时标上具有捕获率和投放率的Holling Ⅲ型捕食系统的周期解

姜全德

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时标上具有捕获率和投放率的Holling Ⅲ型捕食系统的周期解

    通讯作者: 姜全德, 2015109047@gdip.edu.cn
  • 中图分类号: O175.1

Periodic solutions of Holling Ⅲ predator-prey systems with exploited terms and Stork on time scales

    Corresponding author: JIANG Quan-de, 2015109047@gdip.edu.cn
  • CLC number: O175.1

  • 摘要: 研究时标上具有捕获率和投放率的Holling Ⅲ 型捕食系统,考虑了时滞效应,基于时标Mawhin 重合度理论方法,得到了系统至少有一个正周期解.
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出版历程
  • 收稿日期:  2021-04-26
  • 录用日期:  2021-05-31
  • 网络出版日期:  2021-07-22
  • 刊出日期:  2021-09-15

时标上具有捕获率和投放率的Holling Ⅲ型捕食系统的周期解

    通讯作者: 姜全德, 2015109047@gdip.edu.cn
  • 广东轻工职业技术学院 财贸学院 数学教研室,广东 广州 510300

摘要: 研究时标上具有捕获率和投放率的Holling Ⅲ 型捕食系统,考虑了时滞效应,基于时标Mawhin 重合度理论方法,得到了系统至少有一个正周期解.

English Abstract

  • 近年来,很多学者研究了连续或离散情况下的捕食系统[1-15]. 同时,时标理论也吸引了不少学者. 2006年,Bohner等在文献[1]中首次应用重合度延拓定理研究了时标上微分方程的周期解的存在性问题. 目前,关于时标上研究捕食系统的周期解已有不少结果[2-3,9,14].

    2009年,Wang等在文献[2] 中研究了以下捕食系统

    $ \qquad \left\{ \begin{array}{l} \dot x(t) = x(t)({r_1}(t) - {b_1}(t)x(t)) - \dfrac{{{c_1}(t){x^2}(t)}}{{{x^2}(t) + {k^2}}}{y^m}(t), \\ \dot y(t) = y(t)( - {r_2}(t) - {b_2}(t)y(t)) + \dfrac{{{c_2}(t){x^2}(t)}}{{{x^2}(t) + {k^2}}}{y^m}(t), \end{array} \right.$

    其中 $x(t), y(t)$ 分别表示食饵和捕食者的密度,${r_i}, {b_i}, {c_i}$ 都是正的周期为 $\omega $ 的函数,分别为幼年食饵的内蕴增长率、俘获率、成年捕食者的自然死亡率;$k > 0, 0 < m \leqslant 1.$

    如果系统在人为干预情况下,有投放食饵和捕获捕食者的情况,又因现实生活中各种现象存在时间延迟,所以考虑时滞因素. 系统(1)可以写成

    $ \qquad \left\{ \begin{array}{l} \dot x(t) = x(t)({r_1}(t) - {b_1}(t)x(t - {\tau _{\rm{1}}}(t))) - \dfrac{{{c_1}(t){x^2}(t)}}{{{x^2}(t) + {k^2}}}{y^m}(t) + s(t), \\ \dot y(t) = y(t)( - {r_2}(t) - {b_2}(t)y(t - {\tau _2}(t))) + \dfrac{{{c_2}(t){x^2}(t)}}{{{x^2}(t) + {k^2}}}{y^m}(t) - h(t), \end{array} \right.$

    其中 $s(t), h(t)$ 都是正的周期为 $\omega $ 的函数,分别表示食饵种群的投放率和捕食种群的收获率.

    本文将在时标上研究下列系统

    $ \qquad \left\{ \begin{array}{l} u_1^\Delta (t) = {r_1}(t) - {b_1}(t)\exp ({u_1}(t - {\tau _{\rm{1}}}(t))) - \dfrac{{{c_1}(t)\exp ({u_1}(t)){{(\exp ({u_2}(t)))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} + s(t)\exp ( - {u_1}(t)), \\ u_{\rm{2}}^\Delta (t) = - {r_2}(t) - {b_2}(t)\exp ({u_2}(t - {\tau _2}(t))) + \dfrac{{{c_2}(t)(\exp ({u_1}(t)))^{2}{{(\exp ({u_2}(t)))}^{m - 1}}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} - h(t)\exp ( - {u_2}(t)), t \in {{\bf{T}}} \end{array} \right.$

    的多周期解的存在性问题,其中 ${\bf{T}}$ 为任意时标.

    如果令 $x(t) = \exp ({u_1}(t)), y(t) = \exp ({u_2}(t))$,当 ${\bf{T}} = {\bf{R}}$(实数集)时,系统(3)就变成了系统(2).当 ${\bf{T}} = {\bf{Z}}$(整数集)时,系统(3)就变成了离散系统

    $ \qquad \left\{ \begin{array}{l} x(t + 1) = x(t)\exp \left\{ {r_1}(t) - {b_1}(t)x(t - {\tau _{\rm{1}}}(t)) - \dfrac{{{c_1}(t)x(t)}}{{{x^2}(t) + {k^2}}}{y^m}(t) + \dfrac{{s(t)}}{{x(t)}}\right\} , \\ y(t{\rm{ + 1}}) = y(t)\exp \left\{ - {r_2}(t) - {b_2}(t)y(t - {\tau _2}(t)) + \dfrac{{{c_2}(t){x^2}(t)}}{{{x^2}(t) + {k^2}}}{y^{m - 1}}(t) - \dfrac{{h(t)}}{{x(t)}}\right\} . \end{array} \right.$

    • 定义 1[15] 前跳算子 $\sigma :\sigma {(t){\buildrel \Delta \over = }}\inf \left\{ {s \in {{\bf{T}}}:s > t} \right\}, \forall t \in {{\bf{T}}}$,后跳算子$\rho :\rho {(t){\buildrel \Delta \over = }}\sup \{ s \in {\bf{T}}:s < t\} ,\forall t \in {\bf{T}},$ 阶梯函数 $\mu :\mu (t) = \sigma (t) - t.$

      定义 2[15] 假如 $t < {\rm{sup}}{\bf{T}}$ 而且 $\sigma (t) = t$,则称 $t$ 是右稠点. 假如 $t < {\rm{inf}} {\bf{T}}$ 而且 $\;\rho (t) = t$,则称 $t$ 是左稠点.

      定义 3[15] 设函数 $f:{\bf{T}} \to {\bf{R}}, t \in {\bf{T}}$,如果对于任意的 $\varepsilon > 0$,存在实数 $t$$t$ 的邻域 $U$,使得 $\left| {[f(\sigma (t)) - f(s)] - {f^\Delta }(t)[\sigma (t) - s]} \right| \leqslant \varepsilon \left| {\sigma (t) - s} \right|, \forall s \in U$,则 ${f^\Delta }$ 称作是 $f$$t$ 点的delta(或Hilger)导数.

      定义 4[15] 设 $f$${\bf{T}}$ 上的函数,如果 $f$ 在右稠点连续并在左稠点存在有限的左极限,则称 $f$ 为右稠连续的,所有右稠连续的函数集记为 ${C_{rd}}$. 如果 ${\rm{1 + }}\mu (t)f(t) \ne 0, t \in {\bf{T}}$,则称 $f$ 是正规的. 正规右稠连续函数集记为 $\mathcal{R}$, ${\mathcal{R}^ + }: = \{ f\left| {f \in \mathcal{R}, 1 + \mu (t)f(t) > 0\} .} \right.$

      引理1(Mawhin重合度理论)[9] 设 $X,Z$ 均为赋范线性空间,${\rm{DomL}} \subset X \to Z$ 是线性映射,$N:X \to Z$ 为连续映射,如果 $\dim {\rm{Ker}}L = {\rm{co}}\dim {\rm{Im}} L < {\rm{ + }}\infty $,且 ${\rm{Im}} L$$Z$ 中的闭子集,则称 $L$ 为指标为零的Fredholm映射. 如果 $L$ 是指标为零的Fredholm映射且存在连续投影 $P:X \to X$$Q:Z \to Z$,使得 ${\rm{Im}} P = {\rm{Ker}}L, $$ {\rm{Ker}}Q = {\rm{Im}} L, X = {\rm{Ker}}L \oplus {\rm{Ker}}P, Z = {\rm{Im}} L \oplus {\rm{Im}} Q,$$L$ 的限制 ${L_p}$${\rm{Dom}}L \cap {\rm{Ker}}P$${\rm{Im}} L$ 的一一映射,故 ${L_p}$ 可逆,并设其逆映射为 ${K_p}:{\rm{Im}} L \to {\rm{Dom}}L \cap {\rm{Ker}}P.$$\Omega $$X$ 中的有界开集,如果 $QN:\bar \Omega \to Z$${K_p}(I - Q)N:\bar \Omega \to X$ 都是紧的,则称 $N$$\bar \Omega $ 上是 $L - $ 紧的. 由于 ${\rm{Im}} Q$${\rm{Ker}}L$ 同构,故存在同构映射 $J:{\rm{Im}} Q \to L$.

      引理 2(Mawhin 延拓定理)[9] 设 $\Omega \subset X$ 均为一个有界开集,且 $L$ 为指标为零的Fredholm映射,且 $N:X \to Z$$\bar \Omega $ 是紧的. 假设:

      (a)对任意的 $ \lambda \in (\rm{0},\rm{1)}$,当 $x \in \partial {\rm{Dom}}L$ 时,$Lx \ne \lambda Nx$;

      (b)对任意 $x \in \partial \Omega \cap {\rm{Ker}}L$$QNx \ne 0$;

      (c)$\deg \{ JQN, \Omega \cap {\rm{Ker}}L, 0\} \ne 0$,则算子方程 $Lx = Nx$$\bar \Omega \cap {\rm{Dom}}L$ 内至少存在一个解.

      为叙述方便,引入记号

      $ \qquad \begin{split} \kappa = \min [0, + \infty ) \cap {{\bf{T}}},\;\;{I_\omega } = [\kappa , \kappa + \omega ] \cap {\bf{T}}, \bar f = \frac{1}{\omega }\int_\kappa ^{\kappa + \omega } {f(s)\Delta s} , {f^L} = \mathop {\min }\limits_{t \in {I_\omega }} f(t), {f^M} = \mathop {\max }\limits_{t \in {I_\omega }} f(t), \end{split} $

      其中 $f \in {C_{rd}}({\bf{T}})$ 且周期为 $\omega $.

      引理 3[9] 设 ${t_1},{t_2} \in {I_\omega }, t \in {{\bf{T}}}$,若 $g$${\bf{T}} \to {\bf{R}}$ 的周期为 $\omega $ 的函数,则

      $ \qquad g(t) \leqslant g({t_1}) + \int_\kappa ^{\kappa + \omega } {\left| {{g^\Delta }(s)} \right|} \Delta s, g(t) \geqslant g({t_2}) - \int_\kappa ^{\kappa + \omega } {\left| {{g^\Delta }(s)} \right|} \Delta s.$

    • 首先定义

      $ \qquad {l^\omega } = \{ ({u_1},{u_2}) \in ({\bf{T}},{{\bf{R}}^2}):{u_i}(t + \omega ) = {u_i}(t), i = 1,2, \forall t \in {\bf{T}}\} $

      $ \qquad \left\| u \right\| = \left\| {{{({u_1},{u_2})}^{\rm{T}}}} \right\| = \sum\limits_{i = 1}^2 {\max \left| {{u_i}(t)} \right|} , t \in {I_\omega },$

      所以 ${({u_1},{u_2})^{\rm{T}}} \in {l^\omega }$${l^\omega }$ 是Banach空间.令

      $ \qquad \begin{split} l_0^\omega &= \{ ({u_1},{u_2}) \in {l^\omega }: \overline {{u_i}} = 0,i = 1,2\} ,\\ l_c^\omega &= \{ ({u_1},{u_2}) \in {l^\omega }:({u_1}(t),{u_2}(t)) \in {{\bf{R}}^2},\forall t \in {\bf{T}}\} .\end{split}$

      $l_0^\omega $$l_c^\omega $ 都是$l^\omega $的闭线性子空间,${l^\omega } = l_0^\omega \oplus l_c^\omega , \dim l_c^\omega = 2.$

      定理 1 假设以下条件

      $ \qquad \overline {{c_2}} > \overline {{r_2}} ; c_2^M > r_2^L; \overline {{r_1}} > \overline {{c_1}} \exp \left(\ln \left[\frac{{ - \overline {{r_2}} + \overline {{c_2}} }}{{\overline {{b_2}} }}\right] + 2\overline {{c_2}} \omega \right)$

      成立,则系统(3)至少存在一个周期为 $\omega $ 的解.

      证明 令 $X = Z = {l^\omega } = \{ ({u_1},{u_2}) \in C({{\bf{T}}},{{\bf{R}}^2}):{u_i}(t + \omega ) = {u_i}(t),i = 1,2,\forall t \in {\bf{T}}\} $,且

      $ \qquad \left\| u \right\| = \left\| {{{({u_1},{u_2})}^{\rm{T}}}} \right\| = \sum\limits_{i = 1}^2 {\max \left| {{u_i}(t)} \right|} , t \in {I_\omega },$

      ${({u_1},{u_2})^{\rm{T}}} \in {l^\omega }$,所以 $X,Z$ 是赋予范数 $\left\| \cdot \right\|$ 的Banach空间. 定义

      $ \qquad \begin{split} N\left[ \begin{array}{l} {u_1}\\ {u_2} \end{array} \right] = &\left[ \begin{array}{l} {N_1}\\ {N_2} \end{array} \right] =\\ &\left[ \begin{array}{l} {r_1}(t) - {b_1}(t)\exp ({u_1}(t - {\tau _{\rm{1}}}(t))) - \dfrac{{{c_1}(t)\exp ({u_1}(t)){{(\exp ({u_2}(t)))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} + s(t)\exp ( - {u_1}(t))\\ - {r_2}(t) - {b_2}(t)\exp ({u_2}(t - {\tau _2}(t))) + \dfrac{{{c_2}(t)(\exp {({u_1}(t))})^2{{(\exp ({u_2}(t)))}^{m - 1}}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} - h(t)\exp ( - {u_2}(t)) \end{array} \right], \end{split} $

      $ \qquad L\left[ \begin{array}{l} {u_1}\\ {u_2} \end{array} \right] = \left[ \begin{array}{l} u_1^\Delta \\ u_2^\Delta \end{array} \right],P\left[ \begin{array}{l} {u_1}\\ {u_2} \end{array} \right] = \left[ \begin{array}{l} \overline {{u_1}} \\ \overline {{u_2}} \end{array} \right],u = \left[ \begin{array}{l} {u_1}\\ {u_2} \end{array} \right] \in X,Q\left[ \begin{array}{l} {z_1}\\ {z_2} \end{array} \right] = \left[ \begin{array}{l} \overline {{z_1}} \\ \overline {{z_2}} \end{array} \right], z = \left[ \begin{array}{l} {z_1}\\ {z_2} \end{array} \right] \in Z. $

      $Lu = {u^\Delta }, Pu = \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {u(t)} \Delta t, u \in X, Qz = \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {z(t)} \Delta t, z \in {\rm{Z}}$,所以

      $ \qquad {\rm{Ker}}L = l_c^\omega = \{ {({u_1}(t),{u_2}(t))^{\rm{T}}} \in X: {({u_1}(t),{u_2}(t))^{\rm{T}}} \in {{\bf{R}}^2}\} = {{\bf{R}}^2},$

      $ \qquad {\rm{Im}} L = l_0^\omega = \left\{ {({u_1},{u_2})^{\rm{T}}} \in X: \int_\kappa ^{\kappa + \omega } {u_i}(t)\Delta t = 0, i = 1,2\right\}, $

      从而

      $ \qquad \dim {\rm{Ker}}L = 2 = {\rm{co}}\dim L.$

      由于${\rm{Im}} L$$Z$ 中是闭的,所以 $L$ 是指标为零的Fredholm映射. 显然 $P,Q$ 是连续投影,且使得

      $ \qquad {\rm{Im}} P = {{\bf{R}}^2} = {\rm{Ker}}L, {\rm{Im}} L = {\rm{Ker}}Q = {\rm{Im}} (I - Q),$

      而且 $L$ 的逆映射 ${K_p}: {\rm{Im}} L \to {\rm{Ker}}P \cap {\rm{Dom}}L$ 是存在的,即

      $ \qquad {K_p}(z) = \int_\kappa ^t {z(s)\Delta s} - \frac{1}{\omega }\int_\kappa ^{\kappa + \omega } {\int_\kappa ^t {z(s)\Delta s\Delta t,} } $

      $ \qquad {K_p}\left[ \begin{array}{l} {u_1} \\ {u_2} \end{array} \right] = \left[ \begin{array}{l} \displaystyle\int_\kappa ^t {{N_1}(s)\Delta s - \dfrac{1}{\omega } \displaystyle\int_\kappa ^{\kappa + \omega } { \displaystyle\int_\kappa ^t {{N_1}\Delta s\Delta t} } } \\ \displaystyle\int_\kappa ^t {{N_2}(s)\Delta s - \dfrac{1}{\omega } \displaystyle\int_\kappa ^{\kappa + \omega } { \displaystyle\int_\kappa ^t {{N_2}\Delta s\Delta t} } } \end{array} \right],\;\;QN\left[ \begin{array}{l} {u_1} \\ {u_2} \end{array} \right] = \left[ \begin{array}{l} \dfrac{1}{\omega } \displaystyle\int_\kappa ^{\kappa + \omega } {{N_1}\Delta s} \\ \dfrac{1}{\omega } \displaystyle\int_\kappa ^{\kappa + \omega } {{N_2}\Delta s} \end{array} \right], $

      $ \qquad \begin{array}{l} {K_p}(I - Q)N\left[ \begin{array}{l} {u_1} \\ {u_2} \end{array} \right] = \\ \;\;\;\;\;\;\;\;\left[ \begin{array}{l} \displaystyle\int_\kappa ^t {{N_1}(s)\Delta s - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\displaystyle\int_\kappa ^t {{N_1}\Delta s\Delta t} - \left[t - \kappa - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } (t - \kappa )\Delta t\right]\overline {{N_1}} } } \\ \displaystyle\int_\kappa ^t {{N_2}(s)\Delta s - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\displaystyle\int_\kappa ^t {{N_2}\Delta s\Delta t - \left[t - \kappa - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } (t - \kappa )\Delta t\right]\overline {{N_2}} } } } \end{array} \right] = \\ \;\;\;\;\;\;\;\;\left[ \begin{array}{l} \displaystyle\int_\kappa ^t {N_1}(s)\Delta s - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\displaystyle\int_\kappa ^t {{N_1}\Delta s\Delta t} - \left(\dfrac{t}{\omega } - \dfrac{\kappa }{\omega } - \dfrac{1}{2}\right)\displaystyle\int_\kappa ^{\kappa + \omega } {{N_1}\Delta s} } \\ \displaystyle\int_\kappa ^t {N_2}(s)\Delta s - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\displaystyle\int_\kappa ^t {{N_2}\Delta s\Delta t - \left(\dfrac{t}{\omega } - \dfrac{\kappa }{\omega } - \dfrac{1}{2}\right)\displaystyle\int_\kappa ^{\kappa + \omega } {{N_2}\Delta s} } } \end{array} \right]. \end{array} $

      $\Omega $$X$ 中的有界开集,显然 $QX$${K_p}(I - Q)N$ 是连续的. 因为 $X$ 是Banach 空间,于是根据Arzela-Ascoli定理知 $\overline {{K_p}(I - Q)N(\overline \Omega )} $$\Omega $ 上是紧的,并且 $QN(\overline \Omega )$ 是有界的,所以 $N$$\overline \Omega $ 上是 $L - $ 紧的.

      考虑算子方程 $Lx = \lambda Nx$,即

      $ \qquad \left\{ \begin{array}{l} u_1^\Delta (t) = \lambda \left[{r_1}(t) - {b_1}(t)\exp ({u_1}(t - {\tau _{\rm{1}}}(t))) - \dfrac{{{c_1}(t)\exp ({u_1}(t)){{(\exp ({u_2}(t)))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} + s(t)\exp ( - {u_1}(t))\right], \\ u_{\rm{2}}^\Delta (t) = \lambda \left[ - {r_2}(t) - {b_2}(t)\exp ({u_2}(t - {\tau _2}(t))) + \dfrac{{{c_2}(t)\exp {{({u_1}(t))}^2}{{(\exp ({u_2}(t)))}^{m - 1}}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} - h(t)\exp ( - {u_2}(t))\right], \end{array} \right.$

      因为 ${\left[ {{u_1}(t), {u_2}(t)} \right]^{\rm{T}}} \in X$,所以存在 ${\xi _i}, {\eta _i} \in [\kappa , \kappa + \omega ] (i = 1,2)$ 使得

      $ \qquad {u_i}({\xi _i}) = \mathop {\min }\limits_{t \in {I_\omega }} {u_i}(t), {u_i}({\eta _i}) = \mathop {\max }\limits_{t \in {I_\omega }} {u_i}(t), i = 1,2.$

      对系统(7)的第2个方程从 $\kappa $$\kappa + \omega $ 积分,得

      $ \qquad \begin{array}{l} \displaystyle\int_\kappa ^{\kappa {\rm{ + }}\omega } \left[ - {r_2}(t) - {b_2}(t)\exp ({u_2}(t - {\tau _2}(t))) {\rm{ + }} \dfrac{{{c_2}(t)(\exp {({u_1}(t))})^2{{(\exp ({u_2}(t)))}^{m - 1}}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} - h(t)\exp ( - {u_2}(t))\right]\Delta t = 0 , \end{array} $

      所以,有

      $ \qquad \int_\kappa ^{\kappa {\rm{ + }}\omega } [ - {r_2}(t) - {b_2}(t)\exp ({u_2}(t - {\tau _2}(t))) + {c_2}(t)]\Delta t > 0 . $

      再由(10)式得

      $ \qquad \int_\kappa ^{\kappa {\rm{ + }}\omega } [ - {r_2}(t) - {b_2}(t)\exp ({u_2}({\xi _{\rm{2}}})) + {c_2}(t)]\Delta t > 0 . $

      所以,有

      $ \qquad \exp ({u_2}({\xi _2})) < \frac{{\overline {{c_2}} - \overline {{r_2}} }}{{\overline {{b_2}} }},$

      $ \qquad {u_2}({\xi _2}) < \ln \frac{{\overline {{c_2}} - \overline {{r_2}} }}{{\overline {{b_2}} }}.$

      又因为 $u_1^\Delta ({\xi _1}) = 0,$ 所以有

      $ \qquad {r_1}({\xi _{\rm{1}}}) - {b_1}({\xi _{\rm{1}}})\exp ({u_1}({\xi _{\rm{1}}} - {\tau _{\rm{1}}}({\xi _{\rm{1}}}))) - \frac{{{c_1}({\xi _{\rm{1}}})\exp ({u_1}({\xi _{\rm{1}}})){{(\exp ({u_2}({\xi _{\rm{1}}})))}^m}}}{{{{(\exp ({u_1}({\xi _{\rm{1}}})))}^2} + {k^2}}} +s({\xi _{\rm{1}}})\exp ( - {u_1}({\xi _{\rm{1}}})){\rm{ = 0}}, $

      于是 $r_1^M - b_1^L\exp ({u_1}({\xi _1})) + \dfrac{{{s^M}}}{{\exp ({u_1}({\xi _1}))}} > 0. $ 进一步计算,有

      $ \qquad \exp ({u_1}({\xi _1})) < \frac{{r_1^M + \sqrt {{{(r_1^M)}^2} + 4b_1^L{s^M}} }}{{2b_1^L}},$

      故得

      $ \qquad {u_1}({\xi _1}) < \ln \frac{{r_1^M + \sqrt {{{(r_1^M)}^2} + 4b_1^L{s^M}} }}{{2b_1^L}}.$

      又因为 $u_{\rm{2}}^\Delta ({\eta _{\rm{2}}}) = 0,$ 所以有

      $ \qquad \begin{array}{l} - {r_2}({\eta _{\rm{2}}}) - {b_2}({\eta _{\rm{2}}})\exp ({u_2}({\eta _{\rm{2}}} - {\tau _2}({\eta _{\rm{2}}}))) {\rm{ + }} \dfrac{{{c_2}({\eta _{\rm{2}}})\exp {{({u_1}({\eta _{\rm{2}}}))}^2}{{(\exp ({u_2}({\eta _{\rm{2}}})))}^{m - 1}}}}{{{{(\exp ({u_1}({\eta _{\rm{2}}})))}^2} + {k^2}}} - h({\eta _{\rm{2}}})\exp ( - {u_2}({\eta _{\rm{2}}})) = 0 . \end{array} $

      于是 $r_2^L + \dfrac{{{h^L}}}{{\exp ({u_2}({\eta _2}))}} < c_2^M$. 再进一步计算,得

      $ \qquad \exp ({u_2}({\eta _2})) > \frac{{{h^L}}}{{c_2^M - r_2^L}},$

      ${u_2}({\eta _2}) > \ln \dfrac{{{h^L}}}{{c_2^M - r_2^L}}.$ 所以

      $ \qquad \begin{split} \displaystyle\int_\kappa ^{\kappa {\rm{ + }}\omega } {\left| {u_{\rm{1}}^\Delta (t)} \right|} \Delta s =& \lambda \displaystyle\int_\kappa ^{\kappa + \omega } {\left| {{r_{\rm{1}}}(t)} \right.} - {b_{\rm{1}}}(t)\exp ({u_{\rm{1}}}(t - {\tau _{\rm{1}}}(t))){\rm{ + }} \dfrac{{{c_{\rm{1}}}(t)\exp ({u_1}(t)){{(\exp ({u_2}(t)))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}}+ {\rm{ s}}(t)\exp ( - {u_1}(t)\left. ) \right|\Delta t < \\ &2\left[\displaystyle\int_\kappa ^{\kappa + \omega } {{r_1}} (t)\Delta t + \displaystyle\int_\kappa ^{\kappa + \omega } s (t)\exp ( - {u_1}(t))\Delta t\right] = 2(\overline {{r_1}} + \overline s )\omega ,\\[-12pt] \end{split} $

      $ \qquad \begin{split} \int_\kappa ^{\kappa {\rm{ + }}\omega } {\left| {u_{\rm{2}}^\Delta (t)} \right|} \Delta s =& \lambda \int_\kappa ^{\kappa + \omega } {\left| { - {r_2}(t)} \right.} - {b_2}(t)\exp ({u_2}(t - {\tau _2}(t))) {\rm{ + }} \dfrac{{{c_2}(t)(\exp {({u_1}(t))})^2{{(\exp ({u_2}(t)))}^{m - 1}}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} - \\ &h(t)\exp ( - {u_2}(t)\left. ) \right|\Delta t < 2\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_2}(t)\exp {{({u_1}(t))}^2}{{(\exp ({u_2}(t)))}^{m - 1}}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}}} \Delta t = 2\overline {{c_2}} \omega . \end{split} $

      从而,有

      $ \qquad {u_2}(t) < {u_2}({\xi _2}) + \int_\kappa ^{\kappa + \omega } {\left| {u_2^\Delta (t)} \right|} \Delta t < \ln \frac{{ - \overline {{r_2}} + \overline {{c_2}} }}{{\overline {{b_2}} }} + 2\overline {{c_2}} \omega \buildrel \Delta \over = {B_1},$

      $ \qquad {u_2}(t) > {u_2}({\eta _2}) - \int_\kappa ^{\kappa + \omega } {\left| {u_2^\Delta (t)} \right|} \Delta t > \ln \frac{{{h^L}}}{{c_2^M - r_2^L}} - 2\overline {{c_2}} \omega \buildrel \Delta \over = {B_2}.$

      所以 $\mathop {\max }\limits_{t \in [\kappa ,\kappa + \omega ]} \left| {{u_2}(t)} \right| < \max \{ \left| {{B_1}} \right|,\left| {{B_2}} \right|\} \buildrel \Delta \over = {H_2}$,其中 ${H_2}$$\lambda $ 无关.

      对系统(7)的第1个方程从 $\kappa $$\kappa {\rm{ + }}\omega $ 积分,得

      $ \qquad \begin{array}{l} \displaystyle\int_\kappa ^{\kappa {\rm{ + }}\omega } {[{r_1}(t)} - {b_1}(t)\exp ({u_1}(t - {\tau _{\rm{1}}}(t))) - \dfrac{{{c_1}(t)\exp ({u_1}(t)){{(\exp ({u_2}(t)))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}} + s(t)\exp ( - {u_1}(t))]\Delta t = 0. \\ \end{array} $

      所以,有

      $ \qquad \begin{split} &\int_\kappa ^{\kappa {\rm{ + }}\omega } {[{r_1}(t)} - {b_1}(t)\exp ({u_1}({\eta _{\rm{1}}})) - {c_1}(t)\exp ({u_2}({\eta _2}))]\Delta t < 0,\\ &\exp ({u_1}({\eta _1})) > \frac{{\overline {{r_1}} - \overline {{c_1}} \exp ({u_2}({\eta _2}))}}{{\overline {{b_1}} }},\\ &{u_1}({\eta _1}) > \ln \frac{{\overline {{r_1}} - \overline {{c_1}} \exp ({u_2}({\eta _2}))}}{{\overline {{b_1}} }}. \end{split} $

      ${u_1}({\eta _1}) > \ln \dfrac{{\overline {{r_1}} - \overline {{c_1}} \exp ({B_1})}}{{\overline {{b_1}} }}. $ 由(13),(14)式得

      $ \qquad {u_{\rm{1}}}(t) < {u_{\rm{1}}}({\xi _{\rm{1}}}) + \int_\kappa ^{\kappa + \omega } {\left| {u_{\rm{1}}^\Delta (t)} \right|} \Delta t < \ln \frac{{r_1^M + \sqrt {{{(r_1^M)}^2} + 4b_1^L{s^M}} }}{{{{2b}}_1^L}} + 2(\overline {{r_1}} + \overline s )\omega \buildrel \Delta \over = {A_1}, $

      $ \qquad {u_{\rm{1}}}(t) > {u_1}({\eta _{\rm{1}}}) - \int_\kappa ^{\kappa + \omega } {\left| {u_{\rm{1}}^\Delta (t)} \right|} \Delta t > \ln \frac{{\overline {{r_1}} - \overline {{c_1}} \exp ({B_1})}}{{\overline {{b_1}} }} - 2(\overline {{r_1}} + \overline s )\omega \buildrel \Delta \over = {A_{\rm{2}}} .$

      如果常值向量 ${({u_1},{u_2})^{\rm{T}}}$ 是系统(1)的解,则

      $ \qquad QN\left[ \begin{array}{l} {u_1} \\ {u_2} \end{array} \right] = \left[ \begin{array}{l} \overline {{r_1}} - \overline {{b_1}} \exp ({u_1}) - \displaystyle\int_\kappa ^{\kappa + \omega } {\frac{{{c_1}(t)\exp ({u_1}){{(\exp ({u_2}))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}}\Delta t} + \overline s \exp ( - {u_1}) \\ - \overline {{r_2}} - \overline {{b_2}} \exp ({u_2}) + \displaystyle\int_\kappa ^{\kappa + \omega } {\frac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} - \overline h \exp ( - {u_2}) \end{array} \right],$

      考虑方程组

      $ \qquad \left\{ \begin{array}{l} \overline {{r_1}} - \overline {{b_1}} \exp ({u_1}) - \displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_1}(t)\exp ({u_1}){{(\exp ({u_2}))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}}\Delta t} + \overline s \exp ( - {u_1}){\rm{ = 0,}}\\ - \overline {{r_2}} - \overline {{b_2}} \exp ({u_2}) + \displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} - \overline h \exp ( - {u_2}){\rm{ = 0,}} \end{array} \right. $

      ${H_0}$ 充分大,使得当 $H = {H_0} + {H_1} + {H_2}$

      $ \qquad \left\| u \right\| = \left\| {{{({u_1},{u_2})}^{\rm{T}}}} \right\| = \mathop {\max }\limits_{} \{ \left| {{u_1}} \right|,\left| {{u_2}} \right|\} < H $

      成立.

      $\Omega {\rm{ = \{ }}u(t)\left\| u \right\| = \left\| {{{({u_1},{u_2})}^{\rm{T}}}} \right\| = \mathop {\max }\limits_{} \{ \left| {{u_1}} \right|,\left| {{u_2}} \right|\} < H .$

      $u(t) = {({u_1}(t),{u_2}(t))^{\rm{T}}} \in \partial \Omega \cap {\rm{Ker}}L, u(t)$${{\bf{R}}^2}$ 中的常向量且 $\left\| {{{({u_1},{u_2})}^T}} \right\| = H .$

      若系统(22)至少有一个解,则有

      $ \qquad \begin{array}{l} QN\left[ \begin{array}{l} {u_1} \\ {u_2} \end{array} \right] = \left[ \begin{array}{l} \overline {{r_1}} - \overline {{b_1}} \exp ({u_1}) - \displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_1}(t)\exp ({u_1}){{(\exp ({u_2}))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}}\Delta t} + \overline s \exp ( - {u_1})\\ - \overline {{r_2}} - \overline {{b_2}} \exp ({u_2}) + \displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} - \overline h \exp ( - {u_2}) \end{array} \right] \ne \left[ \begin{array}{l} 0\\ 0 \end{array} \right], \end{array}$

      若系统(3)无解,显然也有

      $ \qquad QN\left[ \begin{array}{l} {u_1} \\ {u_2} \end{array} \right] \ne \left[ \begin{array}{l} 0 \\ 0 \end{array} \right].$

      定义映射

      $ \qquad \begin{split} \phi ({u_1},{u_2},\mu ) =& \left[ \begin{array}{l} \overline {{r_1}} - \overline {{b_1}} \exp ({u_1})\\ - \overline {{b_2}} \exp ({u_2}) + \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} \end{array} \right] {\rm{ + }} \\ &\mu \left[ \begin{array}{l} \overline {{r_1}} - \overline {{b_1}} \exp ({u_1}) - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_1}(t)\exp ({u_1}){{(\exp ({u_2}))}^m}}}{{{{(\exp ({u_1}(t)))}^2} + {k^2}}}\Delta t} + \overline s \exp ( - {u_1})\\ -\overline {{r_2}} - \overline h \exp ( - {u_2}) \end{array} \right],\mu \in [0,1]. \end{split} $

      ${({u_1}(t),{u_2}(t))^{\rm{T}}} \in \partial \Omega \cap {\rm{Ker}}L{\rm{ = }}\partial \Omega \cap {{\bf{R}}^2}$ 时,${({u_1},{u_2})^{\rm{T}}}$${\bf{R}^2}$ 中的常向量且 $\left\| {{{({u_1},{u_2})}^{\rm{T}}}} \right\| = H .$ 易知 $\phi ({u_1},{u_2},\mu ) \ne $$ 0$,且代数方程

      $ \qquad \left\{ \begin{array}{l} \overline {{r_1}} - \overline {{b_1}} \exp ({u_1}) = 0, \\ - \overline {{b_2}} \exp ({u_2}) + \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\frac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} = 0 \end{array} \right.$

      有唯一解 ${({u_1}(t),{u_2}(t))^{\rm{T}}}$,因为 ${\rm{Im}} Q = {\rm{Ker}}L$,取 $J = I$,由拓扑度的同伦不变性得

      $ \qquad \deg \{ JQN{({u_1},{u_2})^{\rm{T}}}; \Omega \cap {\rm{Ker}}L; {(0,0)^{\rm{T}}}\}= \deg \{ \phi ({u_1},{u_2},1); \Omega \cap {\rm{Ker}}L; {(0,0)^{\rm{T}}}\} =\deg \{ \phi ({u_1},{u_2},0); \Omega \cap {\rm{Ker}}L; {(0,0)^{\rm{T}}}\} = $

      $ \qquad \deg \{ {(\overline {{r_1}} - \overline {{b_1}} \exp ({u_1}), - \overline {{b_2}} \exp ({u_2}) + \frac{1}{\omega }\int_\kappa ^{\kappa + \omega } {\frac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} )^{\rm{T}}};\Omega \cap KerL; {(0,0)^T}\} = $

      $ \qquad \begin{split} &\;\;\;\;{\rm{sign}}\;\det \left[ \begin{array}{l} - \overline {{b_1}} \exp ({u_1}) \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;0 \\ \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} \qquad - \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{(m - 1){c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} \end{array}\right] =\\ &\;\;\;\;{\rm{sign\; det}}\left(\dfrac{{\overline {{b_1}} \exp ({u_1})}}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{(m - 1){c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} \right) \ne 0. \end{split} $

      至此,我们已经证明了 $\Omega $ 满足引理(1)的所有条件,所以方程 $Lx = Nx$, ${\rm{Dom}}L \cap \overline \Omega $ 内至少存在一个解,即系统(3)在 ${\rm{Dom}}L \cap \overline \Omega $ 至少存在一个周期为 $\omega $ 解.

      更进一步,因为 $ {({u_1}(t),{u_2}(t))^{\rm{T}}} \in \partial \Omega \cap {\rm{Ker}}L,$ 是方程组

      $ \qquad \left\{ \begin{array}{l} \overline {{r_1}} - \overline {{b_1}} \exp ({u_1}) = 0,\\ - \overline {{b_2}} \exp ({u_2}) + \dfrac{1}{\omega }\displaystyle\int_\kappa ^{\kappa + \omega } {\dfrac{{{c_2}(t)(\exp {({u_1})})^2{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}\Delta t} = 0 \end{array} \right.$

      的解且满足:

      $ \qquad \exp ({u_1}) = \frac{{\overline {{r_1}} }}{{\overline {{b_1}} }}, \exp ({u_2}) = \frac{1}{{\overline {{b_2}} \omega }}\int_\kappa ^{\kappa + \omega } {\frac{{{c_2}{{(\exp ({u_1}))}^2}{{(\exp ({u_2}))}^{m - 1}}}}{{{{(\exp ({u_1}))}^2} + {k^2}}}} \Delta t > 0.$

      所以,系统(3)至少存在一个周期为 $\omega $ 的解 $\exp ({u_1}), \exp ({u_2})$. 证毕.

    • 本文研究时标上具有捕获率和投放率的 Holling-Ⅲ 型捕食系统,考虑了时滞效应,基于时标Mawhin 重合度理论方法,得到了系统至少有一个正周期解.

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