自Lotka和Volterra分别对捕食-食饵数学模型做了开创性的研究工作以来, 捕食-食饵数学模型被众多学者广泛研究, 参见文献[1-17].捕食者与被捕食者之间的动力学关系, 基于它们的普遍存在性以及重要性, 已经持续了很长一段时间并将继续被深入研究, 参见文献[11].上个世纪, 许多学者建立了大量的生态系统, 用来描述不同物种之间以及物种与外界环境之间的相互作用.其中, 最经典并且被广泛研究的是Lotka-Volterra竞争系统, 参见文献[12-16].根据实验数据, Holling^[17]提出了三种单调型的功能函数

$ f(x) = c(t)x, \ \mbox{ 或} \ f(x) = \frac{{c(t)x}}{{m +x}}, \ \mbox{ 或 }\ f(x) = \frac{{c(t){x^2}}}{{m + {x^2}}}, $

这些功能函数已经被广泛用在许多模型里加以研究.然而, 有实验显示, 在微生物的层面非单调反应很可能会发生:当营养浓度达到很高的一个水平的时候, 对于特定的增长速度很可能会发生抑制反应.当微生物被用在废物分解或者污水净化方面, 这种抑制反应的现象很常见, 因此, 提出了Holling Ⅳ型功能反应函数

$ f(x) = {\frac{{c(t)x}}{{\alpha + \beta x + {x^2}}}}, $

并用来模拟在高浓度下的抑制反应^[18].与前三种功能反应函数相比, Holling Ⅳ型功能反应函数更合理.基于这个原因, Chen^[19]研究了如下带有Holling Ⅳ型功能反应函数的捕食-食饵模型正周期解的存在性

$ \left\{ {\begin{array}{*{20}{l}} {\mathop {{N_1}}\limits^. (t) = {N_1}(t)\left[ {{b_1}(t) - {a_1}(t){N_1}(t - {\tau _1}(t))- {\frac{{c(t){N_2}(t - \sigma (t))}}{{(N_{\rm{1}}^{\rm{2}}(t)/i) + {N_1}(t) + a}}}} \right], }\\ {\mathop {{N_2}}\limits^. (t) = {N_2}(t)\left[ { - {b_2}(t) + {\frac{{{a_2}(t){N_1}(t - {\tau _2}(t))}}{{(N_{\rm{1}}^{\rm{2}}(t - {\tau _2}(t))/i) + {N_1}(t - {\tau _2}(t)) + a}}}} \right], } \end{array}} \right. $

其中$ {N_1}(t) $和$ {N_2}(t) $分别代表食饵与捕食者在$ t $时刻的种群密度; $ {a_l}(t), {b_l}(t) $和$ {\tau _l}(t)\ (l = 1, 2) $都是以$ \omega>0 $为周期的连续函数, 并且$ a, i $都是正常数.利用Mawhin迭合度理论, 得到了该系统正周期解存在性的充分条件.

众所周知, 在种群动态中, 许多进化过程在经历了相当长一段时间的连续变化之后, 很可能经历一个短时间的突变.例如, 由于食物的供给, 季节性的变化, 狩猎以及一年一度的收获等等, 综合这些因素, 得到了脉冲微分方程.关于脉冲微分方程的理论, 参见文献[20].考虑到脉冲, Du和Feng^[21]研究了带有脉冲时滞的具有Beddington-DeAngelis型功能反应函数的中立型捕食竞争模型正周期解的存在性, 利用Mawhin迭合度理论, 给出了正周期解存在的充分条件.

根据上述文献, 该文研究如下带有脉冲时滞和更一般的Holling Ⅳ型功能反应函数的中立型捕食-食饵模型

(1.1) $ \begin{equation} \left\{ {\begin{array}{*{20}{l}} {\left. {\begin{array}{*{20}{l}} {\mathop {{N_1}}\limits^.(t) = {N_1}(t)\Bigg[ {{b_1}(t) - {a_1}(t){N_1}(t - {\tau _1}(t)) - \rho (t)\mathop {{N_1}}\limits^. (t - {\tau _3}(t))} }\\ { {\ \ \ \ \ \ \ \ \ \ { \ - \frac{{c(t){N_2}(t - \sigma (t))}}{{N_1^2(t) + \alpha {N_1}(t) + \beta }}}} \Bigg], }\\ {\mathop {{N_2}}\limits^. (t) = {N_2}(t)\left[ { - {b_2}(t) + {\frac{{{a_2}(t){N_1}(t - {\tau _2}(t))}}{{N_1^2(t - {\tau _2}(t)) + \alpha {N_1}(t - {\tau _2}(t)) + \beta }}}} \right], } \end{array}} \right\}}\\ {\left. {\begin{array}{*{20}{l}} {\Delta {N_1}({t_k}) = {N_1}(t_k^ + ) - {N_1}({t_k}) = {c_{1k}}{N_1}({t_k}), }\\ {\Delta {N_2}({t_k}) = {N_2}(t_k^ + ) - {N_2}({t_k}) = {c_{2k}}{N_2}({t_k}), } \end{array}} \right\}t = {t_k}, k = 1, 2, \cdots, } \end{array}} \right.t \ne {t_k}, k = 1, 2, \cdots, \end{equation} $

满足初始条件

(1.2) $ {N_1}(t) = \varphi (t), \mathop {{N_1}}\limits^. (t) = \mathop \varphi \limits^. (t), \varphi \in {C^1}([ - \tau, 0], [0, + \infty )), \varphi (0) > 0, \\ {N_2}(t) = \psi (t), \mathop {{N_2}}\limits^. (t) = \mathop \psi \limits^. (t), \psi \in {C^1}([ - \tau, 0], [0, + \infty )), \psi (0) > 0, $

其中, $ N_1 $和$ N_2 $分别代表食饵与捕食者在$ t $时刻的种群密度; $ a_1(t) $$ \in C({{\Bbb R}}, (0, + \infty )) $, $ a_2(t), $$ c(t) \in C({{\Bbb R}}, [0, + \infty )) $, $ \sigma (t), {\tau _1}(t), {\tau _2}(t) \in {C^1}({{\Bbb R}}, [0, + \infty )) $, $ \rho (t)\in {C^1}({{\Bbb R}}, {{\Bbb R}} ) $, $ {\tau _3}(t) \in {C^2}({{\Bbb R}}, $$ [0, + \infty )) $; $ {c_{ik}} > - 1, i = 1, 2, k \in {N^ + } $; $ {b_1}(t), {b_2}(t) \in C({{\Bbb R}}, {{\Bbb R}} ) $, $ \tau = \mathop {\max }\limits_{t \in [0, \omega ]} \{ {\tau _1}(t) $, $ {\tau _2}(t) $, $ {\tau _3}(t)\} $, 其中, $ c(t) $, $ \sigma(t) $, $ \rho(t) $, $ {a_i(t)} $, $ {b_i(t)} $和$ \tau_\ell(t) $$ ( i=1, 2 ; \ell=1, 2, 3) $都是以$ \omega $为正周期的连续函数, $ \alpha $和$ \beta $都是正的常数.并且满足$ \int_0^\omega {c(t){\rm d}t > 0} $和$ \int_0^\omega {{b_i}(t){\rm d}t > 0} $的条件.此外, 由于环境是随机变动的, 所以, 增长函数$ {b_1}(t) $和$ {b_2}(t) $不一定总是正的.本文利用Mawhin迭合度定理和一些分析技巧, 给出了系统$ (1.1) $–$ (1.2) $存在正周期解的充分条件.这些条件是时滞依赖的, 从而本文的研究和文献[18-19, 22-23]不同, 因为这些文献研究的结论都与时滞无关.

为了得到系统$ (1.1) $–$ (1.2) $正周期解的存在性, 首先介绍一些相关的引理和Mawhin迭合度定理.

定义2.1^[21]  假设$ {({N_1}(t), {N_2}(t))^{\rm T}} \in ([ - \tau, \infty ), (0, +\infty ), (0, +\infty )) $, 满足下述条件

(ⅰ) $ {N_1}(t), {N_2}(t) $在每一个区间$ (0, {t_1}] $和$ ({t_k}, {t_{k + 1}}], k \in {N^ + } $上, 都是绝对连续的;

(ⅱ) 对于$ {\forall} $$ {t_k} $, $ k \in {N^ + } $, $ {({N_1}(t_k^ + ), {N_2}(t_k^ + ))^{\rm T}} $和$ {({N_1}(t_k^ - ), {N_2}(t_k^ - ))^{\rm T}} $存在, 且

$ {({N_1}(t_k^ - ), {N_2}(t_k^ - ))^{\rm T}} = {({N_1}({t_k}), {N_2}({t_k}))^{\rm T}}; $

(ⅲ) $ {({N_1}(t), {N_2}(t))^{\rm T}} $在$ [0, \infty )\backslash \{ {t_k}\} $上几乎处处满足系统$ (1.1) $–$ (1.2) $, 并且当$ t = {t_k} $, $ k\in {N^+} $时, 有$ {N_1}(t_k^ + ) - {N_1}({t_k}) = {c_{1k}}{N_1}({t_k}) $和$ {N_2}(t_k^ + ) - {N_2}({t_k}) = {c_{2k}}{N_2}({t_k}) $成立.

则称$ {({N_1}(t), {N_2}(t))^{\rm T}} $是初值问题$ (1.1) $–$ (1.2) $在$ [ - \tau, \infty ) $上的解.

为了叙述方便, 作如下假设.

(H$ _1) $  $ {\{ {t_k}\} _{k \in N}} $是一个严格单调递增序列, 且$ {t_1} > 0 $, $ \mathop {\lim }\limits_{k \to \infty } {t_k} = +\infty $;

(H$ _2) $  $ \{ {c_{ik}}\} $是实序列且$ {c_{ik}} > - 1 $, $ \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{ik}}), i = 1, 2 $是$ \omega $ -周期函数.

下面简要介绍迭合度相关知识.

设$ X $和$ Y $是两个实Banach空间, $ L:DomL \subset X \to Y $是一个线性算子, $ N:X \to Y $是一个连续映射, 若$ \dim KerL = co\dim {\mathop{\rm Im}\nolimits} L < + \infty $, 且$ {\mathop{\rm Im}\nolimits} L $是$ Y $的闭子空间, 则称$ L $是指标为零的Fredholm算子^[25].如果$ L $是指标为零的Fredholm算子, 存在连续映射$ P:X \to X $和$ Q:Y \to Y $, 满足$ {\mathop{\rm Im}\nolimits} P = KerL $, $ {\mathop{\rm Im}\nolimits} L = KerQ = {\mathop{\rm Im}\nolimits} (I - Q) $, 则$ L\left| {_{DomL \cap KerP}:(I - P)X \to {\mathop{\rm Im}\nolimits} L} \right. $是可逆的, 把它的逆算子记为$ {K_p} $.设$ \Omega \subset X $为$ X $中的有界开子集, 如果算子$ QN(\bar{\Omega}) $是有界的, 并且算子$ {K_p}(I - Q)N(\bar{\Omega}) \in X $是紧的, 则称算子$ N $在$ N \in \bar{\Omega} $上是$ L $ -紧的.

引理2.1^[24-25]  假设$ X $和$ Y $是两个实Banach空间, $ L:DomL\subset X \to Y $是指标为零的Fredholm算子, $ \Omega \subset X $是一个有界开集, $ N:\bar{\Omega}\to Y $在$ \bar{\Omega} $上是$ L $ -紧的, 且假设下列条件成立

$ [C_1] $ 当$ x \in \partial \Omega \cap DomL $, $ \lambda \in(0, 1) $时, 有$ Lx \ne\lambda Nx $;

$ [C_2] $ 当$ x \in \partial \Omega \cap KerL $时, 有$ Nx \notin {\mathop{\rm Im}\nolimits} L $;

$ [C_3] $$ \deg \{ JQN, \Omega \cap KerL, 0\} \ne 0 $, 其中$ J:{\mathop{\rm Im}\nolimits} Q \to KerL $是同构映射.

则方程$ Lx = Nx $在$ \bar{\Omega} \cap DomL $内至少存在一个解.

在假设(H$ _1) $和(H$ _2) $条件下, 首先考虑没有脉冲情况下的带有更一般的Holling Ⅳ型功能反应函数的中立型捕食竞争系统

(2.1) $ \begin{equation} {\left\{ \begin{array}{l} \mathop {{\rm{ }}x}\limits^. (t) = x(t)\left[ {{b_1}(t) - {A_1}(t)x(t - {\tau _1}(t)) - \gamma (t)\mathop {{\rm{ }}x}\limits^. (t - {\tau _3}(t)) - {\frac{{C(t)y(t - \sigma (t))}}{{{p^2}(t){x^2}(t) + \alpha p(t)x(t) + \beta }}} }\right], \\ \mathop {{\rm{ }}y}\limits^. (t) = y(t)\left[ { - {b_2}(t) + {\frac{{{A_2}(t)x(t - {\tau _2}(t))}}{{{q^2}(t){x^2}(t - {\tau _2}(t)) + \alpha q(t)x(t - {\tau _2}(t)) + \beta }}}} \right], \end{array} \right.} \end{equation} $

满足初始条件

(2.2) $ \begin{eqnarray} \begin{array}{l} x(t) = \varphi (t), \mathop x\limits^. (t) = \mathop \varphi \limits^. (t), \varphi \in {C^1}([ - \tau, 0], [0, \infty )), \varphi (0) > 0, \\ y(t) = \psi (t), \mathop y\limits^. (t) = \mathop \psi \limits^. (t), \psi \in {C^1}([ - \tau, 0], [0, \infty )), \psi (0) > 0, \end{array} \end{eqnarray} $

其中

(2.3) ${A_1}(t) = {a_1}(t )\mathop \prod \limits_{0 < {t_k} < t - {\tau _1}(t)} (1 + {c_{1k}}), \qquad \gamma(t) = \rho (t)\mathop \prod \limits_{0 < {t_k} < t - {\tau _3}(t)} (1 + {c_{1k}}), \\ C(t) = c(t)\mathop \prod \limits_{0 < {t_k} < t - \sigma (t)} (1 + {c_{2k}}), \qquad\ \ \ \ p(t) = \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}}), \\ {A_2}(t) = {a_2}(t)\mathop \prod \limits_{0 < {t_k} < t - {\tau _2}(t)} (1 + {c_{1k}}), \qquad\ q(t) = \mathop \prod \limits_{0 < {t_k} < t - {\tau _2}(t)} (1 + {c_{1k}}).$

下面的引理给出了初值问题$ (1.1) $–$ (1.2) $与初值问题$ (2.1) $–$ (2.2) $的关系.

引理2.2  假设(H$ _1) $和(H$ _2) $成立, 则有

(ⅰ) 如果$ {(x(t), y(t))^{\rm T}} $是系统$ (2.1) $–$ (2.2) $的解, 则$ {({N_1}(t), {N_2}(t))^{\rm T}} $是系统$ (1.1) $–$ (1.2) $的解, 其中

$ {N_1}(t) = \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}})x(t), \qquad {N_2}(t) = \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{2k}})y(t); $

(ⅱ) 如果$ {({N_1}(t), {N_2}(t))^{\rm T}} $是系统$ (1.1) $–$ (1.2) $的解, 则$ {(x(t), y(t))^{\rm T}} $是系统$ (2.1) $–$ (2.2) $的解, 其中

$ x(t) = \mathop \prod \limits_{0 < {t_k} < t} {(1 + {c_{1k}})^{ - 1}}{N_1}(t), \qquad y(t) = \mathop \prod \limits_{0 < {t_k} < t} {(1 + {c_{2k}})^{ - 1}}{N_2}(t). $

证  (ⅰ) 如果$ {({x}(t), {y}(t))^{\rm T}} $是系统$ (2.1) $–$ (2.2) $的解, 则有

$ {N_1}(t) = \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}})x(t), \qquad {N_2}(t) = \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{2k}})y(t), $

在每一个区间$ ({t_k}, {t_{k+ 1}}] $上都是绝对连续的.对于任何$ t \ne {t_k}, k \in {N^ + } $, 有

(2.4) $ \begin{eqnarray} && \mathop {{N_1}}\limits^. (t) - {N_1}(t)\left[ {{b_1}(t) - {a_1}(t){N_1}(t - {\tau _1}(t)) - \rho (t)\mathop {{N_1}}\limits^. (t - {\tau _3}(t)) - \frac{{c(t){N_2}(t - \sigma (t))}}{{{N_1}^2(t) + \alpha {N_1}(t) + \beta }}} \right]\\ & = & \prod\limits_{0 < {t_k} < t} {(1 + {c_{1k}})} \mathop {{\rm{ }}x}\limits^. (t) - \prod\limits_{0 < {t_k} < t} {(1 + {c_{1k}})} x(t)\Bigg[{b_1}(t) - {a_1}(t)\prod\limits_{0 < {t_k} < t - {\tau _1}(t)} {(1 + {c_{1k}})} x(t - {\tau _1}(t)) \\ && - \rho (t)\prod\limits_{0 < {t_k} < t - {\tau _3}(t)} {(1 + {c_{1k}})} \mathop {{\rm{ }}x}\limits^. (t - {\tau _3}(t))\\ && { - \frac{{c(t)\prod\limits_{0 < {t_k} < t - \sigma (t)} {(1 + {c_{2k}})} y(t - \sigma (t))}}{{{{\Big(\prod\limits_{0 < {t_k} < t} {(1 + {c_{1k}})}\Big )}^2}{x^2}(t) + \alpha \prod\limits_{0 < {t_k} < t} {(1 + {c_{1k}})} x(t) + \beta }}} \Bigg]\\ & = & \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}})\left\{ {\mathop x\limits^. (t) - } x(t)\left[ {{b_1}(t) - {a_1}(t)\mathop \prod \limits_{0 < {t_k} < t - {\tau _1}(t)} (1 + {c_{1k}})x(t - {\tau _1}(t))} \right.\right. \\ && - \rho (t)\mathop \prod \limits_{0 < {t_k} < t - {\tau _3}(t)} (1 + {c_{1k}})\mathop x\limits^. (t - {\tau _3}(t))\\ && \left. {\left. { - \frac{{c(t)\mathop \prod \limits_{0 < {t_k} < t - \sigma (t)} (1 + {c_{2k}})y(t - \sigma (t))}}{{{{\Big(\mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}})\Big)}^2}{x^2}(t) + \mathop \alpha \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}})x(t) + \beta}}} \right]} \right\}\\ & = & \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}})\Bigg\{{\mathop x\limits^. (t) - x(t)} \Bigg[ {{b_1}(t) - {A_1}(t)x(t - {\tau _1}(t)) - \gamma (t)\mathop x\limits^. (t - {\tau _3}(t))} \\ && \ - \left. {\left. {\frac{{C(t)y(t - \sigma (t))}}{{{{{p^2}(t)}}{x^2}(t) + \alpha p(t)x(t) + \beta }}} \right]} \right\}\\ & = & 0. \end{eqnarray} $

类似地, 有

(2.5) $ \begin{eqnarray} && \mathop {{N_2}}\limits^. (t) - {N_2}(t)\left[ {{-b_2}(t) +\frac{{a_2(t){N_1}(t - {\tau_2} (t))}}{{{{{N_1}^2(t-{\tau_2} (t))}}+ \alpha {N_1}(t-{\tau_2} (t)) + \beta}}} \right]{}\\ & = & \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{2k}})\mathop y\limits^. (t) - \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{2k}})y(t)\Bigg[ {{-b_2}(t) } {} \\ && {+ \frac{{a_2(t)\mathop \prod \limits_{0 < {t_k} < t - {\tau_2} (t)} (1 + {c_{1k}})x(t -{\tau_2} (t) )}}{{{{\Big(\mathop \prod \limits_{0 < {t_k} < t-{\tau_2} (t)} (1 + {c_{1k}})\Big)}^2}{x^2}(t-{\tau_2} (t)) + \mathop \alpha \prod \limits_{0 < {t_k} < t-{\tau_2} (t)} (1 + {c_{1k}})x(t-{\tau_2} (t)) + \beta}}} \Bigg]\\ & = & \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{2k}})\left\{ {\mathop y\limits^. (t) - } y(t) \Bigg[ {{-b_2}(t) } \right. {}\\ && \left. {\left. { + \frac{{a_2(t)\mathop \prod \limits_{0 < {t_k} < t - {\tau_2} (t)} (1 + {c_{1k}})x(t - {\tau_2} (t))}}{{{{\Big(\mathop \prod \limits_{0 < {t_k} < t- {\tau_2} (t)} (1 + {c_{1k}})\Big)}^2}{x^2}(t- {\tau_2} (t)) + \mathop \alpha \prod \limits_{0 < {t_k} < t- {\tau_2} (t)} (1 + {c_{1k}})x(t- {\tau_2} (t)) + \beta}}} \right]} \right\}\\ & = & \prod\limits_{0 < {t_k} < t} {(1 + {c_{2k}})} \left\{ {\mathop y\limits^. (t) - y(t)\left[ { - {b_2}(t) + \frac{{{A_2}(t)x(t - {\tau _2}(t))}}{{{q^2}(t){x^2}(t - {\tau _2}(t)) + \alpha q(t)x(t - {\tau _2}(t)) + \beta }}} \right]} \right\}\\ & = & 0. \end{eqnarray} $

另一方面, 对任何$ t = t_k, k \in {N^ + } $, 有

$ {N_1}(t_k^ + ) = \mathop {\lim }\limits_{t \to t_k^ + } \mathop \prod \limits_{0 < {t_j} < t} (1 + {c_{1j}})x(t) = \mathop \prod \limits_{0 < {t_j} < t_k^ + } (1 + {c_{1j}})x(t_k^ + ) = \mathop \prod \limits_{0 < {t_j} \le {t_k}} (1 + {c_{1j}})x({t_k}), $

$ {N_2}(t_k^ + ) = \mathop {\lim }\limits_{t \to t_k^ + } \mathop \prod \limits_{0 < {t_j} < t} (1 + {c_{2j}})y(t) = \mathop \prod \limits_{0 < {t_j} < t_k^ + } (1 + {c_{2j}})y(t_k^ + ) = \mathop \prod \limits_{0 < {t_j} \le {t_k}} (1 + {c_{2j}})y({t_k}) $

和

$ {N_1}({t_k}) = \mathop \prod \limits_{0 < {t_j} < {t_k}} (1 + {c_{1j}})x({t_k}), \qquad {N_2}({t_k}) = \mathop \prod \limits_{0 < {t_j} < {t_k}} (1 + {c_{2j}})y({t_k}). $

因此

(2.6) $ \begin{equation} {N_1}(t_k^ + ) = (1 + {c_{1k}}){N_1}({t_k}), \qquad {N_2}(t_k^ + ) = (1 + {c_{2k}}){N_2}({t_k}). \end{equation} $

此外

$ {N_1}(t_k^{\rm{ - }}) = \mathop {\lim }\limits_{t \to t_k^{\rm{ - }}} \prod\limits_{0 < {t_j} < t} {(1 + {c_{1j}})} x(t) = \prod\limits_{0 < {t_j} \le {t_{k{\rm{ - }}1}}} {(1 + {c_{1j}})} x({t_k}), $

$ {N_2}(t_k^{\rm{ - }}) = \mathop {\lim }\limits_{t \to t_k^{\rm{ - }}} \prod\limits_{0 < {t_j} < t} {(1 + {c_{2j}})} y(t) = \prod\limits_{0 < {t_j} \le {t_{k{\rm{ - }}1}}} {(1 + {c_{2j}})} y({t_k}). $

所以有

(2.7) $ \begin{eqnarray} {N_1}(t_k^{\rm{ - }}) = {N_1}({t_k}), \qquad {N_2}(t_k^{\rm{ - }}) = {N_2}({t_k}). \end{eqnarray} $

由$ (2.1) $–$ (2.7) $式可知, $ (N_1(t), N_2(t))^{\rm T} $是系统$ (1.1) $–$ (1.2) $的一个解.

(ⅱ) 由定义2.1可知

$ {N_1}(t) = \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{1k}})x(t), \qquad {N_2}(t) = \mathop \prod \limits_{0 < {t_k} < t} (1 + {c_{2k}})y(t) $

在每一个区间$ ({t_k}, {t_{k + 1}}], k \in {N^ + } $上都是绝对连续的.由(2.6)式, 对$ {\forall} $$ k \in {N^ + } $, 有

$ x(t_k^ + ) = \mathop \prod \limits_{0 < {t_j} \le {t_k}} {(1 + {c_{1j}})^{ - 1}}{N_1}(t_k^ + ) = \mathop \prod \limits_{0 < {t_j} < {t_k}} {(1 + {c_{1j}})^{ - 1}}{N_1}({t_k}) = x({t_k}), $

$ y(t_k^ + ) = \mathop \prod \limits_{0 < {t_j} \le {t_k}} {(1 + {c_{2j}})^{ - 1}}{N_2}(t_k^ + ) = \mathop \prod \limits_{0 < {t_j} < {t_k}} {(1 + {c_{2j}})^{ - 1}}{N_2}({t_k}) = y({t_k}), $

并且有

$ x(t_k^ - ) = \mathop \prod \limits_{0 < {t_j} \le {t_{k - 1}}} {(1 + {c_{1j}})^{ - 1}}{N_1}(t_k^ - ) = x({t_k}), $

$ y(t_k^ - ) = \mathop \prod \limits_{0 < {t_j} \le {t_{k - 1}}} {(1 + {c_{2j}})^{ - 1}}{N_2}(t_k^ - ) = y({t_k}). $

所以$ x(t) $和$ y(t) $在区间$ [ - \tau, \infty) $上是连续的.进一步容易验证, $ x(t) $和$ y(t) $在区间$ [ - \tau, \infty) $上是绝对连续的.类似于(ⅰ) 的证明, 容易验证

$ x(t) = \mathop \prod \limits_{0 < {t_k} < t} {(1 + {c_{1k}})^{ - 1}}{N_1}(t), \qquad y(t) = \mathop \prod \limits_{0 < {t_k} < t} {(1 + {c_{2k}})^{ - 1}}{N_2}(t) $

是系统$ (2.1) $–$ (2.2) $的解.

为了方便起见, 下面给出一些记号和几个正的常数.

$ {f^L} = \mathop {\min }\limits_{t \in [0, \omega ]} f(t), \; \; {f^M} = \mathop {\max }\limits_{t \in [0, \omega ]} f(t), \; \; {\left| f \right|_0} = \mathop {\max }\limits_{t \in [0, \omega ]} \{ \left| {f(t)} \right|\} , \; \; \hat f = {\frac{\ 1\ }{\ \omega\ }}{\int_0^\omega} {\left| {f(t)} \right|} {\rm d}t , $

$ \bar f = {\frac{1}{\omega }}{\int_0^\omega} {f(t)} {\rm d}t , \; \; {l_ \pm } = {\frac{{(A_2^M{e^G} - \alpha b_2^L{q^L}) \pm \sqrt {{{(A_2^M{e^G} - \alpha b_2^L{q^L})}^2} - 4\beta b_2^{2L}{q^{2L}}} }}{{2b_2^L{q^{2L}}}}}, $

$ {u_ \pm } = { \frac{{(A_2^L{e^{ - G}} - \alpha b_2^M{q^M}) \pm \sqrt {{{(A_2^L{e^{ - G}} -\alpha b_2^M{q^M})}^2} - 4\beta b_2^{2M}{q^{2M}}} }}{{2b_2^M{q^{2M}}}}}, $

$ H = \ln \left( {{\frac{{2b_1^M}}{{{D^L}}}}}\right) + {B^M}{\frac{{2b_1^M}}{{{D^L(1 - {{({{\mathop \tau \limits^. }_3})}^M})}}}} + \left( {{{\hat b}_1} + {{\bar b}_1}} \right)\omega , \; \; D(t) = {\frac{{{A_1}({\varphi ^{ - 1}}(t))}}{{1 - {{\mathop \tau \limits^. }_1}({\varphi ^{ - 1}}(t))}}}, $

$ B(t) = {\frac{\gamma }{{1 - {{\mathop \tau \limits^. }_3}(t)}} }, \; \; G \equiv {\frac{{\left( {{{\hat b}_1} + {{\bar b}_1}} \right)\omega }}{{1 - \gamma {e^H}}}} , \; \; {K_1} = G + \ln {l_ + } , \; \; {K_2} = \mathop {\max }\limits_{t \in [0, \omega ]} \{ {K_1}, H\} , $

其中$ f $是$ \omega $ -周期的连续函数, $ {{\varphi ^{ - 1}}} $是$ \varphi = t - {\tau _1}(t) $的逆函数. $ A_1(t) $由(2.3)式给出, 常数$ \gamma $由下面定理中给出.

定理3.1  假设(H$ _1) $, (H$ _2) $和如下条件成立

(H$ _3) $  $ A_2^M < \left( {\alpha-2\sqrt {\beta} } \right)b_2^L{q^L}{e^{ - G}} $, $ 1 - \gamma {e^{{K_2}}} > 0 $, $ {{\bar b}_1} - {{\bar A}_1}{e^{{K_2}}} > 0 ; $

(H$ _4) $  $ \gamma(t) = \gamma $是常数; $ {\mathop \tau \limits^. {_1}}(t) < 1, $对$ {\forall} $$ t \in {{\Bbb R}} $有$ {\mathop \tau \limits^. {_3}}(t) < 1 $, $ {\mathop \tau \limits^{..} {_3}}(t) = 0 $.

则系统$ (2.1) $–$ (2.2) $至少存在一个$ \omega $ -周期解.

证  因为

$ \begin{eqnarray*} x(t)& = & x(0)\exp \left\{ {\int_0^t {\Bigg[ {{b_1}(s) - {A_1}(s)x(s - {\tau _1}(s)) - \gamma \mathop x\limits^. (s - {\tau _3}(s))} } } \right.\\ && \left. {\left. { - {\frac{{C(s)y(s - \sigma (s))}}{{{p^2}(s){x^2}(s) + \alpha p(s)x(s) + \beta }}}} \right]{\rm d}s} \right\}, \\ y(t) & = & y(0)\exp \left\{{\int_0^t {\left[ { - {b_2}(s) + \frac{{{A_2}(s)x(s - {\tau _2}(s))}}{{{q^2}(s){x^2}(s - {\tau _2}(s)) + \alpha q(s)x(s - {\tau _2}(s)) + \beta }}} \right]{\rm d}s} } \right\}, \end{eqnarray*} $

所以, 对$ \forall t\in {{\Bbb R}} $, 系统$ (2.1) $的解都是正的.做变换

$ x(t) = {e^{u(t)}}, \ \ \ \ \ y(t) = {e^{v(t)}}, $

则系统$ (2.1) $可变换成如下系统

(3.1) $ \begin{equation} \left\{ \begin{array}{l} { } \mathop u\limits^. (t) = {b_1}(t) - {A_1}(t){e^{u(t - {\tau _1}(t))}} - \gamma{e^{u(t - {\tau _3}(t))}}\mathop u\limits^. (t - {\tau _3}(t)) - {\frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} + \alpha p(t){e^{u(t)}} + \beta}}}, \\ { } \mathop v\limits^. (t) = - {b_2}(t) + {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} + \alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta}}}. \end{array} \right. \end{equation} $

令

$ X = Y = \left\{ {z(t) = {{(u(t), v(t))}^{\rm T}} \in C({{\Bbb R}}, {{{\Bbb R}} ^2}):z(t) \equiv z(t + \omega )} \right\}, $

$ \left| z \right| = \left| u \right| + \left| v \right|, \ \ \ \ {\left| z \right|_\infty } = \mathop {\max }\limits_{t \in [0, \omega ]} \left| z \right|, \ \ \ \ \ \left\| z \right\| = {\left| z \right|_\infty } + {\left| {\mathop z\limits^. } \right|_\infty }, $

则$ (X, \left\| . \right\|) $和$ (Y, {\left| .\right|_\infty }) $都是Banach空间.

定义算子$ L, P $和$ Q $如下

$ L:DomL{\cap}X \to Y, \ \ \ \ Lz = {\frac{{{\rm d}z}}{{{\rm d}t}}}; \ \ \ \ \ p(z) = {\frac{1}{\omega }}{\int_0^\omega} {z(t){\rm d}t; \ \ \ \ \ Q(z) = {\frac{1}{\omega }}} {\int_0^\omega} {z(t){\rm d}t;} $

其中, $ DomL = \left\{ {z\left| {z \in X:z(t) \in {C^1}({{\Bbb R}}, {{{\Bbb R}} ^2})} \right.} \right\} $.定义算子$ N:X \to Y $如下

$ Nz = \left( \begin{array}{c} { } {b_1}(t) - {A_1}(t){e^{u(t - {\tau _1}(t))}} - \gamma {e^{u(t - {\tau _3}(t))}}\mathop u\limits^. (t - {\tau _3}(t)) - {\frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} + \alpha p(t){e^{u(t)}} + \beta}}}\\ { } - {b_2}(t) + {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t) {e^{2u(t - {\tau _2}(t))}} +\alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta}}} \end{array} \right). $

容易得到

$ KerL = \left\{ {z = {{(u, v)}^{\rm T}} \in X:{{(u(t), v(t))}^{\rm T}} = {{({c_1}, {c_2})}^{\rm T}} \in {{{\Bbb R}} ^2}, t \in {{\Bbb R}} } \right\} $

和

$ {\mathop{\rm Im}\nolimits} L = \left\{ {z \in Y: \int_0^\omega {z(t){\rm d}t} = 0} \right\} . $

因为$ {\rm Im}L $在$ Y $上是闭的, $ P $和$ Q $都是连续的投影算子, 且满足$ {\rm Im}P = KerL $, $ {\rm Im}L = KerQ = {\rm Im}(I-Q) $, $ dimKerL = codim{\rm Im}L = 2 $, 所以$ L $是指标为零的Fredholm算子, 故算子$ L $有唯一的逆映射.记$ L $的广义逆为$ {K_p}:{\mathop{\rm Im}\nolimits} L \to KerP \cap DomL $.经过计算可得

$ {K_p}(z) = \int_0^t {z(s)} {\rm d}s - {\frac{1}{\omega }}\int_0^\omega {\int_0^t {z(s)} {\rm d}s{\rm d}t} = \int_\omega ^t {z(s)} {\rm d}s + {\frac{1}{\omega }}\int_0^\omega {sz(s)} {\rm d}s. $

从而, 有

$ \begin{eqnarray*} QNz & = & \left( \begin{array}{l} { } {\frac{1}{\omega }}\int_o^\omega {\Bigg[ {{b_1}(t) - {A_1}(t){e^{u(t - {\tau _1}(t))}} - \gamma {e^{u(t - {\tau _3}(t))}}\mathop {{\rm{ }}u}\limits^. (t - {\tau _3}(t))} } \\ { } {{} - {\frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} + \alpha p(t){e^{u(t)}} + \beta }}}} \Bigg]{\rm d}t\\ {\frac{1}{\omega }}\int_0^\omega {\left[ { - {b_2}(t) + {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} + \alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta }}}} \right]} {\rm d}t \end{array} \right)\nonumber\\ & = & \left( \begin{array}{l} { } {\frac{1}{\omega }}\int_o^\omega {\Bigg[ {{b_1}(t) - {A_1}(t){e^{u(t - {\tau _1}(t))}} + \mathop {{\rm{ }}B}\limits^. (t){e^{u(t - {\tau _3}(t))}}} } \\ { } { {}- {\frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} + \alpha \alpha p(t){e^{u(t)}} + \beta }}}} \Bigg]{\rm d}s\\ { } {\frac{1}{\omega }}\int_0^\omega {\left[ { - {b_2}(t) + {\rm{ }}{\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} + \alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta }}}} \right]} {\rm d}s \end{array} \right), \end{eqnarray*} $

其中$ B(t) = {\frac{\gamma }{{1 - {{\mathop \tau \limits^. }_3}(t)}}} $.由假设条件(H$ _4) $, 可得$ \mathop B\limits^. (t) = 0 $.因此

$ QNz = \left( \begin{array}{l} {\frac{1}{\omega }}\int_0^\omega {\left[ {{b_1}(t) - {A_1}(t){e^{u(t - {\tau _1}(t))}} - {\frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} + \alpha p(t){e^{u(t)}} + \beta}}} }\right]} {\rm d}t\\ { } {\frac{1}{\omega }}\int_0^\omega {\left[ { - {b_2}(t) + {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} + \alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta}}}} \right]} {\rm d}t \end{array} \right), $

且

$ \begin{eqnarray*} {K_p}(I - Q)Nz& = & \left( \begin{array}{l} \int_\omega ^t {\left[ {{b_1}(s) - {A_1}(s){e^{u(s - {\tau _1}(s))}} - {\frac{{C(s){e^{v(s - \sigma (s))}}}}{{{p^2}(s){e^{2u(s)}} +\alpha p(s){e^{u(s)}} + \beta}}}} \right]{\rm d}s} \\ \int_\omega ^t {\left[ { - {b_2}(s) + {\frac{{{A_2}(s){e^{u(s - {\tau _2}(s))}}}}{{{q^2}(s){e^{2u(s - {\tau _2}(s))}} + \alpha q(s){e^{u(s - {\tau _2}(s))}} + \beta}}}} \right]{\rm d}s} \end{array} \right)\\ && +{ \frac{1}{\omega }}\left( \begin{array}{l} { } \int_0^\omega {s\left[ {{b_1}(s) - {A_1}(s){e^{u(s - {\tau _1}(s))}} + B(s){e^{u(s - {\tau _3}(s))}}} \right.} \\ { } \left. { {} - {\frac{{C(s){e^{v(s - \sigma (s))}}}}{{{p^2}(s){e^{2u(s)}} + \alpha p(s){e^{u(s)}} + \beta }}}} \right]{\rm d}s\\ [4mm] \int_0^\omega {s\left[ { - {b_2}(s) + {\frac{{{A_2}(s){e^{u(s - {\tau _2}(s))}}}}{{{q^2}(s){e^{2u(s - {\tau _2}(s))}} + \alpha q(s){e^{u(s - {\tau _2}(s))}} + \beta }}}} \right]{\rm d}s} \end{array} \right)\\ && + \left( {{\frac{1}{2}} - {\frac{t}{\omega }}} \right)\left( \begin{array}{l} { } {\int_o^\omega } {\Bigg[ {{b_1}(s) - {A_1}(s){e^{u(s - {\tau _1}(s))}}} } \\ { {}- {\frac{{C(s){e^{v(s - \sigma (s))}}}}{{{p^2}(s){e^{2u(s)}} + \alpha p(s){e^{u(s)}} + \beta }}}} \Bigg]{\rm d}s\\ {\int_0^\omega} {\left[ { - {b_2}(s) + {\frac{{{A_2}(s){e^{u(s - {\tau _2}(s))}}}}{{{q^2}(s){e^{2u(s - {\tau _2}(s))}} + \alpha q(s){e^{u(s - {\tau _2}(s))}} + \beta }}}} \right]} {\rm d}s \end{array} \right). \end{eqnarray*} $

根据Lebesgue控制收敛定理可知, $ QN $和$ K_p(I-Q)N $都是连续的.由Arzela-Ascoli定理, 对于任意的有界开集$ \Omega \in X $, $ {K_p}(I - Q)N(\bar \Omega ) $是紧的, $ QN(\bar \Omega ) $是有界的.因此$ N $在$ \bar \Omega $上是$ L $ -紧的.

由算子方程

$ Lz = {\lambda} Nz , {\quad} {\lambda}\in(0, 1), $

可得

(3.2) $ \begin{equation} \left\{ \begin{array}{l} \mathop u\limits^. (t) = \lambda \Bigg[ {{b_1}(t) - {A_1}(t){e^{u(t - {\tau _1}(t))}} - \gamma {e^{u(t - {\tau _3}(t))}}\mathop u\limits^. (t - {\tau _3}(t))} { - {\frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} + \alpha p(t){e^{u(t)}} + \beta }}}} \Bigg], \\ \mathop v\limits^. (t) = \lambda \left[ { - {b_2}(t) + {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} + \alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta }}}} \right], \end{array} \right. \end{equation} $

对某个$ {\lambda}\in(0, 1) $, 假设$ z(t) = {(u(t), v(t))^T} \in X $是系统$ (3.2) $的一个解.对于方程$ (3.2) $, 在$ [0, {\omega}] $上进行积分, 有

(3.3) $ \begin{equation} \int_0^\omega {\left[ {{A_1}(t){e^{u(t - {\tau _1}(t))}} + \frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} +\alpha p(t){e^{u(t)}} + \beta}}} \right]} {\rm d}t = {{\bar b}_1}\omega, \end{equation} $

(3.4) $ \begin{equation} \int_0^\omega {\left[ {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} + \alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta}}} \right]} {\rm d}t = {{\bar b}_2}\omega . \end{equation} $

由等式(3.2)–(3.4), 有

(3.5) $ \begin{eqnarray} && \int_0^\omega {\left| {\frac{\rm d}{{{\rm d}t}}\left[ {u(t) + \lambda B(t){e^{u(t - {\tau _3}(t))}}} \right]} \right|} {\rm d}t\\ & = & \lambda \int_0^\omega {\left| {\left[ {{b_1}(t) - {A_1}(t){e^{u(t - {\tau _1}(t))}} - \frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} +\alpha p(t){e^{u(t)}} + \beta}}} \right]} \right|} {\rm d}t\\ &\leq& \int_0^\omega {\left| {{b_1}(t)} \right|} {\rm d}t + \int_0^\omega {\left[ {{A_1}(t){e^{u(t - {\tau _1}(t))}} + \frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} +\alpha p(t){e^{u(t)}} + \beta}}} \right]} {\rm d}t\\ & = & \left( {{{\hat b}_1} + {{\bar b}_1}} \right)\omega. \end{eqnarray} $

由(3.3)式, 可得

$ \begin{eqnarray*} \int_0^\omega {{A_1}(t){e^{u(t - {\tau _1}(t))}}} {\rm d}t & = & \int_{ - {\tau _1}(0)}^{\omega - {\tau _1}(\omega )} {\frac{{{A_1}({\varphi ^{ - 1}}(t)){e^{u(t)}}}}{{1 - \mathop {{\tau _1}}\limits^. ({\varphi ^{ - 1}}(t))}}}{\rm d}t = \int_0^\omega {D(t){e^{u(t)}}} {\rm d}t< {{\bar b}_1}\omega . \end{eqnarray*} $

根据广义积分中值定理, 存在$ {\eta _1} \in \left[ {0, \omega } \right] $, 使得

(3.6) $ \begin{eqnarray} \int_0^\omega {D(t){e^{u(t)}}} {\rm d}t & = & D({\eta _1})\int_0^\omega {{e^{u(t)}}} {\rm d}t = D({\eta _1})\int_0^\omega {\left( {1 - \mathop {{\tau _3}}\limits^. (t)} \right){e^{u(t - {\tau _3}(t))}}} {\rm d}t < {{\bar b}_1}\omega . \end{eqnarray} $

由不等式(3.6), 存在$ {\eta _2} \in\left[ {0, \omega } \right] $, 有

(3.7) $ \begin{equation} D({\eta _1})\int_0^\omega {\left( {1 - \mathop {{\tau _3}}\limits^. (t)} \right){e^{u(t - {\tau _3}(t))}}} {\rm d}t = D({\eta _1})\left( {1 - \mathop {{\tau _3}}\limits^. (\eta_2)} \right)\int_0^\omega {{e^{u(t - {\tau _3}(t))}}} {\rm d}t<{{\bar b}_1}\omega. \end{equation} $

根据(3.6)和(3.7)式, 得

$ D({\eta _1})\int_0^\omega {{e^{u(t)}}} {\rm d}t + D({\eta _1})\left( {1 - \mathop {{\tau _3}}\limits^. (\eta_2)} \right)\int_0^\omega {{e^{u(t - {\tau _3}(t))}}} {\rm d}t < 2{{\bar b}_1}\omega. $

进而可知, 存在$ \xi \in[0, \omega] $, 使得

$ D({\eta _1}){e^{u(\xi)}} + D({\eta _1})\left( {1 - \mathop {{\tau _3}}\limits^. (\eta_2)} \right){e^{u({\xi} - {\tau _3}({\xi}))}} < 2{b_1}(\xi ). $

由于$ \int_0^\omega {{b_1}(t)} {\rm d}t > 0 $, 可得$ b_1^M > 0 $, 从而有

(3.8) $ \begin{equation} u(\xi) < \ln \left( {\frac{{2b_1^M}}{{{D^L}}}} \right) \end{equation} $

和

(3.9) $ \begin{equation} {e^{u(\xi - {\tau _3}(\xi ))}} < \frac{{2b_1^M}}{{{D^L}\left( {1 - {{\left( {\mathop {{\tau _3}}\limits^. } \right)}^M}} \right)}}. \end{equation} $

根据不等式(3.5), (3.8)和(3.9), 可得

$ \begin{eqnarray*} u(t) + \lambda B(t){e^{u(t - {\tau _3}(t))}} &\le& u(\xi ) + \lambda B(\xi ){e^{u(\xi - {\tau _3}(\xi ))}} + \int_0^\omega {\left| {{\frac{\rm d}{{{\rm d}t}}}\left[ {u(t) + \lambda B(t){e^{u(t - {\tau _3}(t))}}} \right]} \right|} {\rm d}t \\ & <& \ln \left( {{\frac{{2b_1^M}}{{{D^L}}}}} \right) + {B^M}{\frac{{2b_1^M}}{{{D^L}\left( {1 - {{\left( {\mathop {{\tau _3}}\limits^. } \right)}^M}} \right)}}} + \left( {{{\hat b}_1} + {{\bar b}_1}} \right)\omega \equiv H. \end{eqnarray*} $

因为$ B(t){e^{u(t - {\tau _3}(t))}} > 0 $, 故有

(3.10) $ \begin{eqnarray} u(t) < H . \end{eqnarray} $

根据方程(3.2), (3.3)和不等式(3.10), 可得

$ \begin{eqnarray*} \int_0^\omega {\left| {\mathop u\limits^. (t)} \right|} {\rm d}t &\le& \int_0^\omega {\left| {{b_1}(t)} \right|} {\rm d}t + \int_0^\omega {\left[ {{A_1}(t){e^{u(t - {\tau _1}(t))}} + {\frac{{C(t){e^{v(t - \sigma (t))}}}}{{{p^2}(t){e^{2u(t)}} +\alpha p(t){e^{u(t)}} + \beta}}}} \right]} {\rm d}t\\ &&+ \int_0^\omega {\left| {\gamma {e^{u(t - {\tau _3}(t))}}\mathop u\limits^. (t - {\tau _3}(t))} \right|} {\rm d}t\\ & \le& \left( {{{\hat b}_1} + {{\bar b}_1}} \right)\omega + \gamma {e^H}\int_0^\omega {\left| {\mathop u\limits^. (t)} \right|} {\rm d}t. \end{eqnarray*} $

再由(H$ _3) $, 可得

(3.11) $ \begin{equation} \int_0^\omega {\left| {\mathop u\limits^. (t)} \right|} {\rm d}t \le \frac{{\left( {{{\hat b}_1} + {{\bar b}_1}} \right)\omega }}{{1 - \gamma {e^H}}} \equiv G . \end{equation} $

因为$ z = (u, v)^{\rm T}{\in}X $, 所以存在$ {\underline{\xi}}, \bar \xi, {\underline{\eta}}, \bar \eta \in [0, \omega ] $, 使得

(3.12) $ \begin{equation} u({\underline{\xi}}) = \mathop {\min }\limits_{t \in [0, \omega ]} u(t), {} u(\bar \xi) = \mathop {\max }\limits_{t \in [0, \omega ]} u(t), \end{equation} $

(3.13) $ \begin{equation} v({\underline{\eta}}) = \mathop {\min }\limits_{t \in [0, \omega ]} v(t), {} v(\bar \eta) = \mathop {\max }\limits_{t \in [0, \omega ]} v(t). \end{equation} $

根据(3.4)式, 可知

(3.14) $ \begin{eqnarray} \int_0^\omega {\left[ {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} +\alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta}}} \right]} {\rm d}t = {{\bar b}_2}\omega = \int_0^\omega {{b_2}(t)} {\rm d}t. \end{eqnarray} $

结合(3.12)和(3.14)式, 可得

$ b_2^L \le \frac{{A_2^M{e^{u(\bar \xi )}}}}{{{q^{2L}}{e^{2u({\underline{\xi}})}} + \alpha {q^L}{e^{u({\underline{\xi}} )}} + \beta}} . $

即

$ u(\bar \xi ) \ge \ln \left[ {\frac{{b_2^L\left( {{q^{2L}}{e^{2u({\underline{\xi}} )}} +\alpha {q^L}{e^{u({\underline{\xi}} )}} + \beta} \right)}}{{A_2^M}}} \right]. $

结合(3.11)式, 可得

$ u(t) \ge u(\bar \xi ) - \int_0^\omega {\left| {\mathop u\limits^. (t)} \right|} {\rm d}t > \ln \left[ {\frac{{b_2^L\left( {{q^{2L}}{e^{2u({\underline{\xi}})}} + \alpha {q^L}{e^{u({\underline{\xi}} )}} + \beta} \right)}}{{A_2^M}}} \right] - G . $

进一步, 有

$ u({\underline{\xi}} ) > \ln \left[ {\frac{{b_2^L\left( {{q^{2L}}{e^{2u({\underline{\xi}} )}} +\alpha {q^L}{e^{u({\underline{\xi}} )}} + \beta} \right)}}{{A_2^M}}} \right] - G, $

即

$ b_2^L{q^{2L}}{e^{2u({\underline{\xi}})}} + (\alpha b_2^L{q^L} - A_2^M{e^G}){e^{u({\underline{\xi}})}} + \beta b_2^L < 0 . $

再结合(H$ _3) $, 可得

(3.15) $ \begin{eqnarray} \ln {l_ - } < u({\underline{\xi}} ) < \ln {l_ + }. \end{eqnarray} $

类似的, 根据(3.14)式有

$ \frac{{A_2^L{e^{u({\underline{\xi}} )}}}}{{{q^{2M}}{e^{2u(\bar \xi )}} +\alpha {q^M}{e^{u(\bar \xi )}} + \beta}} \le b_2^M. $

因此有

$ u({\underline{\xi}}) \le \ln \left[ {\frac{{b_2^M\left( {{q^{2M}}{e^{2u(\bar \xi )}} + \alpha {q^M}{e^{u(\bar \xi )}} + \beta} \right)}}{{A_2^L}}} \right] . $

再结合不等式(3.11), 可得

$ u(t) \le u({\underline{\xi}}) + \int_0^\omega {\left| {\mathop u\limits^. (t)} \right|} {\rm d}t < \ln \left[{ {\frac{{b_2^M\left( {{q^{2M}}{e^{2u(\bar \xi )}} +\alpha {q^M}{e^{u(\bar \xi )}} + \beta} \right)}}{{A_2^L}}} }\right] + G . $

特别的, 有

$ u(\bar \xi ) < \ln \left[ {\frac{{b_2^M\left( {{q^{2M}}{e^{2u(\bar \xi )}} + \alpha{q^M}{e^{u(\bar \xi )}} + \beta} \right)}}{{A_2^L}}} \right] + G, $

即

$ b_2^M{q^{2M}}{e^{2u(\bar \xi )}} + (\alpha b_2^M{q^M} - A_2^L{e^{ - G}}){e^{u(\bar \xi )}} + \beta b_2^M > 0 . $

结合假设条件(H$ _3) $, 可得

(3.16) $ \begin{eqnarray} u(\bar \xi ) > \ln {u_ + } \ \mbox{ 或者} \ u(\bar \xi ) < \ln {u_ - }. \end{eqnarray} $

由不等式(3.11)和(3.15), 可得

(3.17) $ \begin{eqnarray} u(t) \le u({\underline{\xi}}) + \int_0^\omega {\left| {\mathop u\limits^. (t)} \right|} {\rm d}t < G + \ln {l_ + }: = {K_1} . \end{eqnarray} $

由(3.10)和(3.17)式, 可得

(3.18) $ \begin{eqnarray} u(t) \le \max \left\{ {{K_1}, H} \right\}: = {K_2} . \end{eqnarray} $

由(3.11)和(3.16)式, 可得

(3.19) $ \begin{eqnarray} u(t) \ge u(\bar \xi ) - \int_0^\omega {\left| {\mathop u\limits^. (t)} \right|} {\rm d}t > \ln {u_ + } - G: = {K_3}. \end{eqnarray} $

再由(3.18)和(3.19)式, 有

(3.20) $ \begin{eqnarray} {\left| u \right|_0} = \mathop {\max }\limits_{t \in [0, \omega ]} \left\{ {\left| {u(t)} \right|} \right\} \le \max \left\{ {\left| {{K_1}} \right|, \left| {{K_2}} \right|, \left| H \right|} \right\} : = {K_4}. \end{eqnarray} $

另一方面, 由(3.3)和(3.18)式, 可得

(3.21) $ \begin{equation} {{\bar b}_1}\omega \ge \frac{{\bar C\omega {e^{v({\underline\eta})}}}}{{{p^{2M}}{e^{2{K_2}}} +\alpha {p^M}{e^{{K_2}}} + \beta}} \end{equation} $

和

(3.22) $ \begin{equation} {{\bar b}_1}\omega \le \omega {{\bar A}_1}{e^{{K_2}}} + \frac{{\bar C\omega {e^{v(\bar \eta )}}}}{\beta}. \end{equation} $

由(3.21)式, 可得

(3.23) $ \begin{equation} {{v({\underline\eta})}} \le \ln \left[ {\frac{{{{\bar b}_1}}}{{\bar C}}\left( {{p^{2M}}{e^{2{K_2}}} + \alpha {p^M}{e^{{K_2}}} + \beta} \right)} \right]. \end{equation} $

结合条件(H$ _3) $和(3.22)式, 有

(3.24) $ \begin{equation} v(\bar \eta ) \ge \ln \left[ {\frac{{\beta\left( {{{\bar b}_1} - {{\bar A}_1}{e^{{K_2}}}} \right)}}{{\bar C}}} \right]. \end{equation} $

再由(3.2)和(3.4)式, 可得

(3.25) $ \begin{eqnarray} \int_0^\omega {\left| {\mathop v\limits^. (t)} \right|} {\rm d}t & \le& \int_0^\omega {\left| {{b_2}(t)} \right|} {\rm d}t + \int_0^\omega {\frac{{{A_2}(t){e^{u(t - {\tau _2}(t))}}}}{{{q^2}(t){e^{2u(t - {\tau _2}(t))}} +\alpha q(t){e^{u(t - {\tau _2}(t))}} + \beta}}} {\rm d}t \\ & = & \left( {{{\hat b}_2} + {{\bar b}_2}} \right)\omega . \end{eqnarray} $

根据不等式(3.23)–(3.25), 可得

(3.26) $ \begin{eqnarray} v(t) &\leq& v({\underline\eta}) + \int_0^\omega {\left| {\mathop v\limits^. (t)} \right|} {\rm d}t{}\\ &<& \ln \left[ {\frac{{\bar b}}{{\bar C}}\left( {{p^{2M}}{e^{2{K_2}}} + \alpha {p^M}{e^{{K_2}}} + \beta} \right)} \right] + \left( {{{\hat b}_2} + {{\bar b}_2}} \right)\omega : = {K_5}, \end{eqnarray} $

且有

(3.27) $ \begin{eqnarray} v(t) &\geq& v(\bar \eta ) - \int_0^\omega {\left| {\mathop v\limits^. (t)} \right|} {\rm d}t > \ln \left[ {\frac{{\beta \left( {{{\bar b}_1} - {{\bar A}_1}{e^{{K_2}}}} \right)}}{{\bar C}}} \right] - \left( {{{\hat b}_2} + {{\bar b}_2}} \right)\omega : = {K_6} . \end{eqnarray} $

再由(3.26)和(3.27)式, 可得

(3.28) $ \begin{eqnarray} {\left| v \right|_0} = \mathop {\max }\limits_{t \in [0, \omega ]} \left\{ {\left| {v(t)} \right|} \right\} < \max \left\{ {\left| {{K_5}} \right|, \left| {{K_6}} \right|} \right\}: = {K_7}. \end{eqnarray} $

结合(3.2), (3.11), (3.18)和(3.28)式, 可得

$ {\left| {\mathop u\limits^. } \right|_0} = \mathop {\max }\limits_{t \in [0, \omega ]} \left\{ {\left| {\mathop u\limits^. (t)} \right|} \right\} \le {\left| {{b_1}} \right|_0} + {\left| {{A_1}} \right|_0}{e^{{K_2}}} + \gamma {e^{{K_2}}}{\left| {\mathop u\limits^. } \right|_0} +{ \frac{{{{\left| C \right|}_0}{e^{{K_7}}}}}{{{p^{2L}}{e^{2{K_3}}} +\alpha {p^L}{e^{{K_3}}} + \beta}}}, $

$ {\left| {\mathop v\limits^. } \right|_0} = \mathop {\max }\limits_{t \in [0, \omega ]} \left\{ {\left| {\mathop v\limits^. (t)} \right|} \right\} \le {\left| {{b_2}} \right|_0} + {\frac{{{{\left| {{A_2}} \right|}_0}}}{{{\alpha q^L}}}}: = {K_8} . $

从而, 有

$ {\left| {\mathop u\limits^. } \right|_0} \le {\frac{1}{{1 - \gamma {e^{{K_2}}}}}}\left( {{{\left| {{b_1}} \right|}_0} + {{\left| {{A_1}} \right|}_0}{e^{{K_2}}} +{ \frac{{{{\left| C \right|}_0}{e^{{K_7}}}}}{{{p^{2L}}{e^{2{K_3}}} +\alpha {p^L}{e^{{K_3}}} + \beta}}}} \right): = {K_9} . $

因此, 由(3.20)和(3.28)式, 可得

(3.29) $ \begin{eqnarray} \left\| z \right\| = \left\| {{{(u, v)}^T}} \right\| = {\left| z \right|_\infty } + {\left| {\mathop z\limits^. } \right|_\infty } < {K_4} + {K_7} + {K_8} + {K_9}: = {K_{10}}. \end{eqnarray} $

显然, $ \ln {l_ \pm }, \ln {u_ \pm }, {K_2} $和$ {K_{10}} $都不依赖于$ \lambda $.令$ K = {K_{10}} + \tilde K. $当$ \tilde K $取足够大, 可使得如下代数方程

(3.30) $ \begin{eqnarray} \left\{ \begin{array}{l} {{\bar b}_1} - {{\bar A}_1}{e^u} - \overline {{\frac{{C(t){e^v}}}{{{p^2}(t){e^{2u}} +\alpha p(t){e^u} + \beta}}}} = 0, \\ - {{\bar b}_2} + \overline {{\frac{{{A_2}(t){e^u}}}{{{q^2}(t){e^{2u}} +\alpha q(t){e^u} + \beta}}} } = 0 \end{array} \right. \end{eqnarray} $

的解满足$ \left\| {{z^*}} \right\| = \left\| {{{({u^*}, {v^*})}^{\rm T}}} \right\| = \left| {{u^*}} \right| + \left| {{v^*}} \right| < K $.

令

$ \Omega = \left\{ {z(t) = {{(u(t), v(t))}^{\rm T}}\left| {z(t) \in X, \left\| {z(t) } \right\|< K} \right.} \right\} . $

下面验证引理2.1的三个条件成立.

(a) 由上述计算可知, 对$ {\forall} $$ \lambda {\in}(0, 1) , z \in \partial \Omega \cap DomL, $有$ Lz \ne \lambda Nz $成立.

(b) 当$ {(u(t), v(t))^{\rm T}} \in \partial \Omega \cap KerL $, 且$ {(u(t), v(t))^{\rm T}} $在$ R^2 $中是范数为$ K $的一个常向量, 记作: $ {(u, v)^{\rm T}} $.如果$ QN{(u, v)^{\rm T}} = 0 $, 则$ {(u, v)^{\rm T}} $是方程(3.30)的一个常数解.由上面讨论可知, $ \left\| {{{(u, v)}^{\rm T}}} \right\| < K $, 这与$ \left\| {{{(u, v)}^{\rm T}}} \right\| = K $矛盾.所以对$ {\forall} $$ z \in \partial \Omega\cap KerL $, 有$ QNz \ne 0 $.

(c) 下面验证条件$ [C_3] $成立.定义$ \phi:DomL\cap KerL\times[0, 1]\rightarrow X $为

$ \phi (u, v, \mu ) = \left( \begin{array}{l} \bar C{e^v} - {{\bar A}_1}{e^u}\\ - {{\bar b}_2} + {{\bar A}_2}{e^u} \end{array} \right) + \mu \left( \begin{array}{l} {{\bar b}_1} - \bar C{e^v} - \overline {{\frac{{C(t){e^v}}}{{{p^2}(t){e^{2u}} + \alpha p(t){e^u} + \beta }}}} \\ - {{\bar A}_2}{e^u} + \overline {{\frac{{{A_2}(t){e^u}}}{{{q^2}(t){e^{2u}} + \alpha q(t){e^u} + \beta }}}} \end{array} \right), $

其中参数$ \mu\in[0, 1] $.常向量$ {(u, v)^{\rm T}} \in\partial \Omega \cap KerL = \partial \Omega \cap {R^2} $, 且$ \left\| {{{(u, v)}^{\rm T}}} \right\| = K $.下证当$ {(u, v)^{\rm T}} \in \partial \Omega \cap KerL $时, 有$ \phi (u, v, \mu )\neq 0 $成立.否则, 假设存在常向量$ {(u, v)^{\rm T}} $, 范数为$ \left\| {{{(u, v)}^{\rm T}}} \right\| = K $时, 满足$ \phi (u, v, \mu ) = 0 $.由$ \phi (u, v, \mu ) = 0 $, 类似于(3.8), (3.9), (3.15), (3.16), (3.23)和(3.24)式的推导, 可得$ \left\| {{{(u, v)}^{\rm T}}} \right\| < K $, 这就产生矛盾.假设$ J $是从$ ImQ $到$ KerL $的恒等同构映射, 由根据拓扑度的同伦不变性, 可得

$ \begin{eqnarray*} &&\deg \{ JQN{(u, v)^{\rm T}}, \Omega \cap KerL, {(0, 0)^{\rm T}}\}\\ & = &\deg \{ QN{(u, v)^{\rm T}}, \Omega \cap KerL, {(0, 0)^{\rm T}}\} \\ & = & \deg \{ \phi (u, v, 1), \Omega \cap KerL, {(0, 0)^{\rm T}}\}\\ & = & \deg \{ \phi (u, v, 0), \Omega \cap \ KerL, {(0, 0)^{\rm T}}\}\\ & = & \deg \{ {(\bar C{e^v} - {{\bar A}_1}{e^u}, - {{\bar b}_2} + {{\bar A}_2}{e^u})^{\rm T}}, \Omega \cap KerL, {(0, 0)^{\rm T}}\}. \end{eqnarray*} $

易得, 下面的代数方程

$ \left\{ \begin{array}{l} \bar C{e^v} - {{\bar A}_1}{e^u} = 0, \\ - {{\bar b}_2} + {{\bar A}_2}{e^u} = 0 \end{array} \right. $

有唯一解$ {({u^*}, {v^*})^{\rm T}} \in \Omega \cap KerL $.因此有

$ \begin{eqnarray*} \deg \{ JQN{(u, v)^{\rm T}}, \Omega \cap KerL, {(0, 0)^{\rm T}}\} & = & {\mathop{\rm sgn}} \left| {\begin{array}{*{20}{c}} { - {{\bar A}_1}{e^{{u^*}}}}&{\bar C{e^{{v^*}}}}\\ {{{\bar A}_2}{e^{{u^*}}}}&0 \end{array}} \right| = {\mathop{\rm sign}} ( - \bar C{{\bar A}_2}{e^{{v^*} + {u^*}}})\neq0. \end{eqnarray*} $

所以引理2.1中的所有条件均满足, 因此系统(2.1)–(2.2)至少存在一个正的$ \omega $ -周期解.进而由引理2.3可知, 系统(1.1)–(1.2)至少存在一个正的$ \omega $ -周期解.

该文主要研究一类带有脉冲和Holling-IV型功能反应函数的中立型捕食-食饵种群动力学模型(1.1)–(1.2).通过运用Mawhin迭合度理论和一些分析技巧, 得到了系统(1.1)–(1.2)正周期解的存在性.今后, 我们将进一步讨论系统(1.1)–(1.2)正周期解的全局吸引性以及多周期解的存在性.