the Lebesgue dominated convergence theorem, we get
$$\begin{aligned}& g_{i1} ( t + T_{7} ) - g_{i1} ( T_{7} ) \\& \quad = \int _{T_{7}}^{t + T_{7}} g_{i1} ( s ) \Biggl( r_{i} ( s ) - \sum_{k = 1}^{n} b_{ik} ( s ) g_{k1} \bigl( s - \tau_{k} ( s ) \bigr) - \sum_{k = 1}^{m} \frac{c_{ik} ( s ) g _{i1} ( s ) }{g_{i1}^{2} ( s ) + f_{ik} ( s ) }g _{k2}^{\alpha } ( s ) \Biggr) \,ds, \\& g_{j2} ( t + T_{7} ) - g_{j2} ( T_{7} ) \\& \quad = \int _{T_{7}}^{t + T_{7}} g_{j2} ( s ) \Biggl( - r_{j} ( s ) - \sum_{k = 1}^{m} p_{jk} ( s ) g_{k2} ( s ) + \sum _{k = 1}^{n} \frac{q_{kj} ( s ) g_{k1}^{2} ( s ) }{g _{k1}^{2} ( s ) + f_{kj} ( s ) }g_{j2}^{\alpha - 1} ( s ) \Biggr) \,ds. \end{aligned}$$
Since \(T_{7} \in R\) is arbitrarily given, \(g ( t ) \) is a solution of system (1.2) on R.
It is easy to know \(m_{i} \le g_{i1} ( t ) \le M_{i}\), \(n_{j} \le g_{j2} ( t ) \le N_{j}\) for any \(t \in R\). Thus, the set \(\Omega \ne \emptyset \), that is, system (1.2) has at least one bounded positive solution. □