本文研究以下具有时滞和反馈控制的三种群非自治比率依赖型食物链模型的解的动力学行为

(1.1) $ \begin{equation} \left\{\begin{array}{l} \dot{x}_{1}(t)=x_{1}(t)\big[r_{1}(t)-a_{11}(t) _{1}\left(t-\tau_{1}\right)-\frac{a_{12}(t) x_{2}(t)}{b_{12}(t)+x_{1}(t)}-d_{1}(t) u_{1}(t)\big], \\ \dot{x}_{2}(t)=x_{2}(t)\big[-r_{2}(t)-a_{22}(t) x_{2}\left(t-\tau_{2}\right)+\frac{a_{21}(t) x_{1}\left(t-\tau_{3}\right)}{b_{12}(t)+x_{1}\left(t-\tau_{3}\right)}-\frac{a_{23}(t) x_{3}(t)}{b_{23}(t)+x_{2}(t)}+d_{2}(t) u_{2}(t)\big], \\ \dot{x}_{3}(t)=x_{3}(t)\big[-r_{3}(t)-a_{33}(t) x_{3}\left(t-\tau_{4}\right)+\frac{a_{32}(t) x_{2}\left(t-\tau_{5}\right)}{b_{23}(t)+x_{2}\left(t-\tau_{5}\right)}+d_{3}(t) u_{3}(t)\big], \\ \dot{u}_{1}(t)=e_{1}(t)-f_{1}(t) u_{1}(t)+q_{1}(t) x_{1}(t), \\ \dot{u}_{2}(t)=e_{2}(t)-f_{2}(t) u_{2}(t)-q_{2}(t) x_{2}(t), \\ \dot{u}_{3}(t)=e_{3}(t)-f_{3}(t) u_{3}(t)-q_{3}(t) x_{3}(t), \end{array}\right. \end{equation} $

带有如下初始条件

(1.2) $ \begin{equation} x_{i}(t)=\phi_{i}(t) \geq 0, t \in[-\tau, 0), \phi_{i}(0)>0, u_{i}(0)=u_{i 0}>0, i=1, 2, 3, \end{equation} $

其中$ x_{i}(t), (i=1, 2, 3) $表示第$ i $个种群的密度, $ r_{1}(t) $是食饵种群的内禀增长率, $ r_{i}(t), i=2, 3 $是捕食者的死亡率. $ a_{ii}(t), (i=1, 2, 3) $表示第i个种群的内部竞争系数, $ a_{12}(t), a_{23}(t) $表示食饵被捕食者捕获的比率, $ a_{21}(t), a_{32}(t) $表示捕食者捕获食饵后的吸收率, $ b_{12}(t), b_{23}(t) $为捕食者的捕获率. $ u_{i}(t), (i=1, 2, 3) $是反馈控制函数, $ d_{i}(t), e_{i}(t), q_{i}(t), f_{i}(t), (i=1, 2, 3) $是控制参数. $ \tau_{1}, \tau_{2}, \tau_{4} $是常数, 分别表示因为食饵、捕食者和顶级捕食者拥挤引起的负反馈时滞. $ \tau_{3}, \tau_{5} $是由于怀孕而导致的时滞, 也就是说, 只有成熟的成年捕食者才能为捕食者提供食物. $ \tau=\max\{\tau_1, \tau_2, \tau_3, \tau_4, \tau_5\} $, 模型中的所有系数都是定义在$ [0, +\infty) $上的正连续和有界函数.

生物种群的持久性、全局吸引力和周期性是人口动力学的一个重要研究方向. 该模型(1.1)–(1.2)描述了三个物种之间的捕食性关系, 其中第二个种群捕食第一个种群, 第三个种群捕食第二个种群. 比率依赖的Lotka-Volterra捕食-食饵模型在参考文献[1, 11, 17, 19, 33, 37]中得到了广泛的研究, 并取得了许多优秀的结果. 2007年, Liang和Pan^[17]通过构造李亚普诺夫函数, 研究了一类比率依赖的Holling-Tanner系统, 得到了该系统正平衡点的全局稳定性的充分条件. 在文献$ [37] $中, 作者利用李亚普诺夫泛函法和线性化方法, 研究了具有Holling二型函数响应的捕食-食饵系统, 取得其正常数平衡点的局部稳定性和全局稳定性的充分条件. Xu等人在文献[33]中研究了一类捕食- 食饵模型的动力学行为, 该模型假设捕食者和食饵都使用基于博弈论的策略来最大化它们的人均种群增长率. 在文献$ [11] $中, 作者研究了捕食者和食饵之间的相互作用及解的动力学行为. 2018年, Louartassi等人在文献[19] 中利用Pontryagin最大值原理研究了对食饵具有捕获和储备区的捕食-食饵模型的动力学行为. 此外, 现实世界中的生态系统往往受到一些不可预测的因素的干扰, 这些因素影响着生态模型的参数, 如死亡率、出生率等. 因此, 我们应该在生态系统中增加反馈控制, 研究反馈控制对系统持久性和全局稳定性的影响, 为生态系统的保护提供一些理论方法. 越来越多的学者对具有反馈控制的Lotka-Volterra捕食-食饵模型进行了讨论, 参见文献[13, 25, 31, 38]. Gopalsamy和翁在文献[13]中研究了具有反馈控制的两种群竞争模型的正平衡的全局吸引性. 2009年, Nie等人在文献[25]中研究了反馈控制对非自治捕食-食饵Lotka-Volterra系统动态行为的影响, 通过构造适当的李亚普诺夫泛函, 他们得到了模型的任何正解的全局稳定性的可验证的充分条件. 在文献[38]中, Zhang等人考虑了一类具有反馈控制的Lotka-Volterra随机系统, 结合Jensen不等式, 构造合适的李亚普诺夫函数, 得到了保证系统在均方意义下全局耗散的充分条件. 值得一提的是, 王等人在文献[32]中考虑了一类具有反馈控制的Lotka-Volterra比率依赖捕食-食饵系统, 并建立了保证该系统的持久性和正解的全局吸引力的充分条件. 最后, 实际上, 任何生物或环境参数都会自然地受到时间波动的影响. 因此, 更现实的描述种群之间相互作用的数学模型应该考虑时滞的影响. 基于越来越多的生理和生物学证据, 一些学者认识到在种群相互作用中不可避免地存在一定的时间延迟, 并发现较长的时间延迟可能会影响系统的稳定性, 参见文献[20, 22, 23, 26, 36]. Xu和Chen在文献[36]中研究了一类时滞三种群非自治的Lotka-Volterra捕食-食饵系统, 并利用时滞微分不等式和李亚普诺夫稳定性理论证明了该系统在适当的条件下是一致持久的和全局稳定的. 在文献[23]中, Nakata和Moroya考虑了一类具有时滞的非自治Lotka-Volterra互助模型, 建立了保证该系统持久的若干充分条件. 2010年, Shen和You在文献[26]中研究了一类三种群食物链比率依赖模型, 利用微分不等式获得了保证捕食者物种灭绝和系统持久性的充分条件. 特别是, Xu等人在文献[35]中, 通过构造合适的李亚普诺夫泛函, 研究了具有常系数且没有反馈控制的系统正平衡(1.1)–(1.2)的全局渐近稳定性. 有关更多类似的结果, 参见文献[3, 6, 10, 18, 34].

此外, 我们都知道, 任何真实的生态和生物系统都总是受到环境噪声^{[9, 14, 24]}的影响. 噪声不可避免地出现于远离平衡态的真实系统中, 特别是在Lotka-Volterra系统中, 因为生态系统和一般的生物和物理系统都是开放的系统, 它们不可避免地与环境相互作用. 系统与外部环境的相互作用导致了周期性的确定性外力, 在种群动力系统中, 这种随机噪声主要是乘法类型的随机扰动. 此外, 乘法噪声负责噪声诱导跃迁^[15], 类似于相变现象, 发生在噪声强度较大的某些特定值, 使阶参量的概率密度函数在噪声强度的某些值下突然改变其形状. 因此, 对具有随机噪声扰动的Lotka-Volterra生态系统的研究无疑具有重要的物理数学意义, 它也使研究环境噪声对低压系统的积极影响成为可能^{[7, 27, 30]}. 一些关于环境噪声对生态系统动力学性质的影响以及噪声在跨学科物理模型中的积极作用的结果, 可参见文献[5, 8, 12, 21, 28, 29].

然而, 据作者所知, 还没有人考虑同时具有时滞、比率依赖性和反馈控制的一般非自治的Lotka-Volterra系统. 在上述工作的激励下, 我们提出并研究上述三种群具有时滞和反馈控制的非自治比率依赖型Lotka-Volterra食物链模型. 本文的结构如下: 在第二节中, 给出了系统(1.1)–(1.2)的持久性条件. 第三节通过构造非负李亚普诺夫函数, 得到了保证系统正解全局吸引的充分条件. 第四节利用一些新的分析方法和Brouwer不动点理论, 得到了相应周期系统正周期解的存在性、唯一性和稳定性的一些充分条件. 最后, 在第五部分对上述系统进行了数值仿真, 验证了所得的理论结果. 另外, 对具有乘法噪声的上述系统进行了数值仿真, 发现该模型一些新的有趣的动力学行为.

我们首先介绍以下定义和符号, 以获得系统$ (1.1) $的持久性结果. 如果$ g(t) $是一个定义在$ [0, +\infty) $上的连续有界函数, 则记

$ g^{m}=\sup \{g(t) \mid 0 \leq t<\infty\}, \quad {g}^{l}=\inf \{g(t) \mid 0 \leq t<\infty\} . $

定义2.1  如果存在正常数$ M_{i}, N_{i}, m_{i}, n_{i}, (i=1, 2, 3) $和$ T $, 使得对系统(1.1)–(1.2)的任意正解$ Z(t)=\left(x_{1}(t), x_{2}(t), x_{3}(t), u_{1}(t), u_{2}(t), u_{3}(t)\right) $, 当$ t>T $时, 有$ m_{i} \leq x_{i}(t) \leq M_{i}, n_{i} \leq u_{i}(t) \leq N_{i} $, 则称系统$ (1.1) $是持久的.

作为文献[4]中引理2.1的直接推论, 我们有下述引理.

引理2.1  如果$ a>0, b>0 $且$ \dot{x} \geq b-a x $, 当$ t \geq 0 $和$ x(0)>0 $时, 我们有$ \liminf\limits_{t \rightarrow +\infty} x(t) \geq b / a $. 如果$ a>0, b>0 $且$ \dot{x} \leq b-a x $, 当$ t \geq 0 $和$ x(0)>0 $时, 我们有$ \limsup\limits_{t \rightarrow +\infty} x(t) \leq b / a $.

以下两个引理来自文献[23], 它们将被用于证明系统$ (1.1) $的持久性.

引理2.2  对初值$ y(t)=\phi(t) \geq 0 $, $ t \in[-m \tau, 0) $及$ \phi(0)>0 $, 成立

$ \dot{y}_(t) \leq y(t)\left(\lambda-\sum\limits_{l=0}^{m} \mu^{l} y(t-l \tau)\right)+D, $

其中$ \lambda>0, \mu^{l} \geq 0(l=0, 1, 2, \cdots, m), \mu=\sum\limits_{l=0}^{m} \mu^{l}>0 $及$ D \geq 0 $是常数, 那么存在一个正的常数$ M_{y}<\infty $使得

(2.1) $ \begin{equation} \limsup \limits_{t \rightarrow +\infty} y(t) \leq M_{y}=-\frac{D}{\lambda}+\left(\frac{D}{\lambda}+y^{*}\right) \exp (\lambda m \tau)<+\infty, \end{equation} $

其中$ y=y^{\ast } $是方程$ y(\lambda-\mu y)+D=0 $的唯一解.

引理2.3  假设$ y(t)>0 $, 并成立

$ \dot{y}_(t) \geq y(t)\left(\lambda-\sum\limits_{l=0}^{m} \mu^{l} y(t-l \tau)\right), $

如果$ (2.1) $式成立, 则存在一个正常数$ m_{y}>0 $, 使得对$ \mu=\sum\limits_{l=0}^{m} \mu^{l}>0 $, 有

$ \liminf\limits_{t \rightarrow +\infty} y(t) \geq m_{y}=\frac{\lambda}{\mu} \exp \left\{\left(\lambda-\mu M_{y}\right) m \tau\right\}>0 . $

对于系统$ (1.1) $, 我们设

$ \begin{eqnarray} M_{1}&=&\frac{r_{1}^{m}}{a_{11}^{l}} \exp \left(r_{1}^{m} \tau_{1}\right), N_{1}=\frac{e_{1}^{m}+q_{1}^{m} M_{1}}{f_{1}^{l}}, N_{2}=\frac{e_{2}^{m}}{f_{2}^{l}}, N_{3}=\frac{e_{3}^{m}}{f_{3}^{l}}, \\ M_{2}&=&\frac{\left(a_{21}^{m} M_{1}\right) /\left(b_{12}^{l}+M_{1}\right)+d_{2}^{m} N_{2}-r_{2}^{l}}{a_{22}^{l}} \exp \left[\left(\left(a_{21}^{m} M_{1}\right) /\left(b_{12}^{l}+M_{1}\right)+d_{2}^{m} N_{2}-r_{2}^{l}\right) \tau_{2}\right], \\ M_{3}&=&\frac{\left(a_{32}^{m} M_{2}\right) /\left(b_{23}^{l}+M_{2}\right)+d_{3}^{m} N_{3}-r_{3}^{l}}{a_{33}^{l}} \exp \left[\left(\left(a_{32}^{m} M_{2}\right) /\left(b_{23}^{l}+M_{2}\right)+d_{3}^{m} N_{3}-r_{3}^{l}\right) \tau_{4}\right], \\ m_{1}&=&\frac{r_{1}^{l}-a_{12}^{m} M_{2} / b_{12}^{l}-d_{1}^{m} N_{1}}{a_{11}^{m}} \exp \left[\left(r_{1}^{l}-a_{12}^{m} M_{2} / b_{12}^{l}-d_{1}^{m} N_{1}-a_{11}^{m} M_{1}\right) \tau_{1}\right], \\ n_{1}&=&\frac{e_{1}^{l}+q_{1}^{l} m_{1}}{f_{1}^{m}}, n_{2}=\frac{e_{2}^{l}-q_{2}^{m} M_{2}}{f_{2}^{m}}, n_{3}=\frac{e_{3}^{l}-q_{3}^{m} M_{3}}{f_{3}^{m}}, \\ m_{2}&=&\frac{\left(a_{21}^{l} m_{1}\right) /\left(b_{12}^{m}+m_{1}\right)+d_{2}^{l} n_{2}-a_{23}^{m} M_{3} / b_{23}^{l}-r_{2}^{m}}{a_{22}^{m}} \exp \left[\left(\left(a_{21}^{l} m_{1}\right) /\left(b_{12}^{m}+m_{1}\right)\right.\right. \\ &&\left.\left.+d_{2}^{l} n_{2}-a_{23}^{m} M_{3} / b_{23}^{l}-r_{2}^{m}-a_{22}^{m} M_{2}\right) \tau_{2}\right], \\ m_{3}&=&\frac{\left(a_{32}^{l} m_{2}\right) /\left(b_{23}^{m}+m_{2}\right)+d_{3}^{l} n_{3}-r_{3}^{m}}{a_{33}^{m}}\exp[((a_{32}^{l} m_{2})/(b_{23}^{m}+m_{2})+d_{3}^{l} n_{3} -r_{3}^{m}-a_{3}^{m} M_{3}) \tau_{4}] . \end{eqnarray} $

定理2.1  若系统$ (1.1) $满足以下条件及初始条件$ (1.2) $

$ {\rm (H_{1})}\; r_{2}^{l}<\left(a_{21}^{m} M_{1}\right) /\left(b_{12}^{l}+M_{1}\right)+d_{2}^{m} N_{2}, $

$ {\rm (H_{2})}\; r_{3}^{l}<\left(a_{32}^{m} M_{2}\right) /\left(b_{23}^{l}+M_{2}\right)+d_{3}^{m} N_{3}, $

$ {\rm (H_{3})}\; r_{1}^{l}>a_{12}^{m} M_{2} / b_{12}^{l}+d_{1}^{m} N_{1}, $

$ {\rm (H_{4})}\; e_{2}^{l}>q_{2}^{m} M_{2}, $

$ {\rm (H_{5})}\; e_{3}^{l}>q_{3}^{m} M_{3}, $

$ {\rm (H_{6})}\; \left(a_{21}^{l} m_{1}\right) /\left(b_{12}^{m}+m_{1}\right)+d_{2}^{l} n_{2}>a_{23}^{m} M_{3} / b_{23}^{l}+r_{2}^{m}, $

$ {\rm (H_{7})}\; r_{3}^{m}<\left(a_{32}^{l} m_{2}\right) /\left(b_{23}^{m}+m_{2}\right)+d_{3}^{l} n_{3}, $

那么系统$ (1.1) $是持久的.

证  由系统$ (1.1) $的第一个方程, 我们得到

$ \begin{eqnarray} \dot{x}_{1}(t) &=&x_{1}(t)\left[r_{1}(t)-a_{11}(t) x_{1}\left(t-\tau_{1}\right)-\frac{a_{12}(t) x_{2}(t)}{b_{12}(t)+x_{1}(t)}-d_{1}(t) u_{1}(t)\right] \\ & \leq & x_{1}(t)\left[r_{1}(t)-a_{11}(t) x_{1}\left(t-\tau_{1}\right)\right] \leq x_{1}(t)\left[r_{1}^{m}-a_{11}^{l} x_{1}\left(t-\tau_{1}\right)\right] . \end{eqnarray} $

根据引理2.2, 我们得到

(2.2) $ \begin{equation} \limsup \limits_{t \rightarrow +\infty} x_{1}(t) \leq \frac{r_{1}^{m}}{a_{11}^{l}} \exp \left(r_{1}^{m} \tau_{1}\right)=M_{1} . \end{equation} $

根据系统$ (1.1) $的第四个方程, 我们可得

$ \dot{u}_{1}(t)=e_{1}(t)-f_{1}(t) u_{1}(t)+q_{1}(t) x_{1}(t) \leq e_{1}^{m}-f_{1}^{l} u_{1}(t)+q_{1}^{m} M_{1}, $

根据引理$ 2.1 $, 我们有

(2.3) $ \begin{equation} \limsup \limits_{t \rightarrow +\infty}(t) \leq \frac{e_{1}^{\pi}+q_{1}^{n} M_{1}}{f_{1}^{l}}=N_{1} . \end{equation} $

根据系统$ (1.1) $的第五个和第六个方程, 我们可得

$ \dot{u}_{2}(t)=e_{2}(t)-f_{2}(t) u_{2}(t)-q_{2}(t) x_{2}(t) \leq e_{2}(t)-f_{2}(t) u_{2}(t) \leq e_{2}^{n}-f_{2}^{\prime} u_{2}(t), $

$ \dot{u}_{3}(t)=e_{3}(t)-f_{3}(t) u_{3}(t)-q_{3}(t) x_{3}(t) \leq e_{3}(t)-f_{3}(t) u_{3}(t) \leq e_{3}^{n}-f_{3}^{\prime} u_{3}(t) . $

根据引理2.1, 我们可以得到

(2.4) $ \begin{equation} \limsup \limits_{t \rightarrow +\infty} {p} u_{2}(t) \leq e_{2}^{n} / f_{2}^{l}=N_{2}, \end{equation} $

(2.5) $ \begin{equation} \limsup \limits_{t \rightarrow +\infty} u_{3}(t) \leq e_{3}^{n} / f_{3}^{l}=N_{3} . \end{equation} $

根据系统$ (1.1) $的第二和第三个方程, 我们有

$ \begin{eqnarray} \dot{x}_{2}(t)&=&x_{2}(t)\left[-r_{2}(t)-a_{22}(t) x_{2}\left(t-\tau_{2}\right)+\frac{a_{11}(t) x_{1}\left(t-\tau_{3}\right)}{b_{12}(t)+x_{1}\left(t-\tau_{3}\right)}-\frac{a_{23}(t) x_{3}(t)}{b_{23}(t)+x_{2}(t)}+d_{2}(t) u_{2}(t)\right]\\ &\leq & x_{2}(t)\left[-r_{2}(t)-a_{22}(t) x_{2}\left(t-\tau_{2}\right)+\frac{a_{21}(t) x_{1}\left(t-\tau_{3}\right)}{b_{12}(t)+x_{1}\left(t-\tau_{3}\right)}+d_{2}(t) u_{2}(t)\right]\\ &\leq &x_{2}(t)\left[-r_{2}^{l}-a_{22}^{l} x_{2}\left(t-\tau_{2}\right)+\frac{a_{21}^{m} M_{1}}{b_{12}(l)+M_{1}}+d_{2}^{m} N_{2}\right], \\ \dot{x}_{3}(t)&=&x_{3}(t)\left[-r_{3}(t)-a_{33}(t)x_{3} \left(t-\tau_{4}\right)+\frac{a_{32}(t)x_{2}\left(t-\tau_{5}\right)} {b_{23}(t)+x_{2}\left(t-\tau_{5}\right)}+d_{3}(t)u_{3}(t)\right]\\ &\leq& x_{3}(t)\left[-r_{3}^{l}-a_{33}^{l} x_{3}\left(t-\tau_{4}\right)+\frac{a_{32}^{m} M_{2}}{b_{23}^{l}+M_{2}}+d_{3}^{m} N_{3}\right] . \end{eqnarray} $

根据引理$ 2.2 $, 如果$ {\rm (H_{1}), (H_{2})} $成立, 我们有

(2.6) $ \begin{eqnarray} \limsup \limits_{t\rightarrow +\infty} x_{2}(t) &\leq& \frac{(a_{21}^{m} M_{1}) /(b_{12}^{l}+M_{1})+d_{2}^{m} N_{2}-r_{2}^{l}}{a_{22}^{l}} \exp [((a_{21}^{m} M_{1})/(b_{12}^{l}+M_{1}){}\\ &&+d_{2}^{m}N_{2}-r_{2}^{l})\tau_{2}]=M_{2}, \end{eqnarray} $

(2.7) $ \begin{eqnarray} \limsup \limits_{t \rightarrow +\infty} x_{3}(t) &\leq& \frac{\left(a_{32}^{m} M_{2}\right) /\left(b_{23}^{l}+M_{2}\right)+d_{3}^{m} N_{3}-r_{3}^{l}}{d_{33}^{l}}\exp [((a_{32}^{m} M_{2}) /(b_{23}^{l}+M_{2}){}\\ &&+d_{3}^{m}N_{3}-r_{3}^{l})\tau_{4}]=M_{3} . \end{eqnarray} $

另一方面, 根据系统$ (1.1) $的第一个方程, 我们可得

$ \begin{eqnarray} \dot{x}_{1}(t)&=&x_{1}(t)\left[r_{1}(t)-a_{11}(t) x_{1}\left(t-\tau_{1}\right)-\frac{a_{12}(t) x_{2}(t)}{b_{12}(t)+x_{1}(t)}-d_{1}(t) u_{1}(t)\right]\\ &\geq & x_{1}(t)\left[r_{1}^{l}-a_{11}^{m} x_{1}\left(t-\tau_{1}\right)-\frac{a_{12}^{m} M_{2}}{b_{12}^{l}}-d_{1}^{m} N_{l}\right] . \end{eqnarray} $

根据假设$ {\rm(H_{3})} $和引理$ 2.3 $, 我们可以得到

(2.8) $ \begin{equation} \liminf\limits_{t \rightarrow +\infty} x_{1}(t) \geq \frac{r_{1}^{l}-a_{12}^{m} M_{2} / b_{12}^{l}-d_{1}^{m} N_{1}}{a_{11}^{m}}\exp \left[\left(r_{1}^{l}-a_{12}^{m} M_{2} / b_{12}^{l}-d_{1}^{m} N_{1}-a_{11}^{m} M_{1}\right) \tau_{1}\right]=m_{1}. \end{equation} $

根据系统$ (1.1) $的第四个方程, 我们有

$ \dot{u}_{1}(t)=e_{1}(t)-f_{1}(t) u_{1}(t)+q_{1}(t) x_{1}(t) \geq e_{1}^{l}-f_{1}^{m} u_{1}(t)+q_{1}^{l} m_{1} . $

根据引理$ 2.1 $, 可得

(2.9) $ \begin{equation} \liminf\limits_{H+\infty}u_{1}(t)\geq\frac{e_{1}^{l}+q_{1}^{l}m_{1}}{f_{1}^{m}}=n_{1} . \end{equation} $

由系统$ (1.1) $的第五和第六方程, 我们可得

$ \dot{u}_{2}(t)=e_{2}(t)-f_{2}(t) u_{2}(t)-q_{2}(t) x_{2}(t) \geq e_{2}^{l}-f_{2}^{m} u_{2}(t)-q_{2}^{m} M_{2}, $

$ \dot{u}_{3}(t)=e_{3}(t)-f_{3}(t) u_{3}(t)-q_{3}(t) x_{3}(t) \geq e_{3}^{l}-f_{3}^{m} u_{3}(t)-q_{3}^{m} M_{3} . $

如果$ {\rm (H_{4}), (H_{5})} $成立, 根据引理$ 2.1 $, 我们可得

(2.10) $ \begin{equation} \liminf\limits_{t \rightarrow +\infty} u_{2}(t) \geq \frac{e_{2}^{l}-q_{2}^{m} M_{2}}{f_{2}^{m}}=n_{2}, \end{equation} $

(2.11) $ \begin{equation} \liminf\limits_{t \rightarrow +\infty} u_{3}(t) \geq \frac{e_{3}^{l}-q_{3}^{m} M_{3}}{f_{3}^{m}}=n_{3} . \end{equation} $

根据系统$ (1.1) $的第二和第三个方程, 我们有

$ \begin{eqnarray} \dot{x}_{2}(t)&=&x_{2}(t)\left[-r_{2}(t)-a_{22}(t) x_{2}\left(t-\tau_{2}\right)+\frac{a_{21}(t) x_{1}\left(t-\tau_{3}\right)}{b_{12}(t)+x_{1}\left(t-\tau_{3}\right)}-\frac{a_{23}(t) x_{3}(t)}{b_{23}(t)+x_{2}(t)}+d_{2}(t) u_{2}(t)\right]\\ &\geq & x_{2}(t)\left[-r_{2}^{m}-a_{22}^{m} x_{2}\left(t-\tau_{2}\right)+\frac{a_{21}^{l} m_{1}}{b_{12}^{m}+m_{1}}-\frac{a_{23}^{m} M_{3}}{b_{23}^{l}}+d_{2}^{l} n_{2}\right], \\ \dot{x}_{3}(t)&=&x_{3}(t)\left[-r_{3}(t)-a_{33}(t) x_{3}\left(t-\tau_{4}\right)+\frac{a_{32}(t) x_{2}\left(t-\tau_{5}\right)}{b_{23}(t)+x_{2}\left(t-\tau_{5}\right)}+d_{3}(t) u_{3}(t)\right]\\ &\geq& x_{3}(t)\left[-r_{3}^{m}-a_{33}^{m} x_{3}\left(t-\tau_{4}\right)+\frac{a_{32}^{l} m_{2}}{b_{23}^{m}+m_{2}}+d_{3}^{l} n_{3}\right] . \end{eqnarray} $

根据引理$ 2.3 $, 如果$ {\rm (H_{6}), (H_{7})} $成立, 我们可得

(2.12) $ \begin{eqnarray} \liminf\limits_{t \rightarrow +\infty} x_{2}(t)& \geq &\frac{\left(a_{21}^{l} m_{1}\right) /\left(b_{12}^{m}+m_{1}\right)+d_{2}^{l} n_{2}-a_{23}^{m} M_{3} / b_{23}^{l}-r_{2}^{m}}{a_{22}^{m}} \exp \left[\left(\left(a_{21}^{l} m_{1}\right) /\left(b_{12}^{m}+m_{1}\right)\right.\right.{}\\ &&\left.\left.+d_{2}^{l} n_{2}-a_{23}^{m} M_{3} / b_{23}^{l}-r_{2}^{m}-a_{22}^{m} M_{2}\right) \tau_{2}\right]=m_{2}, \end{eqnarray} $

(2.13) $ \begin{eqnarray} \liminf\limits_{t \rightarrow +\infty} x_{3}(t)& \geq &\frac{\left(a_{32}^{l} m_{2}\right) /\left(b_{23}^{m}+m_{2}\right)+d_{3}^{l} n_{3}-r_{3}^{m}}{a_{33}^{m}} \exp \left[\left(\left(a_{32}^{l} m_{2}\right) /\left(b_{23}^{m}+m_{2}\right)\right.\right.{}\\ &&\left.\left.+d_{3}^{l} n_{3}-r_{3}^{m}-a_{33}^{m} M_{3}\right) \tau_{4}\right]=m_{3} . \end{eqnarray} $

根据(2.2)–(2.13)式, 可知系统$ (1.1) $是持久的.

在本节中, 我们取得了一些保证捕食-食饵系统$ (1.1) $的正解的全局吸引性的一些充分条件. 首先, 我们给出以下定义和引理.

定义3.1  假设$ (x_{1}(t), $$ x_{2}(t), $$ x_{3}(t), $$ u_{1}(t), $$ u_{2}(t), $$ u_{3}(t)) $和$ (y_{1}(t), y_{2}(t), y_{3}(t), v_{1}(t), v_{2}(t), $$ v_{3}(t)) $是系统$ (1.1) $的任意两个正解, 如果

$ \lim\limits_{t \rightarrow +\infty}\left|x_{i}(t)-y_{i}(t)\right|=0, \lim\limits_{t \rightarrow +\infty}\left|u_{i}(t)-v_{i}(t)\right|=0, (i=1, 2, 3), $

那么系统$ (1.1) $被称为是全局吸引的.

引理3.1^{[16, 引理8.2]}  如果函数$ f(t): {{\mathbb{R}}} ^{+} \rightarrow {{\mathbb{R}}} $是一致连续的, 极限$ \lim\limits_{t \rightarrow +\infty} \int_{0}^{t} f(s) {\rm d}s $存在且有限, 那么有$ \lim\limits_{t \rightarrow +\infty} f(t)=0 $.

定理3.1  假设系统$ (1.1) $除了条件$ {\rm (H_{1})} $–$ {\rm (H_{7})} $之外, 还满足

$ {\rm (H_{8})}{\quad} A_{i}>0, B_{i}>0, {\quad} i= 1, 2, 3, $

其中

$ \begin{eqnarray} A_{1}&=&a_{11}^{l}-\frac{a_{12}^{m} M_{2}}{\left(b_{12}^{l}+m_{1}\right)^{2}}-\left[r_{1}^{m}+a_{11}^{m} M_{1}+\frac{a_{12}^{m} M_{2}}{b_{12}^{l}+m_{1}}+d_{1}^{m} N_{1}\right] a_{11}^{m} \tau_{1}-M_{1}\left(a_{11}^{m}\right)^{2} \tau_{1}\\ &&-\frac{a_{22}^{m} b_{12}^{m}}{\left(b_{12}^{l}+m_{1}\right)^{2}}-M_{2} \frac{a_{22}^{m} a_{22}^{m} \tau_{2}}{b_{12}^{l}+m_{1}}-q_{1}^{m}, \\ A_{2}&=&a_{22}^{l}-\frac{a_{23}^{m} M_{3}}{\left(b_{23}^{l}+m_{2}\right)^{2}}-\left[r_{2}^{m}+a_{22}^{m} M_{2}+\frac{a_{21}^{m} M_{1}}{b_{12}^{l}+M_{1}}+\frac{a_{23}^{m} M_{3}}{b_{23}^{l}+m_{2}}+d_{2}^{m} N_{2}\right] a_{22}^{m} \tau_{2}\\ &&-M_{2}\left(a_{22}^{m}\right)^{2} \tau_{2} -\frac{a_{12}^{m} b_{12}^{m}+a_{12}^{m} M_{1}}{\left(b_{12}^{l}+m_{1}\right)^{2}} -\frac{M_{1} a_{12}^{m} a_{11}^{m} \tau_{1}}{b_{12}^{l}+m_{1}}-\frac{a_{32}^{m} b_{23}^{m}}{\left(b_{23}^{l}+m_{2}\right)^{2}}-\frac{M_{3} a_{32}^{m} a_{33}^{m} \tau_{4}}{b_{23}^{l}+m_{2}}-q_{2}^{m}, \\ A_{3}&=&a_{33}^{l}-\left[r_{3}^{m}+a_{33}^{m} M_{3}+\frac{a_{32}^{m} M_{2}}{b_{23}^{l}+M_{2}}+d_{3}^{m} N_{3}\right] a_{33}^{m} \tau_{4}-M_{3}\left(a_{33}^{m}\right)^{2} \tau_{4} \\ &&-\frac{a_{23}^{m} b_{23}^{m}+a_{23}^{m} M_{2}}{\left(b_{23}^{l}+m_{2}\right)^{2}}-\frac{M_{2} a_{23}^{m} a_{22}^{m} \tau_{2}}{b_{23}^{l}+m_{2}}-q_{3}^{m}, \\ B_{1}&=&f_{1}^{l}-d_{1}^{m}-M_{1} d_{1}^{m} a_{11}^{m} \tau_{1}, B_{2}=f_{2}^{l}-d_{2}^{m}-M_{2} d_{2}^{m} a_{22}^{m} \tau_{2}, B_{3}=f_{3}^{l}-d_{3}^{m}-M_{3} d_{3}^{m} a_{33}^{m} \tau_{4}, \end{eqnarray} $

那么系统$ (1.1) $是全局吸引的.

证  假设$ \left(x_{1}(t), x_{2}(t), x_{3}(t), u_{1}(t), u_{2}(t), u_{3}(t)\right) $和$ \left(y_{1}(t), y_{2}(t), y_{3}(t), v_{1}(t), v_{2}(t), v_{3}(t)\right) $是系统$ (1.1) $的任意两个正解, 那么根据定理$ 2.1 $, 存在正常数$ T $和$ M_{i}, N_{i}, m_{i}, n_{i}(i=1, 2, 3) $, 使得当$ t \geq T $时, 有

$ m_{i} \leq x_{i}(t), y_{i}(t) \leq M_{i}, n_{i} \leq u_{i}(t), v_{i}(t) \leq N_{i}, i=1, 2, 3 . $

我们定义

$ V_{11}(t)=\left|\ln x_{i}(t)-\ln y_{i}(t)\right|, $

计算$ V_{11}(t) $沿着系统$ (1.1) $的右上导数, 我们可得

(3.1) $ \begin{eqnarray} D^{+}V_{11}(t)&=&{\rm sgn}\left\{x_{1}(t)-y_{1}(t)\right\} \bigg[-a_{11}(t)\left(x_{1}\left(t-\tau_{1}\right)-y_{1}\left(t-\tau_{1}\right)\right) {}\\ &&-\frac{(a_{12}(t) b_{12}(t)+a_{12}(t) y_{1}(t))(x_{2}(t)-y_{2}(t))-a_{12}(t) y_{2}(t)(x_{1}(t)-y_{1}(t))}{(b_{12}(t)+x_{1}(t))(b_{12}(t)+y_{1}(t))}{}\\ &&-d_{1}(t)(u_{1}(t)-v_{1}(t))\bigg]{}\\ &=&{\rm sgn}\left\{x_{1}(t)-y_{1}(t)\right\} \bigg[-a_{11}(t)\left(x_{1}(t)-y_{1}(t)\right)+a_{11}(t) \int_{t-\tau_{1}}^{t}\left(\dot{x}_{1}(\theta)-\dot{y}_{1}(\theta)\right) {\rm d}\theta{}\\ &&-\frac{(a_{12}(t) b_{12}(t)+a_{12}(t) y_{1}(t))(x_{2}(t)-y_{2}(t))-a_{12}(t) y_{2}(t)(x_{1}(t)-y_{1}(t))}{(b_{12}(t)+x_{1}(t))(b_{12}(t)+y_{1}(t))}{}\\ &&-d_{1}(t)(u_{1}(t)-v_{1}(t))\bigg]{}\\ &=&{\rm sgn}\{x_{1}(t)-y_{1}(t)\} \bigg[-a_{11}(t)(x_{1}(t)-y_{1}(t))-d_{1}(t)(u_{1}(t)-v_{1}(t)){}\\ &&-\frac{\left(a_{12}(t) b_{12}(t)+a_{12}(t) y_{1}(t)\right)\left(x_{2}(t)-y_{2}(t)\right)-a_{12}(t) y_{2}(t)\left(x_{1}(t)-y_{1}(t)\right)}{\left(b_{12}(t)+x_{1}(t)\right)\left(b_{12}(t)+y_{1}(t)\right)}{}\\ &&+a_{11}(t) \int_{t-\tau_{1}}^{t}\bigg(x_{1}(\theta)\bigg[r_{1}(\theta)-a_{11}(\theta) x_{1}(\theta-\tau_{1})-\frac{a_{12}(\theta) x_{2}(\theta)}{b_{12}(\theta)+x_{1}(\theta)}-d_{1}(\theta) u_{1}(\theta)\bigg]{}\\ && -y_{1}(\theta)\bigg[r_{1}(\theta)-a_{11}(\theta) y_{1}\left(\theta-\tau_{1}\right)-\frac{a_{12}(\theta) y_{2}(\theta)}{b_{12}(\theta)+y_{1}(\theta)}-d_{1}(\theta) v_{1}(\theta)\bigg]\bigg) {\rm d}\theta\bigg]{}\\ &=&{\rm sgn}\left\{x_{1}(t)-y_{1}(t)\right\} \bigg[-a_{11}(t) \left(x_{1}(t)-y_{1}(t)\right)-d_{1}(t)\left(u_{1}(t)-v_{1}(t)\right) {}\\ &&-\frac{\left(a_{12}(t) b_{12}(t)+a_{12}(t) y_{1}(t)\right)\left(x_{2}(t)-y_{2}(t)\right)-a_{12}(t) y_{2}(t)\left(x_{1}(t)-y_{1}(t)\right)} {\left(b_{12}(t)+x_{1}(t)\right) \left(b_{12}(t)+y_{1}(t)\right)}{}\\ &&+a_{11}(t) \int_{t-\tau_{1}}^{t}\bigg(((x_{1}(\theta)-y_{1}(\theta)) \bigg[r_{1}(\theta)-a_{11}(\theta) y_{1}(\theta-\tau_{1})-\frac{a_{12}(\theta) y_{2}(\theta)}{b_{12}(\theta)+y_{1}(\theta)}{}\\ & & -d_{1}(\theta) v_{1}(\theta)\bigg]+x_{1}(\theta)\bigg[-a_{11}(\theta)(x_{1}\left(\theta-\tau_{1}\right) -y_{1}\left(\theta-\tau_{1}\right)) {}\\ &&-\frac{a_{12}(\theta)}{b_{12}(\theta)+x_{1}(\theta)} (x_{2}(\theta)-y_{2}(\theta))-d_{1}(\theta)(u_{1}(\theta)-v_{1}(\theta))\bigg]\bigg) {\rm d}\theta\bigg]{}\\ & \leq &-\left(a_{11}(t)-\frac{a_{12}(t) y_{2}(t)}{\left(b_{12}(t)+x_{1}(t)\right) \left(b_{12}(t)+y_{1}(t)\right)}\right)\left|x_{1}(t)-y_{1}(t)\right|{}\\ &&+\frac{(a_{12}(t) b_{12}(t)+a_{12}(t) y_{1}(t))}{(b_{12}(t)+x_{1}(t))(b_{12}(t)+y_{1}(t))} |x_{2}(t)-y_{2}(t)|+d_{1}(t)|u_{1}(t)-v_{1}(t)|{}\\ &&+a_{11}(t) \int_{t-\tau_{1}}^{t}\bigg(\bigg[r_{1}(\theta)+a_{11}(\theta) y_{1}(\theta-\tau_{1})+\frac{a_{12}(\theta) y_{2}(\theta)}{b_{12}(\theta)+y_{1}(\theta)}+d_{1}(\theta) v_{1}(\theta)\bigg]{}\\ & &\times |x_{1}(\theta)-y_{1}(\theta)|+x_{1}(\theta) \bigg[a_{11}(\theta)|x_{1}(\theta-\tau_{1})-y_{1}(\theta-\tau_{1})|{}\\ &&+\frac{a_{12}(\theta)}{b_{12}(\theta)+x_{1}(\theta)} \left|x_{2}(\theta)-y_{2}(\theta)\right|+d_{1}(\theta)\left|u_{1} (\theta)-v_{1}(\theta)\right|\bigg]\bigg) {\rm d}\theta . \end{eqnarray} $

接下来, 我们定义

(3.2) $ \begin{eqnarray} V_{12}(t)&=&\int_{t-\tau_{1}}^{t}\int_{s}^{t}a_{11}\left(s+\tau_{1}\right) \bigg(\bigg[r_{1}(\theta)+a_{11}(\theta) y_{1}\left(\theta-\tau_{1}\right)+\frac{a_{12}(\theta) y_{2}(\theta)}{b_{12}(\theta)+y_{1}(\theta)} {}\\ &&+d_{1}(\theta) v_{1}(\theta)\bigg] \left|x_{1}(\theta)-y_{1}(\theta)\right|+x_{1}(\theta) \bigg[a_{11}(\theta)\left|x_{1}\left(\theta-\tau_{1}\right)-y_{1}\left(\theta-\tau_{1}\right)\right| {}\\ &&+\frac{a_{12}(\theta)}{b_{12}(\theta)+x_{1}(\theta)}\left|x_{2}(\theta)-y_{2}(\theta) \right|+d_{1}(\theta)\left|u_{1}(\theta)-v_{1}(\theta)\right|\bigg]\bigg) {\rm d}\theta {\rm d}s, \end{eqnarray} $

于是, 根据$ (3.1) $和$ (3.2) $式, 有

(3.3) $ \begin{eqnarray} \sum\limits_{i=1}^{2} D^{+} V_{1 i}(t) &\leq &-\left(a_{11}(t)-\frac{a_{12}(t) y_{2}(t)}{\left(b_{12}(t)+x_{1}(t)\right)\left(b_{12}(t)+y_{1}(t)\right)}\right)\left|x_{1}(t)-y_{1}(t)\right|{}\\ &&+\frac{\left(a_{12}(t) b_{12}(t)+a_{12}(t) y_{1}(t)\right)}{\left(b_{12}(t)+x_{1}(t)\right)\left(b_{12}(t)+y_{1}(t)\right)}\left|x_{2}(t)-y_{2}(t)\right|+d_{1}(t)\left|u_{1}(t)-v_{1}(t)\right|{}\\ &&+\int_{t-\tau_{1}}^{t} a_{11}\left(s+\tau_{1}\right) {\rm d}s\bigg[r_{1}(t)+a_{11}(t) y_{1}\left(t-\tau_{1}\right)+\frac{a_{12}(t) y_{2}(t)}{b_{12}(t)+y_{1}(t)}{}\\ &&+d_{1}(t) v_{1}(t)\bigg]|x_{1}(t)-y_{1}(t)|+M_{1} \int_{t-\tau_{1}}^{t} a_{11}(s+\tau_{1}) {\rm d}s \times a_{11}(t)|x_{1}(t-\tau_{1}){}\\ & &-y_{1}(t-\tau_{1})|+M_{1} \int_{t-\tau_{1}}^{t} a_{11}(s+\tau_{1}) {\rm d}s \times \frac{a_{12}(t)}{b_{12}(t)+x_{1}(t)}|x_{2}(t)-y_{2}(t)|{}\\ &&+M_{1} \int_{t-\tau_{1}}^{t} a_{11}\left(s+\tau_{1}\right) {\rm d}s \times d_{1}(t)\left|u_{1}(t)-v_{1}(t)\right| . \end{eqnarray} $

定义

(3.4) $ \begin{equation} V_{13}(t)=M_{1} \int_{t-\tau_{1}}^{t} \int_{w}^{w+\tau_{1}} a_{11}\left(s+\tau_{1}\right) a_{11}\left(w+\tau_{1}\right)\left|x_{1}(w)-y_{1}(w)\right| {\rm d}s{\rm d}w. \end{equation} $

设

(3.5) $ \begin{equation} V_{1}(t)=V_{11}(t)+V_{12}(t)+V_{13}(t), \end{equation} $

根据(3.3)–(3.4)式, 计算$ V_{1}(t) $的右上导数, 可得

(3.6) $ \begin{eqnarray} D^{+} V_{1}(t)&\leq &-\bigg[a_{11}^{l}-\frac{a_{12}^{m} M_{2}}{(b_{12}^{l}+m_{1})^{2}}-\bigg(r_{1}^{m}+a_{11}^{m} M_{1}+\frac{a_{12}^{m} M_{2}}{b_{12}^{l}+m_{1}}+d_{1}^{m} N_{1}\bigg)\int_{t-\tau_{1}}^{t} a_{11}^{m}(s+\tau_{1}){\rm d}s{}\\ &&-M_{1} a_{11}^{m} \int_{t}^{t+\tau_{1}} a_{11}^{m}\left(s+\tau_{1}\right) {\rm d}s\bigg]\left|x_{1}(t)-y_{1}(t)\right|{}\\ & &+\left[\frac{a_{12}^{m} b_{12}^{m}+d_{12}^{m} M_{1}}{\left(b_{12}^{l}+m_{1}\right)^{2}}+M_{1} \frac{d_{12}^{m}}{b_{12}^{l}+m_{1}} \int_{t-\tau}^{t} a_{11}^{m}\left(s+\tau_{1}\right) {\rm d}s\right]\left|x_{2}(t)-y_{2}(t)\right|{}\\ & &+\left[d_{1}^{m}+M_{1} d_{1}^{m} \int_{t-\tau_{1}}^{t} a_{11}^{m}\left(s+\tau_{1}\right) {\rm d}s\right]\left|u_{1}(t)-v_{1}(t)\right|{}\\ &\leq &-\bigg[a_{11}^{l}-\frac{d_{12}^{m} M_{2}}{(b_{12}^{l}+m_{1})^{2}}-\bigg(r_{1}^{m}+a_{11}^{m} M_{1}+\frac{a_{12}^{m} M_{2}}{b_{12}^{l}+m_{1}}+d_{1}^{m} N_{1}\bigg) a_{11}^{m} \tau_{1}{}\\ &&-M_{1}(a_{11}^{m})^{2} \tau_{1}\bigg]|x_{1}(t)-y_{1}(t)|+ \bigg(\frac{a_{12}^{m} b_{12}^{m}+a_{12}^{m} M_{1}}{(b_{12}^{l}+m_{1})^{2}}+\frac{M_{1} a_{12}^{m} a_{11}^{m} \tau_{1}}{b_{12}^{l}+m_{1}}\bigg)|x_{2}(t)-y_{2}(t)|{}\\ &&+(d_{1}^{m}+M_{1} d_{1}^{m} a_{11}^{m} \tau_{1})|u_{1}(t)-v_{1}(t)| . \end{eqnarray} $

类似地, 我们定义$ V_{21}(t)=\left|\ln x_{2}(t)-\ln y_{2}(t)\right| $. 计算和估计$ V_{21}(t) $沿着系统$ (1.1) $的右上导数, 可以得出如下结论

(3.7) $ \begin{eqnarray} D^{+}V_{21}(t)&=&{\rm sgn}\{x_{2}(t)-y_{2}(t)\} \bigg[-a_{22}(t)(x_{2}(t-\tau_{2})-y_{2}(t-\tau_{2})){}\\ &&+\frac{a_{21}(t) b_{12}(t)(x_{1}(t-\tau_{3})-y_{1}(t-\tau_{3}))}{(b_{12}(t)+x_{1}(t-\tau_{3}))(b_{12}(t)+y_{1}(t-\tau_{3}))} {}\\ & &-\frac{(a_{23}(t) b_{23}(t)+a_{23}(t) y_{2}(t))(x_{3}(t)-y_{3}(t))-a_{23}(t) y_{3}(t)(x_{2}(t)-y_{2}(t))}{(b_{23}(t)+x_{2}(t))(b_{23}(t)+y_{2}(t))}{}\\ &&+d_{2}(t)(u_{2}(t)-v_{2}(t))\bigg]{}\\ &=&{\rm sgn}\{x_{2}(t)-y_{2}(t)\}\bigg[-a_{22}(t)(x_{2}(t)-y_{2}(t)){}\\ &&+\frac{a_{21}(t)b_{12}(t)(x_{1}(t-\tau_{3})-y_{1}(t-\tau_{3}))}{(b_{12}(t)+x_{1}(t-\tau_{3}))(b_{12}(t)+y_{1}(t-\tau_{3})}{}\\ &&-\frac{(a_{23}(t) b_{23}(t)+a_{23}(t) y_{2}(t))(x_{3}(t)-y_{3}(t))-a_{23}(t) y_{3}(t)(x_{2}(t)-y_{2}(t))}{(b_{23}(t)+x_{2}(t))(b_{23}(t)+y_{2}(t))}{}\\ &&+d_{2}(t)(u_{2}(t)-v_{2}(t))+a_{22}(t)\int_{t-\tau_{2}}^{t}(\dot{x}_{2}(\theta)-\dot{y}_{2}(\theta)) {\rm d}\theta\bigg]{} \\ &=&{\rm sgn}\left\{x_{2}(t)-y_{2}(t)\right\} \bigg[-a_{22}(t)\left(x_{2}(t)-y_{2}(t)\right) {}\\ && +\frac{a_{21}(t) b_{12}(t)\left(x_{1}\left(t-\tau_{3}\right)-y_{1}\left(t-\tau_{3}\right)\right)}{\left(b_{12}(t) +x_{1}\left(t-\tau_{3}\right)\right)\left(b_{12}(t)+y_{1}\left(t-\tau_{3}\right)\right)} {}\\ &&-\frac{(a_{23}(t) b_{23}(t)+a_{23}(t) y_{2}(t))(x_{3}(t)-y_{3}(t))-a_{23}(t) y_{3}(t)(x_{2}(t)-y_{2}(t))}{(b_{23}(t)+x_{2}(t))(b_{23}(t)+y_{2}(t))} {}\\ &&+d_{2}(t)(u_{2}(t)-v_{2}(t))+a_{22}(t)\int_{t-\tau_{2}}^{t} \bigg(x_{2}(\theta) \bigg[-r_{2}(\theta)-a_{22}(\theta)x_{2} (\theta-\tau_{2}) {}\\ &&+\frac{a_{21}(\theta)x_{1}(\theta-\tau_{3})}{b_{12}(\theta)+x_{1}(\theta-\tau_{3})} -\frac{a_{23}(\theta) x_{3}(\theta)}{b_{23}(\theta)+x_{2}(\theta)}+d_{2}(\theta) u_{2}(\theta)\bigg]{}\\ && -y_{2}(\theta)\bigg[-r_{2}(\theta)-a_{22}(\theta) y_{2}(\theta-\tau_{2})+\frac{a_{21}(\theta) y_{1}(\theta-\tau_{3})}{b_{12}(\theta)+y_{1}(\theta-\tau_{3})}{}\\ &&-\frac{a_{23}(\theta) y_{3}(\theta)}{b_{23}(\theta)+y_{2}(\theta)}+d_{2}(\theta) v_{2}(\theta)\bigg]\bigg) {\rm d}\theta\bigg]{}\\ &=&{\rm sgn}\left\{x_{2}(t)-y_{2}(t)\right\} \bigg[-a_{22}(t)\left(x_{2}(t)-y_{2}(t)\right) {}\\ && +\frac{a_{21}(t) b_{12}(t)\left(x_{1}\left(t-\tau_{3}\right)-y_{1}\left(t-\tau_{3}\right)\right)}{\left(b_{12}(t)+x_{1}\left(t-\tau_{3}\right)\right)\left(b_{12}(t)+y_{1}\left(t-\tau_{3}\right)\right)} {}\\ &&-\frac{\left(a_{23}(t) b_{23}(t)+a_{23}(t) y_{2}(t)\right)\left(x_{3}(t)-y_{3}(t)\right)-a_{23}(t) y_{3}(t)\left(x_{2}(t)-y_{2}(t)\right)}{\left(b_{23}(t)+x_{2}(t)\right)\left(b_{23}(t)+y_{2}(t)\right)}{}\\ &&+d_{2}(t)\left(u_{2}(t)-v_{2}(t)\right)+a_{22}(t) \int_{t-\tau_{2}}^{t}\bigg(( x _ { 2 } ( \theta ) - y _ { 2 } ( \theta ) ) {}\\ &&\times\bigg[-r_{2}(\theta)-a_{22}(\theta) y_{2}(\theta-\tau_{2})+\frac{a_{21}(\theta) y_{1}(\theta-\tau_{3})}{b_{12}(\theta)+y_{1}(\theta-\tau_{3})}-\frac{a_{23}(\theta) y_{3}(\theta)}{b_{23}(\theta)+y_{2}(\theta)} {}\\ &&+d_{2}(\theta) v_{2}(\theta)\bigg]+x_{2}(\theta) \bigg[-a_{22}(\theta)(x_{2}(\theta-\tau_{2})-y_{2}(\theta-\tau_{2})) {}\\ && +\frac{a_{21}(\theta)}{b_{12}(\theta)+x_{1}(\theta-\tau_{3})} \times\left(x_{1}\left(\theta-\tau_{3}\right)-y_{1}\left(\theta-\tau_{3}\right)\right){}\\ &&-\frac{a_{23}(\theta)}{b_{23}(\theta)+x_{2}(\theta)}\left(x_{3}(\theta) -y_{3}(\theta)\right)+d_{2}(\theta)\left(u_{2}(\theta)-v_{2}(\theta)\right)\bigg]\bigg) {\rm d}\theta\bigg]{}\\ &\leq &-\left(a_{22}(t)-\frac{a_{23}(t) y_{3}(t)}{\left(b_{23}(t)+x_{2}(t)\right)\left(b_{23}(t)+y_{2}(t)\right)}\right)\left|x_{2}(t)-y_{2}(t)\right|{}\\ &&+\frac{\left(a_{23}(t) b_{23}(t)+a_{23}(t) y_{2}(t)\right)}{\left(b_{23}(t)+x_{2}(t)\right)\left(b_{23}(t)+y_{2}(t)\right)}\times\left|x_{3}(t)-y_{3}(t)\right|+d_{2}(t)\left|u_{2}(t)-v_{2}(t)\right|{}\\ &&+\frac{a_{21}(t) b_{12}(t)}{\left(b_{12}(t)+x_{1}\left(t-\tau_{3}\right)\right)\left(b_{12}(t)+y_{1}\left(t-\tau_{3}\right)\right)}\times\left|x_{1}\left(t-\tau_{3}\right)-y_{1}\left(t-\tau_{3}\right)\right|{}\\ &&+a_{22}(t) \int_{t-\tau_{2}}^{t}\left(\left[r_{2}(\theta)+a_{22}(\theta) y_{2}\left(\theta-\tau_{2}\right)+\frac{a_{21}(\theta) y_{1}\left(\theta-\tau_{3}\right)}{b_{12}(\theta)+y_{1}\left(\theta-\tau_{3}\right)}\right.\right.{}\\ && +\frac{a_{23}(\theta) y_{3}(\theta)}{b_{23}(\theta)+y_{2}(\theta)}+d_{2}(\theta) v_{2}(\theta)\bigg]\left|x_{2}(\theta)-y_{2}(\theta)\right| {}\\ &&+x_{2}(\theta)\bigg[a_{22}(\theta) \left|x_{2}\left(\theta-\tau_{2}\right)-y_{2}\left(\theta-\tau_{2}\right)\right| {}\\ && +\frac{a_{21}(\theta)}{b_{12}(\theta)+x_{1}(\theta-\tau_{3})}|x_{1}(\theta-\tau_{3})-y_{1}(\theta-\tau_{3})| {}\\ && +\frac{a_{23}(\theta)}{b_{23}(\theta)+x_{2}(\theta)}|x_{3}(\theta)-y_{3}(\theta)| +d_{2}(\theta)|u_{2}(\theta)-v_{2}(\theta)|\bigg]\bigg) {\rm d}\theta . \end{eqnarray} $

定义

(3.8) $ \begin{eqnarray} V_{22}(t) &=&\int_{t-\tau_{2}}^{t} \int_{s}^{t} a_{22}\left(s+\tau_{2}\right)\left\{\bigg[r_{2}(\theta)+a_{22}(\theta) y_{2}\left(\theta-\tau_{2}\right)+\frac{a_{21}(\theta) y_{1}\left(\theta-\tau_{3}\right)}{b_{22}(\theta)+y_{1}\left(\theta-\tau_{3}\right)}\right.{}\\ &&\left.+\frac{a_{23}(\theta) y_{3}(\theta)}{b_{23}(\theta)+y_{2}(\theta)}+d_{2}(\theta) v_{2}(\theta)\right]\left|x_{2}(\theta)-y_{2}(\theta)\right| {}\\ &&+x_{2}(\theta)\left[a_{22}(\theta)\left|x_{2}\left(\theta-\tau_{2}\right)-y_{2}\left(\theta-\tau_{2}\right)\right|+\frac{a_{21}(\theta)}{b_{2}(\theta) +x_{1}\left(\theta-\tau_{3}\right)}\left|x_{1}\left(\theta-\tau_{3}\right)\right.\right. {}\\ &&\left.\left.\left.-y_{1}\left(\theta-\tau_{3}\right)\right|+\frac{a_{23}(\theta)}{b_{23}(\theta)+x_{2}(\theta)}\left|x_{3}(\theta)-y_{3}(\theta)\right|+d_{2}(\theta)\left|u_{2}(\theta)-v_{2}(\theta)\right|\right]\right\} {\rm d}\theta {\rm d}s . \end{eqnarray} $

然后, 根据$ (3.7) $和$ (3.8) $式, 我们有

(3.9) $ \begin{eqnarray} \sum\limits_{i=1}^{2} D^{+} V_{2i}(t)& \leq &-\left(a_{22}(t)-\frac{a_{23}(t) y_{3}(t)}{\left(b_{3}(t)+x_{2}(t)\right)\left(b_{23}(t)+y_{2}(t)\right)}\right)\left|x_{2}(t)-y_{2}(t)\right|{}\\ &&+\frac{\left(a_{23}(t) b_{23}(t)+a_{23}(t) y_{2}(t)\right)}{\left(b_{23}(t)+x_{2}(t)\right)\left(b_{27}(t)+y_{2}(t)\right)}\times\left|x_{3}(t)-y_{3}(t)\right|+d_{2}(t)\left|u_{2}(t)-v_{2}(t)\right|{}\\ &&+\frac{a_{21}(t) b_{12}(t)}{\left(b_{12}(t)+x_{1}\left(t-\tau_{3}\right)\right)\left(b_{12}(t)+y_{1}\left(t-\tau_{3}\right)\right)}\times\left|x_{1}\left(t-\tau_{3}\right)-y_{1}\left(t-\tau_{3}\right)\right|{}\\ &&+\int_{t-\tau_{2}}^{t} a_{22}\left(s+\tau_{2}\right) {\rm d}s\left[r_{2}(t)+a_{22}(t) y_{2}\left(t-\tau_{2}\right)+\frac{a_{21}(t) y_{1}\left(t-\tau_{3}\right)}{b_{12}(t)+y_{1}\left(t-\tau_{3}\right)}\right.{}\\ &&\left.+\frac{a_{23}(t) y_{3}(t)}{b_{23}(t)+y_{2}(t)}+d_{2}(t) v_{2}(t)\right]\left|x_{2}(t)-y_{2}(t)\right|+M_{2} \int_{t-\tau_{2}}^{t} a_{22}\left(s+\tau_{2}\right) {\rm d}s {}\\ & &\times a_{22}(t)\times\left|x_{2}\left(t-\tau_{2}\right)-y_{2}\left(t-\tau_{2}\right)\right|+M_{2} \int_{t-\tau_{2}}^{t} a_{22}\left(s+\tau_{2}\right) {\rm d}s {}\\ & &\times \frac{a_{21}(t)}{b_{12}(t)+x_{1}\left(t-\tau_{3}\right)}\left|x_{1}\left(t-\tau_{3}\right)-y_{1}\left(t-\tau_{3}\right)\right|{}\\ &&+M_{2}\int_{t-\tau_{2}}^{t}a_{22}(s+\tau_{2}){\rm d}s\times \frac{a_{23}(t)}{b_{23}(t)+x_{2}(t)}|x_{3}(t)-y_{3}(t)|{}\\ &&+M_{2}\int_{t-\tau_{2}}^{t}a_{22}(s+\tau_{2}){\rm d}s\times d_{2}(t)|u_{2}(t)-v_{2}(t)|. \end{eqnarray} $

定义

(3.10) $ \begin{eqnarray} V_{23}(t)&=&\int_{t-\tau_{3}}^{t} \frac{a_{21}\left(w+\tau_{3}\right) b_{12}\left(w+\tau_{3}\right)}{\left(b_{12}\left(w+\tau_{3}\right)+x_{1}(w)\right)\left(b_{12}\left(w+\tau_{3}\right)+y_{1}(w)\right)}\left|x_{1}(w)-y_{1}(w)\right| {\rm d}w{}\\ &&+M_{2} \int_{t-\tau_{2}}^{t} \int_{w}^{w+\tau_{2}} a_{22}\left(s+\tau_{2}\right) a_{22}\left(w+\tau_{2}\right)\left|x_{2}(w)-y_{2}(w)\right| {\rm d}s {\rm d}w{}\\ & &+M_{2} \int_{t-\tau_{3}}^{t} \int_{w+\tau_{3}-\tau_{2}}^{w+\tau_{3}} a_{22}\left(s+\tau_{2}\right) \frac{a_{21}\left(w+\tau_{3}\right)}{b_{12}\left(w+\tau_{3}\right)+x_{1}(w)}\left|x_{1}(w)-y_{1}(w)\right| {\rm d}s{\rm d}w . \end{eqnarray} $

令

(3.11) $ \begin{equation} V_{2}(t)=V_{21}(t)+V_{22}(t)+V_{23}(t) . \end{equation} $

由$ (3.9) $和$ (3.10) $式可以得到$ V_{2}(t) $的右上导数

(3.12) $ \begin{eqnarray} D^{+} V_{2}(t)&\leq &-\bigg[a_{22}^{l}-\frac{a_{23}^{m} M_{3}}{(b_{23}^{l}+m_{2})^{2}}-\bigg(r_{2}^{m}+a_{22}^{m} M_{2}+\frac{a_{21}^{m} M_{1}}{b_{12}^{l}+M_{1}}+\frac{a_{23}^{m} M_{3}}{b_{23}^{l}+m_{2}}+d_{2}^{m} N_{2}\bigg) {}\\ &&\times \int_{t-\tau_{2}}^{t} a_{22}^{m}(s+\tau_{2}) {\rm d}s-M_{2} a_{22}^{m} \int_{t}^{t+\tau_{2}} a_{22}^{m}(s+\tau_{2}) {\rm d}s\bigg]|x_{2}(t)-y_{2}(t)|{}\\ &&+\left[\frac{a_{21}^{m} b_{12}^{m}}{\left(b_{12}^{l}+m_{1}\right)^{2}}+M_{2} \frac{a_{21}^{m}}{b_{12}^{l}+m_{1}} \int_{t+\tau_{3}-\tau_{2}}^{t+\tau_{3}} a_{22}^{m}\left(s+\tau_{2}\right) {\rm d}s\right]\left|x_{1}(t)-y_{1}(t)\right|{}\\ &&+\left[\frac{a_{23}^{m} b_{23}^{m}+a_{23}^{m} M_{2}}{\left(b_{23}^{l}+m_{2}\right)^{2}}+M_{2} \frac{a_{23}^{m}}{b_{23}^{l}+m_{2}} \int_{t-\tau_{2}}^{t} a_{22}^{m}\left(s+\tau_{2}\right) {\rm d}s\right]\left|x_{3}(t)-y_{3}(t)\right|{}\\ &&+\left[d_{2}^{m}+M_{2} d_{2}^{m} \int_{t-\tau_{2}}^{t} a_{22}^{m}\left(s+\tau_{2}\right) {\rm d}s\right]\left|u_{2}(t)-v_{2}(t)\right|{}\\ & \leq &-\bigg[a_{22}^{l}-\frac{a_{23}^{m} M_{3}}{\left(b_{23}^{l}+m_{2}\right)^{2}}-\left(r_{2}^{m}+a_{22}^{m} M_{2}+\frac{a_{21}^{m} M_{1}}{b_{12}^{l}+M_{1}}+\frac{a_{23}^{m} M_{3}}{b_{23}^{l}+m_{2}}+d_{2}^{m} N_{2}\right) a_{22}^{m} \tau_{2}{}\\ &&-M_{2}\left(d_{22}^{m}\right)^{2} \tau_{2}\bigg]\left|x_{2}(t)-y_{2}(t)\right|+\left[\frac{d_{21}^{m} b_{12}^{m}}{\left(b_{12}+m_{1}\right)^{2}}+\frac{M_{2} d_{21}^{m} d_{22}^{m} \tau_{2}}{b_{12}+m_{1}}\right]\left|x_{1}(t)-y_{1}(t)\right|{}\\ &&+\left[\frac{a_{23}^{m} b_{23}^{m}+a_{23}^{m} M_{2}}{\left(b_{23}^{l}+m_{2}\right)^{2}}+\frac{M_{2} a_{23}^{m} d_{22}^{m} \tau_{2}}{b_{23}^{l}+m_{2}}\right]\left|x_{3}(t)-y_{3}(t)\right|{}\\ &&+\left(d_{2}^{n}+M_{2} d_{2}^{m} d_{22}^{m} \tau_{2}\right)\left|u_{2}(t)-v_{2}(t)\right| . \end{eqnarray} $

类似地, 我们定义

$ V_{31}(t)=\left|\ln x_{3}(t)-\ln y_{3}(t)\right|. $

于是, 我们可得

(3.13) $ \begin{eqnarray} D^{+} V_{31}(t)&=&{\rm sgn}\left\{x_{3}(t)-y_{3}(t)\right\} \bigg[-a_{33}(t)\left(x_{3}\left(t-\tau_{4}\right)-y_{3}\left(t-\tau_{4}\right)\right) {}\\ && +\frac{a_{32}(t) b_{23}(t)\left(x_{2}\left(t-\tau_{5}\right)-y_{2}\left(t-\tau_{5}\right)\right)}{\left(b_{23}(t)+x_{2}\left(t-\tau_{5}\right)\right)\left(b_{23}(t) +y_{2}\left(t-\tau_{5}\right)\right)}+d_{3}(t)\left(u_{3}(t)-v_{3}(t)\right)\bigg]{}\\ &=&{\rm sgn}\left\{x_{3}(t)-y_{3}(t)\right\} \bigg[-a_{33}(t)\left(x_{3}(t)-y_{3}(t)\right)+d_{3}(t)\left(u_{3}(t)-v_{3}(t)\right){}\\ &&+\frac{a_{32}(t) b_{23}(t)\left(x_{2}\left(t-\tau_{5}\right)-y_{2}\left(t-\tau_{5}\right)\right)} {\left(b_{23}(t)+x_{2}\left(t-\tau_{5}\right)\right)\left(b_{23}(t)+y_{2}\left(t-\tau_{5}\right)\right)}+a_{33}(t) \int_{t-\tau_{4}}^{t}\left(\dot{x}_{3}(\theta)-\dot{y}_{3}(\theta)\right) {\rm d}\theta\bigg]{}\\ &=&{\rm sgn}\left\{x_{3}(t)-y_{3}(t)\right\} \bigg[-a_{33}(t)\left(x_{3}(t)-y_{3}(t)\right)+d_{3}(t)\left(u_{3}(t)-v_{3}(t)\right){}\\ &&+\frac{a_{32}(t) b_{23}(t)\left(x_{2}\left(t-\tau_{5}\right)-y_{2}\left(t-\tau_{5}\right)\right)} {\left(b_{23}(t)+x_{2}\left(t-\tau_{5}\right)\right)\left(b_{23}(t)+y_{2}\left(t-\tau_{5}\right)\right)}+a_{33}(t) \int_{t-\tau_{4}}^{t}\left(x_{3}(\theta)\right.{}\\ &&\times\left[-r_{3}(\theta)-a_{33}(\theta) x_{3}\left(\theta-\tau_{4}\right)+\frac{a_{32}(\theta) x_{2}\left(\theta-\tau_{5}\right)}{b_{23}(\theta)+x_{2}\left(\theta-\tau_{5}\right)}+d_{3}(\theta) u_{3}(\theta)\right]{}\\ &&\left.\left.-y_{3}(\theta)\left[-r_{3}(\theta)-a_{33}(\theta) y_{3}\left(\theta-\tau_{4}\right)+\frac{a_{32}(\theta) y_{2}\left(\theta-\tau_{5}\right)}{b_{23}(\theta)+y_{2}\left(\theta-\tau_{5}\right)}+d_{3}(\theta) v_{3}(\theta)\right]\right) {\rm d}\theta\right]{}\\ &=&{\rm sgn}\left\{x_{3}(t)-y_{3}(t)\right\} \bigg[-a_{33}(t)\left(x_{3}(t)-y_{3}(t)\right)+d_{3}(t)\left(u_{3}(t)-v_{3}(t)\right) {}\\ &&+\frac{a_{32}(t) b_{23}(t)\left(x_{2}\left(t-\tau_{5}\right)-y_{2}\left(t-\tau_{5}\right)\right)}{\left(b_{23}(t)+x_{2}\left(t-\tau_{5}\right)\right)\left(b_{23}(t)+y_{2}\left(t-\tau_{5}\right)\right)}+a_{33}(t) \int_{t-\tau_{4}}^{t}\bigg(\left(x_{3}(\theta)-y_{3}(\theta)\right){}\\ &&\times\left[-r_{3}(\theta)-a_{33}(\theta) y_{3}\left(\theta-\tau_{4}\right)+\frac{a_{32}(\theta) y_{2}\left(\theta-\tau_{5}\right)}{b_{23}(\theta)+y_{2}\left(\theta-\tau_{5}\right)}+d_{3}(\theta) v_{3}(\theta)\right]{}\\ &&+x_{3}(\theta) \bigg[-a_{33}(\theta)\left(x_{3}\left(\theta-\tau_{4}\right)-y_{3} \left(\theta-\tau_{4}\right)\right)+\frac{a_{32}(\theta)}{b_{23}(\theta)+x_{2}\left(\theta-\tau_{5}\right)} {}\\ &&\times\left(x_{2}\left(\theta-\tau_{5}\right)-y_{2}\left(\theta-\tau_{5}\right)\right) +d_{3}(\theta)\left(u_{3}(\theta)-v_{3}(\theta)\right)\bigg]\bigg) {\rm d}\theta\bigg]{}\\ &\leq &-a_{33}(t)\left|x_{3}(t)-y_{3}(t)\right|+d_{3}(t)\left|u_{3}(t)-v_{3}(t)\right|{}\\ &&+\frac{a_{32}(t) b_{23}(t)}{\left(b_{23}(t)+x_{2}\left(t-\tau_{5}\right)\right)\left(b_{23}(t)+y_{2}\left(t-\tau_{s}\right)\right)} \left|x_{2}\left(t-\tau_{3}\right)-y_{2}\left(t-\tau_{3}\right)\right|{}\\ &&+a_{33}(t) \int_{t-\tau_{4}}^{t}\left(\left[r_{3}(\theta)+a_{33}(\theta) y_{3}\left(\theta-\tau_{4}\right)+\frac{a_{32}(\theta) y_{2}\left(\theta-\tau_{3}\right)}{b_{23}(\theta)+y_{2}\left(\theta-\tau_{s}\right)}\right.\right.{}\\ &&+d_{3}(\theta) v_{3}(\theta)\bigg] \left|x_{3}(\theta)-y_{3}(\theta)\right|+x_{3}(\theta)\bigg[a_{33}(\theta)\left|x_{3} \left(\theta-\tau_{4}\right)-y_{3}\left(\theta-\tau_{4}\right)\right| {}\\ &&+\frac{a_{32}(\theta)}{b_{23}(\theta)+x_{2}\left(\theta-\tau_{5}\right)}\left|x_{2} \left(\theta-\tau_{5}\right)-y_{2}\left(\theta-\tau_{s}\right)\right|+d_{3}(\theta)\left|u_{3}(\theta) -v_{3}(\theta)\right|\bigg]\bigg) {\rm d}\theta . {}\\ \end{eqnarray} $

接下来, 定义

(3.14) $ \begin{eqnarray} V_{32}(t)&=&\int_{t-\tau_{4}}^{t} \int_{s}^{t} a_{33}\left(s+\tau_{4}\right)\left(\bigg[r_{3}(\theta)+a_{33}(\theta) y_{3}\left(\theta-\tau_{4}\right)+\frac{a_{32}(\theta) y_{2}\left(\theta-\tau_{5}\right)}{b_{23}(\theta)+y_{2}\left(\theta-\tau_{5}\right)}\right.{}\\ &&+d_{3}(\theta) v_{3}(\theta)\bigg]\left|x_{3}(\theta)-y_{3}(\theta)\right|+x_{3}(\theta) \bigg[a_{33}(\theta)\left|x_{3}\left(\theta-\tau_{4}\right)-y_{3}\left(\theta-\tau_{4}\right)\right|{}\\ && +\frac{a_{32}(\theta)}{b_{23}(\theta)+x_{2}\left(\theta-\tau_{5}\right)}\left|x_{2}\left(\theta-\tau_{5}\right) -y_{2}\left(\theta-\tau_{5}\right)\right|+d_{3}(\theta)\left|u_{3} (\theta)-v_{3}(\theta)\right|\bigg]\bigg) {\rm d}\theta {\rm d}s . {}\\ \end{eqnarray} $

于是, 我们有

(3.15) $ \begin{eqnarray} \sum\limits_{i=1}^{2} D^{+} V_{3 i}(t)&\leq &-a_{33}(t)\left|x_{3}(t)-y_{3}(t)\right|+d_{3}(t)\left|u_{3}(t)-v_{3}(t)\right|{}\\ &&+\frac{a_{32}(t) b_{23}(t)}{\left(b_{23}(t)+x_{2}\left(t-\tau_{5}\right)\right)\left(b_{23}(t)+y_{2}\left(t-\tau_{5}\right)\right)}\left|x_{2}\left(t-\tau_{5}\right)-y_{2}\left(t-\tau_{5}\right)\right| {}\\ &&+\int_{t-\tau_{4}}^{t} a_{33}\left(s+\tau_{4}\right) {\rm d}s \times\bigg[r_{3}(t)+a_{33}(t) y_{3}\left(t-\tau_{4}\right)+\frac{a_{32}(t) y_{2}\left(t-\tau_{5}\right)}{b_{23}(t)+y_{2}\left(t-\tau_{5}\right)}{}\\ &&+d_{3}(t) v_{3}(t)\bigg]\left|x_{3}(t)-y_{3}(t)\right|{}\\ &&+M_{3} \int_{t-\tau_{4}}^{t} a_{33}\left(s+\tau_{4}\right) {\rm d}s \times a_{33}(t)\left|x_{3}\left(t-\tau_{4}\right)-y_{3}\left(t-\tau_{4}\right)\right|{}\\ &&+M_{3} \int_{t-\tau_{4}}^{t} a_{33}\left(s+\tau_{4}\right) {\rm d}s \times \frac{a_{32}(t)}{b_{23}(t)+x_{2}\left(t-\tau_{5}\right)}\left|x_{2}\left(t-\tau_{5}\right)-y_{2}\left(t-\tau_{5}\right)\right|{}\\ &&+M_{3} \int_{t-\tau_{4}}^{t} a_{33}\left(s+\tau_{4}\right) {\rm d}s \times d_{3}(t)\left|u_{3}(t)-v_{3}(t)\right| . \end{eqnarray} $

定义

(3.16) $ \begin{eqnarray} V_{33}(t)&=&\int_{t-\tau_{5}}^{t} \frac{a_{32}\left(w+\tau_{5}\right) b_{23}\left(w+\tau_{5}\right)}{\left(b_{23}\left(w+\tau_{5}\right)+x_{2}(w)\right)\left(b_{23}\left(w+\tau_{5}\right)+y_{2}(w)\right)}\left|x_{2}(w)-y_{2}(w)\right| {\rm d}w{}\\ & &+M_{3} \int_{t-\tau_{4}}^{t} \int_{w}^{w+\tau_{4}} a_{33}\left(s+\tau_{4}\right) a_{33}\left(w+\tau_{4}\right)\left|x_{3}(w)-y_{3}(w)\right| {\rm d}s {\rm d}w{}\\ &&+M_{3} \int_{t-\tau_{5}}^{t} \int_{w+\tau_{5}-\tau_{4}}^{w+\tau_{5}} a_{33}\left(s+\tau_{4}\right) \frac{a_{32}\left(w+\tau_{5}\right)}{b_{23}\left(w+\tau_{5}\right)+x_{2}(w)}\left|x_{2}(w)-y_{2}(w)\right| {\rm d}s {\rm d}w . \end{eqnarray} $

设

(3.17) $ \begin{equation} V_{3}(t)=V_{31}(t)+V_{32}(t)+V_{33}(t) . \end{equation} $

于是, 我们有

(3.18) $ \begin{eqnarray} D^{+} V_{3}(t)&\leq &-\left[a_{33}^{l}-\left(r_{3}^{m}+a_{33}^{m} M_{3}+\frac{a_{32}^{m} M_{2}}{b_{23}^{l}+M_{2}}+d_{3}^{m} N_{3}\right) \int_{t-\tau_{4}}^{t} a_{33}^{m}\left(s+\tau_{4}\right) {\rm d}s\right.{}\\ & &\left.-M_{3} a_{33}^{m} \int_{t}^{t+\tau_{4}} a_{33}^{m}\left(s+\tau_{4}\right) {\rm d}s\right]\left|x_{3}(t)-y_{3}(t)\right|{}\\ &&+\left[\frac{a_{32}^{m} b_{23}^{m}}{\left(b_{23}^{l}+m_{2}\right)^{2}}+M_{3} \frac{a_{32}^{m}}{b_{23}^{l}+m_{2}} \int_{t+\tau_{5}-\tau_{4}}^{t+\tau_{5}} a_{33}^{m}\left(s+\tau_{4}\right) {\rm d}s\right]\left|x_{2}(t)-y_{2}(t)\right|{}\\ &&+\left(d_{3}^{m}+M_{3} d_{3}^{m} \int_{t-\tau_{4}}^{t} a_{33}^{m}\left(s+\tau_{4}\right) {\rm d}s\right)\left|u_{3}(t)-v_{3}(t)\right|{}\\ & \leq &-\left[a_{33}^{l}-\left(r_{3}^{m}+a_{33}^{m} M_{3}+\frac{a_{32}^{m} M_{2}}{b_{23}^{l}+M_{2}}+d_{3}^{m} N_{3}\right) a_{33}^{m} \tau_{4}-M_{3}\left(d_{33}^{m}\right)^{2} \tau_{4}\right]{}\\ &&\times\left|x_{3}(t)-y_{3}(t)\right|+\left(\frac{a_{32}^{m} b_{23}^{m}}{\left(b_{23}^{l}+m_{2}\right)^{2}}+\frac{M_{3} a_{32}^{m} a_{33}^{m} \tau_{4}}{b_{23}^{l}+m_{2}}\right)\left|x_{2}(t)-y_{2}(t)\right|{}\\ &&+\left(d_{3}^{m}+M_{3} d_{3}^{m} a_{33}^{m} \tau_{4}\right)\left|u_{3}(t)-v_{3}(t)\right| . \end{eqnarray} $

定义

$ V_{4}(t)=\left|\ln u_{1}(t)-\ln v_{1}(t)\right|, V_{5}(t)=\left|\ln u_{2}(t)-\ln v_{2}(t)\right|, V_{6}(t)=\left|\ln u_{3}(t)-\ln v_{3}(t)\right|. $

计算$ V_{4}(t), V_{5}(t), V_{6}(t) $的右上导数, 我们有

(3.19) $ \begin{eqnarray} D^{+} V_{4}(t)&\leq & {\rm sgn}\left(u_{1}(t)-v_{1}(t)\right)\left[-f_{1}(t)\left(u_{1}(t)-v_{1}(t)\right)+q_{1}(t)\left(x_{1}(t)-y_{1}(t)\right)\right. {}\\ &\leq&-f_{1}^{l}\left|u_{1}(t)-v_{1}(t)\right|+q_{1}^{m}\left|x_{1}(t)-y_{1}(t)\right|, \end{eqnarray} $

(3.20) $ \begin{eqnarray} D^{+} V_{5}(t)&\leq & {\rm sgn}\left(u_{2}(t)-v_{2}(t)\right)\left[-f_{2}(t)\left(u_{2}(t)-v_{2}(t)\right)-q_{2}(t)\left(x_{2}(t)-y_{2}(t)\right)\right.{}\\ &\leq&-f_{2}^{l}\left|u_{2}(t)-v_{2}(t)\right|+q_{2}^{m}\left|x_{2}(t)-y_{2}(t)\right| \end{eqnarray} $

及

(3.21) $ \begin{eqnarray} D^{+} V_{6}(t)&\leq& {\rm sgn}\left(u_{3}(t)-v_{3}(t)\right)\left[-f_{3}(t)\left(u_{3}(t)-v_{3}(t)\right)-q_{3}(t)\left(x_{3}(t)-y_{3}(t)\right)\right. {}\\ &\leq&-f_{3}^{l}\left|u_{3}(t)-v_{3}(t)\right|+q_{3}^{m}\left|x_{3}(t)-y_{3}(t)\right| . \end{eqnarray} $

进一步, 考虑李雅普诺夫泛函

$ V(t)=\sum\limits_{i=1}^{6} V_{i}(t). $

于是, 根据$ (3.6) $, $ (3.12) $, (3.18)–(3.21)式, 我们可得, 当$ t \geq T^{*}=T+\tau $时, 有

(3.22) $ \begin{equation} D^{+}V(t)\leq-\sum\limits_{i=1}^{3}\left(A_{i}\left|x_{i}(t)-y_{i}(t)\right|+B_{i}\left|u_{i}(t)-v_{i}(t)\right|\right) . \end{equation} $

根据定理$ 3.1 $的条件$ {\rm (H_{8})} $, 我们设$ \alpha=\min\left\{A_{1}, A_{2}, A_{3}, B_{1}, B_{2}, B_{3}\right\} $. 因此, 我们有

(3.23) $ \begin{equation} A_{i} \geq \alpha>0, B_{i} \geq \alpha>0, (i=1, 2, 3). \end{equation} $

对$ (3.22) $式两边从$ T^{\ast} $到$ t $积分, 并利用$ (3.23) $可得

(3.24) $ \begin{equation} V(t)+\alpha\int_{T^{*}}^{t}\left(\sum\limits_{i=1}^{3}\left[\left|x_{i}(s)-y_{i}(s)\right|+\left|u_{i}(s)-v_{i}(s)\right|\right]\right) {\rm d}s \leq V\left(T^{*}\right)<+\infty, \end{equation} $

所以$ V_(t) $在区间$ \left[T^{*}, +\infty\right) $上是有界的, 且有

(3.25) $ \begin{equation} \int_{T^{*}}^{\infty}\left(\sum\limits_{i=1}^{3}\left[\left|x_{i}(t)-y_{i}(t)\right|+\left|u_{i}(t)-v_{i}(t)\right|\right]\right) {\rm d}s \leq \frac{V(T^{*})}{\alpha}<+\infty, \end{equation} $

根据$ (3.25) $式, 可得

(3.26) $ \begin{equation} \sum\limits_{i=1}^{3}\left(\left|x_{i}(t)-y_{i}(t)\right|+\left|u_{i}(t)-v_{i}(t)\right|\right) \in L^{1}(T^{*}, +\infty), \end{equation} $

利用定理$ 2.1 $, 我们可得$ \left|x_{i}(t)-y_{i}(t)\right|, \left|u_{i}(t)-v_{i}(t)\right|, (i=1, 2, 3) $及它们的导数在区间$ \left[T^{*}, \right. $$ \left.+\infty\right) $上是有界的, 并且$ \left|x_{i}(t)-y_{i}(t)\right|, \left|u_{i}(t)-v_{i}(t)\right| $在区间$ \left[T^{*}, +\infty\right) $上是一致连续的. 根据引理$ 3.1 $, 我们可得

$ \lim\limits_{t \rightarrow +\infty}\left|x_{i}(t)-y_{i}(t)\right|=0, \lim\limits_{t \rightarrow +\infty}\left|u_{i}(t)-v_{i}(t)\right|=0, i=1, 2, 3 . $

因此, 系统$ (1.1) $是全局吸引的.

在本节中, 我们将利用上述结果和不动点理论, 探讨参数为连续正$ \omega $ -周期函数的周期系统$ (1.1) $的正周期解的存在性、唯一性和稳定性. 首先, 我们给出以下基本引理.

引理4.1^[2]  设$ S\subset R_{n} $是紧凸集. 如果映射$ T:S\rightarrow S $是连续的, 则存在一个不动点$ x^{*} \in S $使得$ T\left(x^{*}\right)=x^{*} $.

定理4.1  设条件$ {\rm (H_{1})} $–$ {\rm (H_{8})} $成立, 则周期系统$ (1.1) $存在唯一的全局渐近稳定的正$ \omega $ -周期解.

证  根据泛函微分方程解的存在唯一性定理, 我们可以定义一个Poincaré映射$ T: {{\mathbb{R}}} _{+}^{6} \rightarrow {{\mathbb{R}}} _{+}^{6} $, 其表达式为: $ T\left(X_{0}\right)=X\left(t, \omega, X_{0}\right) $, 其中$ X\left(t, \omega, X_{0}\right)=\left(x_{1}(t), x_{2}(t), x_{3}(t), \right. $$ \left. u_{1}(t), u_{2}(t), u_{3}(t)\right) $是初始条件为$ (1.2) $的$ \omega $ -周期系统$ (1.1) $的一个正解. 设

$ S=\{(x_{1}, x_{2}, x_{3}, u_{1}, u_{2}, u_{3})\in {{\mathbb{R}}} _{+}^{6}| m_{i}\leq x_{i}\leq M_{i}, n_{i}\leq u_{i}\leq N_{i}, i=1, 2, 3\}, $

那么$ S\subset {{\mathbb{R}}} _{6}^{+} $是一个紧凸集. 根据带有初值$ (1.2) $的$ \omega $ -周期系统$ (1.1) $的解的连续性, 我们可知算子$ T $是连续的. 进一步, 根据定理$ 2.1 $知算子$ T $把$ S $映射到$ S $. 因此, 根据引理$ 4.1 $和定理$ 3.1 $, 知$ \omega $ -周期系统$ (1.1) $存在唯一的正的全局渐近稳定的$ \omega $ -周期解.

基于种群的周期性增长和生长环境的周期性, 我们选择了合适的周期系数来验证上述理论结果. 为此, 我们考虑如下带有多时滞和反馈控制的三种群Lotka-Volterra比率依赖食物链模型:

(5.1) $ \begin{equation} \left\{ \begin{array}{ll} \dot{x}_{1}(t)=& x_{1}(t)\left[(5.5+0.5 \cos \pi t)-(4.5+0.5 \sin \pi t) x_{1}\left(t-\tau_{1}\right)\right.\\ &{ } \left.-\frac{(0.5+0.1 \sin \pi t) x_{2}(t)}{(1.7+0.1 \sin \pi t)+x_{1}(t)}-(0.1+0.05 \sin \pi t) u_{1}(t)\right], \\ \dot{x}_{2}(t)=& x_{2}(t)\left[-(0.00015+0.00005 \cos \pi t)-(10+0.5 \sin \pi t) x_{2}\left(t-\tau_{2}\right)\right.\\ &{ } \left.+\frac{(9.5+0.5 \sin \pi t) x_{1}\left(t-\tau_{3}\right)}{(1.7+0.1 \sin \pi t)+x_{1}\left(t-\tau_{3}\right)}-\frac{(1.5+0.5 \sin \pi t) x_{3}(t)}{(1.6+0.1 \sin \pi t)+x_{2}(t)}\right.\\ &{ } \left.+(0.7+0.1 \sin \pi t) u_{2}(t)\right], \\ \dot{x}_{3}(t)=& x_{3}(t)\left[-(0.00005+0.00001 \cos \pi t)-(11.25+1.25 \sin \pi t) x_{3}\left(t-\tau_{4}\right)\right.\\ &{ } \left.+\frac{(13.5+0.5 \sin \pi t) x_{2}\left(t-\tau_{5}\right)}{(1.6+0.1 \sin \pi t)+x_{2}\left(t-\tau_{5}\right)}+(0.3+0.1 \sin \pi t) u_{3}(t)\right], \\ \dot{u}_{1}(t)=&(1.5+0.2 \cos \pi t)-(0.7+0.2 \sin \pi t) u_{1}(t)+(0.15+0.05 \sin \pi t) x_{1}(t), \\ \dot{u}_{2}(t)=&(4.5+0.1 \cos \pi t)-(4.05+0.05 \sin \pi t) u_{2}(t)-(1.5+0.5 \sin \pi t) x_{2}(t), \\ \dot{u}_{3}(t)=&(3+0.5 \cos \pi t)-(5.45+0.05 \sin \pi t) u_{3}(t)-(1.5+0.5 \sin \pi t) x_{3}(t) . \end{array}\right. \end{equation} $

通过简单的计算, 容易检验周期系统$ (5.1) $的系数满足定理$ 2.1 $、定理$ 3.1 $和定理$ 4.1 $中的所有假设条件. 因此, 周期系统$ (5.1) $是持久的；此外, 系统$ (5.1) $具有唯一正的全局渐近稳定的2 -周期解. 利用MATLAB $ 7.1 $软件包, 我们可以得到周期系统$ (5.1) $的数值解, 如图 1、图 2和图 3所示. 从图 1可以看出, 具有时滞$ \tau_{1}=0.01, \tau_{2}=0.02, \tau_{3}=0.03, \tau_{4}=0.02, \tau_{5}=0.01 $以及初始条件$ (5.2) $的周期系统$ (5.1) $的持久性.

(5.2) $ \begin{equation} \left\{ \begin{array}{l} x_{1}(t)=x_{2}(t)=x_{3}(t)=\sin t+0.5, \quad t \in[-0.03, 0], \\ u_{1}(0)=0.5, u_{2}(0)=0.5, u_{3}(0)=0.5 . \end{array} \right. \end{equation} $

根据图 2, 很容易发现系统$ (5.1) $具有全局吸引性. 系统$ (5.1) $的解的动态特性如图 3所示.

为了观察噪声对三种群食物链Lotka-Volterra系统$ (1.1) $的动力学行为的影响, 我们在系统$ (1.1) $的前三个方程中的每一个上添加了一个乘法噪声源, 并获得了如下具有乘法噪声源及时滞的随机三种群食物链模型

(5.3) $ \begin{equation} \left\{\begin{array}{ll} \dot{x}_{1}(t)=& x_{1}(t)\left[(5.5+0.5 \cos \pi t)-(4.5+0.5 \sin \pi t) x_{1}\left(t-\tau_{1}\right)\right.\\ &{ } \left.-\frac{(0.5+0.1 \sin \pi t) x_{2}(t)}{(1.7+0.1 \sin \pi t)+x_{1}(t)}+D_{1} \xi_{1}(t)\right], \\ \dot{x}_{2}(t)=& x_{2}(t)\left[-(0.00015+0.00005 \cos \pi t)-(10+0.5 \sin \pi t) x_{2}\left(t-\tau_{2}\right)\right.\\ &{ } \left.+\frac{(9.5+0.5 \sin \pi t) x_{1}\left(t-\tau_{3}\right)}{(1.7+0.1 \sin \pi t)+x_{1}\left(t-\tau_{3}\right)}-\frac{(1.5+0.5 \sin \pi t) x_{3}(t)}{(1.6+0.1 \sin \pi t)+x_{2}(t)}+D_{2} \xi_{2}(t)\right], \\ \dot{x}_{3}(t)=& x_{3}(t)\left[-(0.00005+0.00001 \cos \pi t)-(11.25+1.25 \sin \pi t) x_{3}\left(t-\tau_{4}\right)\right.\\ &{ } \left.+\frac{(13.5+0.5 \sin \pi t) x_{2}\left(t-\tau_{5}\right)}{(1.6+0.1 \sin \pi t)+x_{2}\left(t-\tau_{5}\right)}+D_{3} \xi_{3}(t)\right], \end{array}\right. \end{equation} $

其中$ \zeta _{i}(t) $是具有零均值的$ \delta $ -型高斯白噪声源, 即: $ \langle \delta_{i}(t)\rangle=0 $, 与$ \langle\delta _{i}(t)\delta _{i}({t}')\rangle= D_{i}\delta \left ( t-{t}' \right )\delta _{ij} $, 其中$ D_{i} $表示噪声强度、时滞$ \tau_{1}= 0.01, \tau_{2}= 0.02, \tau_{3}= 0.02, \tau_{4}= 0.02, \tau_{5}= 0.01 $, 初始条件如下:

(5.4) $ \begin{equation} x_{1}(t)=x_{2}(t)=x_{3}(t)=\sin t+0.5, \quad t\in[-0.03, 0], \end{equation} $

利用MATLAB $ 7.1 $软件, 我们可以获得不同噪声强度下模型(5.3)–(5.4)的一些数值解, 如图 4至图 10所示.

从图 4中, 我们可以发现, 当噪声强度非常低时, 新的随机模型(5.3)–(5.4)具有几乎确定的周期解. 此外, 图 5显示, 当噪声强度$ D_{1} $增加到$ 0.04 $且噪声强度$ D_{2} $和$ D_{3} $非常低时, 捕食种群$ x_{1}(t) $的周期性被破坏. 图 6显示, 当噪声强度$ D_{2} $增加到$ 0.085 $且噪声强度$ D_{1} $和$ D_{3} $非常低时, 捕食者种群$ x_{2}(t) $的周期性被破坏, 图 7显示, 当噪声强度$ D_{3} $增加到$ 0.05 $且噪声强度$ D_{1} $和$ D_{2} $非常低时, 捕食种群$ x_{3}(t) $的周期性被破坏. 最后, 图 8显示, 当噪声强度$ D_{1} $增加到$ 3.2 $且噪声强度$ D_{2} $和$ D_{3} $非常低时, 捕食者种群和食饵种群$ x_{i}(t)(i=1, 2, 3) $都会灭绝. 图 9显示, 当噪声强度$ D_{2} $增加到$ 2.502 $且噪声强度$ D_{1} $和$ D_{3} $非常低时, 捕食者种群$ x_{i}(t)(i=2, 3) $都会灭绝, 图 10显示, 当噪声强度$ D_{3} $增加到$ 2.216 $, 且噪声强度$ D_{1} $和$ D_{2} $非常低时, 捕食者种群$ x_{3}(t) $趋于灭绝. 通过将图 4至图 10所示结果与图 1至图 3所示结果进行比较, 可以明显看出, 随机环境噪声会影响三种群食物链系统的持久性、周期性和稳定性.

在本文中, 我们提出并研究了一类具有时滞和反馈控制的非线性非自治三种群比率依赖型食物链模型. 通过改进一些现有的方法, 尤其是针对具有反馈控制项的时滞微分方程构造Lyapunov函数的技巧, 研究了系统$ (1.1) $的持久性、全局吸引性以及相应周期系统$ (5.1) $的唯一正周期解的全局渐近稳定性等问题. 首先, 利用一些新的分析方法和比较定理, 构造了一个合适的Lyapunov函数, 得到了关于上述问题的一些有趣的结果, 这些结果可以看作是文献[32, 35]的主要结果的推广. 所得结果对相关种群动力学模型的研究及应用具有重要的理论及实际意义. 此外, 所研究的系统的数值模拟表明, 本文所得判据是新的、通用的、易于验证的. 最后, 我们还求得了相应的带有乘法噪声源的时滞随机三种群比率依赖食物链模型的数值解, 得到了该系统一些新的有趣的动力学行为. 通过比较模型(5.1)–(5.2)和模型(5.3)–(5.4)的解的变化过程, 容易发现随机环境噪声会影响具有时滞的周期三种群比率依赖食物链系统的持久性、周期性和稳定性.