冬眠也叫“冬蛰”.某些动物在冬季时生命活动处于极度降低的状态, 是这些动物对冬季外界不良环境条件的一种适应^{[1, 2]}.

然而, 随着社会的发展, 环境污染也变得越来越严重, 由于环境毒素的存在对种群的生存存在严重的危害, 而由于天气环境的变化、污染源对环境脉冲排放环境毒素等造成了环境生物多样性的减少^{[3, 4]}.近年来, 多生物数学家^[5-10]对污染环境下的数学建模产生浓厚的兴趣.同时也有许多学者对种群的阶段结构有所研究^[11-13].

基于上述讨论, 本文讨论污染环境下具冬眠特征的阶段结构单种群动力学模型, 分析其动力学行为, 为污染环境下具冬眠特征的生物资源管理提供决策支持.

常微分方程描述的阶段结构种群模型.

$ \begin{equation} \label{eq:1} \left\{\begin{aligned} \frac{dx(t)}{dt} &= by(t)-(c+d_{1})x(t), \\ \frac{dy(t)}{dt}&=cx(t)-d_{2}y(t), \end{aligned} \right. \end{equation} $

(2.1)

其中$x(t)$表示种群幼体在时刻$t$的密度, $y(t)$表示种群成体在时刻$t$的密度, $b>0$表示种群出生率系数, $c>0$表示种群幼体向种群成体转化的系数, $d_{1}>0$表示种群幼体的死亡系数, $d_{2}>0$表示种群成体的死亡系数.

考虑到种群的脉冲出生, 那么上述模型可修改为

$ \begin{equation} \left\{\begin{aligned} \frac{dx(t)}{dt} &= -(c+d_{1})x(t), t\neq n\tau, \\ \frac{dy(t)}{dt}&=cx(t)-d_{2}y(t), t\neq n\tau, \\ \triangle x(t)&=y(t)(a-bx(t)), t=n\tau, \\ \triangle y(t)&=0, t=n\tau, \end{aligned} \right. \end{equation} $

(2.2)

其中$x(t)$表示种群幼体在时刻$t$的密度, $y(t)$表示种群成体在时刻$t$的密度, $c>0$表示种群幼体向种群成体转化的系数, $d_{1}>0$表示种群幼体的死亡系数, $d_{2}>0$表示种群成体的死亡系数, $a>0$表示种群脉冲出生系数, $b>0$表示种群脉冲出生的种内竞争系数, $\tau>0$表示种群脉冲出生周期.

而污染环境下的毒素具脉冲输入的单种群动力学模型

$ \begin{equation} \left\{\begin{aligned} \frac{dx(t)}{dt} &= x(t)(a-bx(t))-\beta c_{e}(t)x(t), t\neq n\tau, \\ \frac{dc_{e}(t)}{dt}&=-gc_{e}(t), t\neq n\tau, \\ \triangle x(t)&=0, t=n\tau, \\ \triangle c_{e}(t)&=\mu, t=n\tau, \end{aligned} \right. \end{equation} $

(2.3)

其中$x(t)$表示种群在时刻$t$的密度, $c_{e}(t)$表示环境毒素在时刻$t$的浓度, $a>0$表示种群内禀增长率系数, $b>0$表示种内竞争系数. $\beta>0$表示种群受毒素的影响造成种群的死亡系数, $g>0$表示毒素受阳光等生化反应作用的影响的消耗系数, $\mu>0$表示环境变化等影响下毒素输入种群生活环境的浓度量, $\tau>0$表示毒素脉冲输入环境的周期.

考虑到具冬眠特征种群, 建立毒素具脉冲输入与种群脉冲出生的切换阶段结构单种群阶段结构动力学模型

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \frac{dx(t)}{dt}=-(c+d_{1})x(t)-\beta c_{e}(t)x(t), \\ \displaystyle \frac{dy(t)}{dt}=cx(t)-d_{1}y(t)-\beta c_{e}(t)y(t), \\ \displaystyle \frac{dc_{e}(t)}{dt}=-h_{1}c_{e}(t), \\ \end{array} \right\} t\in( n\tau, (n+l)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x(t)=y(t)(a-by(t)), \\ \displaystyle \triangle y(t)=0, \\ \displaystyle \triangle c_{e}(t)=0 , \\ \end{array} \right\} t= (n+l)\tau, n= 1, 2, \cdots, \\ \left. \begin{array}{llc} \displaystyle \frac{dx(t)}{dt}=-d_{2}x(t), \\ \displaystyle \frac{dy(t)}{dt}=-d_{2}y(t), \\ \displaystyle \frac{dc_{e}(t)}{dt}=-h_{2}c_{e}(t), \\ \end{array} \right\} t\in( (n+l)\tau, (n+1)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x(t)=0, \\ \displaystyle \triangle y(t)=0, \\ \displaystyle \triangle c_{e}(t)=\mu , \\ \end{array} \right\} t= (n+1)\tau, n= 1, 2, \cdots, \end{array} \right.\label{a} \end{equation} $

(.24)

其中$x(t)$表示种群幼体在时刻$t$的密度, $y(t)$表示种群成体在时刻的密度, $c_{e}(t)$表示环境毒素在时刻$t$的浓度, $c>0$表示种群幼体转化为成体的转化率, $d_{1}>0$表示种群幼体在非冬眠期的死亡系数, 也表示种群成体在非冬眠期的死亡系数, $\beta>0$表示种群幼体在非冬眠期由于环境毒素而引发的死亡系数，也表示种群成体在非冬眠期由于环境毒素而引发的死亡系数, $h_{1}>0$表示在种群非冬眠期环境毒素因在环境生化作用消耗系数, $a>0$表示种群成体脉冲出生的出生系数, $b>0$表示种群成体出生的种内竞争系数, $d_{2}$表示种群幼体在冬眠期的死亡系数, 也表示种群成体在冬眠期的死亡系数, $h_{2}>0$表示在冬眠期环境毒素因在环境生化作用消耗系数, $\mu>0$表示环境毒素浓度的脉冲输入, $\tau>0$表示种群的脉冲出生周期.

方程组$(2.4)$右边的函数$f=(f_{1}, f_{2}, f_{3})$, 方程组$(2.4)$的解$z:R_{+}\rightarrow R_{+}^{3}$是一个分段连续的函数, $z(t)=(x(t), y(t), c_{e}(t))^{T}$, $R_{+}=[0, +\infty)$, $R_{+}^{3}=\{z\in R^{3}|z>0\}$, $z(t)$在$(n\tau, (n+l)\tau]\times R_{+}^{3}$是连续的, 在$((n+l)\tau, (n+1)\tau]\times R_{+}^{3}$也是连续的.根据文献^[9], $f$的光滑性保证了方程组(2.4)的解的全局存在性和唯一性.

设$V:R_{+}\times R_{+}^{3}\rightarrow R_{+}$, 那么$V$属于$V_{0}$, 如果

Ⅰ)$V$在$(n\tau, (n+l)\tau]\times R_{+}^{3}$对每个$z\in R_{+}^{3} $是连续的, 且

$ \lim\limits_{(t, u)\rightarrow((n+l)\tau^{+}, z)}V(t, u)=V((n+l)\tau^{+}, z) $

与

$ \lim\limits_{(t, u)\rightarrow((n+1)\tau^{+}, z)}V(t, u)=V((n+1)\tau^{+}, z) $

存在;

Ⅱ) $V$在$z$是局部李普希茨的.

引理1 考虑系统$(2.4)$的子系统

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \frac{dc_{e}(t)}{dt} = -h_{1} c_{e}(t), t\in( n\tau, (n+l)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \triangle c_{e}(t)=0, t=(n+l)\tau, \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \frac{dc_{e}(t)}{dt}&=-h_{2}c_{e}(t), t\in( (n+l)\tau, (n+1)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \triangle c_{e}(t)=\mu, t=(n+1)\tau, \\ \end{array} \right. \end{array} \right. \end{equation} $

(3.1)

则$(3.1)$式存在全局渐近稳定的周期解

$ \begin{equation} \overline{c_{e}(t)}=\left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle c^{\ast}_{e}e^{-h_{1}(t-n\tau)}, t\in( n\tau, (n+l)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle c^{\ast}_{e}e^{-h_{1}l\tau-h_{2}(t-(n+l)\tau)}, t\in( (n+l)\tau, (n+1)\tau], \end{array} \right. \end{array} \right. \end{equation} $

(3.2)

其中

$ c^{\ast}_{e}=\frac{\mu}{1-e^{-h_{1}l\tau-h_{2}(1-l)\tau}}. $

证 系统$(3.1)$的第一个方程在$(n\tau, (n+l)\tau]$上积分可得

$ \begin{equation} \left. \begin{array}{llc} \left. \begin{array}{llc} \displaystyle c_{e}(t)=c_{e}(n\tau^{+})e^{-h_{1}(t-n\tau)}, t\in( n\tau, (n+l)\tau]. \end{array} \right. \end{array} \right. \end{equation} $

(3.3)

系统$(3.1)$的第三个方程在$((n+l)\tau, (n+1)\tau]$上积分可得

$ \begin{equation} \left. \begin{array}{llc} \left. \begin{array}{llc} \displaystyle c_{e}(t)=c_{e}((n+l)\tau^{+})e^{-h_{2}(t-(n+l)\tau)}, t\in( (n+l)\tau, (n+1)\tau].\\[6pt] \end{array} \right. \end{array} \right. \end{equation} $

(3.4)

考虑到系统$(3.1)$的脉冲效应可得到频闪映射

$ \begin{equation} \left. \begin{array}{llc} \left. \begin{array}{llc} \displaystyle c_{e}((n+1)\tau^{+})=c_{e}(n\tau^{+})e^{-[h_{1}l+h_{2}(1-l)]\tau}+\mu.\\[6pt] \end{array} \right. \end{array} \right. \end{equation} $

(3.5)

于是容易得到$(3.5)$式的唯一不动点

$ c^{\ast}_{e}=\frac{\mu}{1-e^{-h_{1}l\tau-h_{2}(1-l)\tau}}. $

做记号$c_{e}^{n+1}=c_{e}((n+1)\tau).$令

$ \begin{equation} \left. \begin{array}{llc} \left. \begin{array}{llc} \displaystyle f(x)=xe^{-[h_{1}l+h_{2}(1-l)]\tau}+\mu. \end{array} \right. \end{array} \right. \end{equation} $

(3.6)

由$(3.5)$式可以得

$ \begin{equation} \left. \begin{array}{llc} \left. \begin{array}{llc} \displaystyle c_{e}^{n+1}=f(c_{e}^{n}). \end{array} \right. \end{array} \right. \end{equation} $

(3.7)

对$(3.7)$式求导可得

$ \begin{equation} \left. \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \frac{dc_{e}^{n+1}}{dc_{e}^{n}}|_{c^{n}_{e}=c^{\ast}_{e}}=e^{-h_{1}l\tau-h_{2}(1-l)\tau}<1. \end{array} \right. \end{array} \right. \end{equation} $

(3.8)

所以系统$(3.5)$唯一的不动点

$ c^{\ast}_{e}=\frac{\mu}{1-e^{-h_{1}l\tau-h_{2}(1-l)\tau}}, $

是局部稳定的, 从而是全局渐近稳定的.与参考文献^[13]同理, 可知(3.5)式存在全局渐近稳定的周期解

$ \begin{equation} \overline{c_{e}(t)}=\left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle c^{\ast}_{e}e^{-h_{1}(t-n\tau)}, t\in( n\tau, (n+l)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle c^{\ast}_{e}e^{-h_{1}l\tau-h_{2}(t-(n+l)\tau)}, t\in( (n+l)\tau, (n+1)\tau]. \end{array} \right. \end{array} \right. \end{equation} $

(3.9)

注1 由引理1可知, 对任意的$\varepsilon>0$有

$ (c_{e}^{\ast}e^{-h_{1}l\tau}+c_{e}^{\ast}e^{-h_{1}l\tau-h_{2}(1-l)\tau})-\varepsilon\leq c_{e}(t)\leq (c_{e}^{\ast}+c_{e}^{\ast}e^{-h_{1}l\tau})+\varepsilon. $

考虑系统$(2.4)$的子系统

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \frac{dx(t)}{dt}=-(c+d_{1})x(t), \\ \displaystyle \frac{dy(t)}{dt}=cx(t)-d_{1}y(t), \\ \end{array} \right\} t\in( n\tau, (n+l)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x(t)=y(t)(a-by(t)), \\[6pt] \displaystyle \triangle y(t)=0, \\[6pt] \end{array} \right\} t= (n+l)\tau, n= 1, 2, \cdots, \\ \left. \begin{array}{llc} \displaystyle \frac{dx(t)}{dt}=-d_{2}x(t), \\ \displaystyle \frac{dy(t)}{dt}=-d_{2}y(t), \\ \end{array} \right\} t\in( (n+l)\tau, (n+1)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x(t)=0, \\ \displaystyle \triangle y(t)=0, \\ \end{array} \right\} t= (n+1)\tau, n= 1, 2, \cdots. \end{array} \right. \end{equation} $

(3.10)

由系统$(3.10)$的第一个与第二个方程, 容易得到脉冲点之间的解析解为

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x(t)= \end{array} \right. \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x(n\tau^{+})e^{-(c+d_{1})(t-n\tau)}, t\in(n\tau, (n+l)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle x((n+l)\tau^{+})e^{-d_{2}(t-(n+l)\tau)}, t\in((n+l)\tau, (n+1)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle y(t)= \end{array} \right. \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle (1-e^{-c(t-n\tau)})x(n\tau^{+})e^{-d_{1}(t-n\tau)}-y(n\tau^{+})e^{-d_{1}(t-n\tau)}, t\in(n\tau, (n+l)\tau], \\ \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle y((n+l)\tau^{+})e^{-d_{2}(t-(n+l)\tau)}, t\in((n+l)\tau, (n+1)\tau]. \end{array} \right. \end{array} \right. \end{equation} $

(3.11)

考虑到系统$(3.10)$的第三、第四个方程与第七、第八个方程的脉冲效应, 得到系统$(3.10)$的频闪映射

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x((n+1)\tau^{+})=Ax(n\tau^{+})+Bx(n\tau^{+})-b[Cx(n\tau^{+})+Dy(n\tau^{+})]^{2}e^{-d_{2}(1-l)\tau}, \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle y((n+1)\tau^{+})=Ex(n\tau^{+})+Fy(n\tau^{+}), \end{array} \right. \end{array} \right. \end{equation} $

(3.12)

其中

$ \begin{eqnarray*}&& A=[e^{-(c+d_{1})l\tau}+a(1-e^{cl\tau})e^{-d_{1}l\tau}]e^{-d_{2}(1-l)\tau}, \\ && B=ae^{-[d_{1}l+d_{2}(1-l)]\tau}, \\ && C=(1-e^{-cl\tau})e^{-d_{1}l\tau}<1, \\ && D=e^{-d_{1}l\tau}<1, \\ && E=(1-e^{-cl\tau})e^{-[d_{1}l+d_{2}(1-l)]\tau}<1, \\ && F=e^{-[d_{1}l+d_{2}(1-l)]\tau}<1.\end{eqnarray*} $

计算$(3.12)$式, 可得到两个不动点$P_{1}(0, 0)$与$P_{2}(x^{\ast}, y^{\ast})$, 其中

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x^{\ast}=\frac{(1-F)[(A-1)(1-F)+BE]}{b[C(1-F)+DE]^{2}}\times e^{-d_{2}(1-l)\tau}, (A-1)(1-F)+BE>0, \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle y^{\ast}=\frac{E[(A-1)(1-F)+BE]}{b[C(1-F)+DE]^{2}}\times e^{-d_{2}(1-l)\tau}, (A-1)(1-F)+BE>0. \end{array} \right. \end{array} \right. \end{equation} $

(3.13)

定理2 (ⅰ)如果$(A-1)(1-F)+BE < 0, $那么不动点$P_{1}(0, 0)$是全局渐近稳定的;

(ⅱ)如果$(A-1)(1-F)+BE>0, $那么不动点$P_{2}(x^{\ast}, y^{\ast})$是全局渐近稳定的, 其中$x^{\ast}$和$y^{\ast}$按照$(3.13)$式定义.

证 为了方便计算, 做记号$(x^{n+1}, y^{n+1})=(x((n+1)\tau^{+}), y((n+1)\tau^{+})), $那么差分方程$(3.12)$线性化后可以写为

$ \begin{equation} \left. \begin{array}{llc} \displaystyle \left( \begin{array}{c} x^{n+1} \\ y^{n+1}\\ \end{array} \right) = M \left( \begin{array}{c} x^{n} \\[6pt] y^{n}\\[6pt] \end{array} \right). \end{array} \right. \end{equation} $

(3.14)

显然系统$(3.12)$的不动点$P_{1}(0, 0)$与$P_{2}(x^{\ast}, y^{\ast})$的附近的动力学性质由其线性系统$(3.14)$决定, $M$作为线性系统$(3.14)$的矩阵, 不动点$P_{1}(0, 0)$与$P_{2}(x^{\ast}, y^{\ast})$的稳定性由$M$的特征值小于$1$来决定.当$M$满足如下的Jury判据条件时, 可知$M$的特征值小于1^[13],

$ \begin{equation} 1-{\hbox{tr}}M+{\hbox{det}}M>0. \end{equation} $

(3.15)

(ⅰ)当$(A-1)(1-F)+BE < 0$时, 显然$P_{1}(0, 0)$是系统$(3.12)$唯一的不动点, 于是得到

$ \begin{equation} \left. \begin{array}{llc} \displaystyle M= \left( \begin{array}{cc} A&B \\ E&F\\ \end{array} \right). \end{array} \right. \end{equation} $

(3.16)

由Jury判据条件

$ \begin{eqnarray*} &&1-{\hbox{tr}}M+{\hbox{det}}M=1-(A+F)+(AF-BE)\\ &=&(1-A)(1-F)-BE=-[(A-1)(1-F)+BE]>0.\end{eqnarray*} $

所以系统$(3.12)$唯一的不动点$P_{1}(0, 0)$是局部稳定的, 从而是全局渐近稳定的.

(ⅱ)当$(A-1)(1-F)+BE>0$时, 显然$P_{1}(0, 0)$与$P_{2}(x^{\ast}, y^{\ast})$是系统$(3.12)$的不动点.由(ⅰ)的证明过程容易知道系统$(3.12)$的不动点$P_{1}(0, 0)$是不稳定的.考虑不动点$P_{2}(x^{\ast}, y^{\ast})$, 得到

$ \begin{equation} \left. \begin{array}{llc} \displaystyle M= \left( \begin{array}{cc} A-2bC(Cx^{\ast}+Dy^{\ast})&B-2bD(Cx^{\ast}+Dy^{\ast}) \\ E&F\\ \end{array} \right). \end{array} \right. \end{equation} $

(3.17)

于是

$ \begin{eqnarray*}&& 1-{\hbox{tr}}M+{\hbox{det}}M=1-[(A-2bC(Cx^{\ast}+Dy^{\ast})+F])\\ &&+[A-2bC(Cx^{\ast}+Dy^{\ast})]F-[B-2bD(Cx^{\ast}+Dy^{\ast})]E\\ &=&\frac{(A-1)(1-F)+BE}{b[C(1-F)+DE]^{2}}>0.\end{eqnarray*} $

由Jury判据, 所以系统$(3.12)$的不动点$P_{2}(x^{\ast}, y^{\ast})$是局部稳定的, 从而是全局渐近稳定的.

类似参考文献[13], 可以得到如下的引理.

定理3 (ⅰ)如果$(A-1)(1-F)+BE < 0, $那么系统$(3.12)$的平凡周期解$(0, 0)$是全局渐近稳定的;

(ⅱ)如果$(A-1)(1-F)+BE>0, $那么系统$(3.12)$的正周期解$(\widetilde{x(t)}, \widetilde{y(t)})$是全局渐近稳定的, 其中

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \widetilde{x(t)}= \end{array} \right. \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x^{\ast}e^{-(c+d_{1})(t-n\tau)}, t\in(n\tau, (n+l)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle x^{\ast\ast}e^{-d_{2}(t-(n+l)\tau)}, t\in((n+l)\tau, (n+1)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \widetilde{y(t)}= \end{array} \right. \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle (1-e^{-c(t-n\tau)})x^{\ast}e^{-d_{1}(t-n\tau)}-y^{\ast}e^{-d_{1}(t-n\tau)}, t\in(n\tau, (n+l)\tau], \\ \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle y^{\ast\ast}e^{-d_{2}(t-(n+l)\tau)}, t\in((n+l)\tau, (n+1)\tau], \end{array} \right. \end{array} \right. \end{equation} $

(3.18)

而

$ x^{\ast\ast}=(x^{\ast}+y^{\ast})e^{-d_{1}l\tau}+x^{\ast}e^{-(c+d_{1})l\tau}, y^{\ast\ast}=(x^{\ast})(1-e^{-cl\tau})e^{-d_{1}l\tau}+y^{\ast})e^{-d_{1}l\tau}, $

且$x^{\ast}$与$y^{\ast}$如$(3.13)$式所定义.

注2 由引理1与定理3可知, 当$(A-1)(1-F)+BE>0, $存在$M>0$使得$x(t) < M, y(t) < M, c_{o}(t) < M.$

考虑到

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \frac{dx(t)}{dt}\geq-(c+d_{1})x(t)-\beta(\overline{c_{e}(t)}+\varepsilon)x(t), \\ \displaystyle \frac{dy(t)}{dt}\geq cx(t)-d_{1}y(t)-\beta(\overline{c_{e}(t)}+\varepsilon)y(t), \\ \end{array} \right\} t\in( n\tau, (n+l)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x(t)=y(t)(a-by(t)), \\[6pt] \displaystyle \triangle y(t)=0, \\[6pt] \end{array} \right\} t= (n+l)\tau, n= 1, 2, \cdots, \\ \left. \begin{array}{llc} \displaystyle \frac{dx(t)}{dt}=-d_{2}x(t), \\[6pt] \displaystyle \frac{dy(t)}{dt}=-d_{2}y(t), \\[6pt] \end{array} \right\} t\in( (n+l)\tau, (n+1)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x(t)=0, \\[6pt] \displaystyle \triangle y(t)=0, \\[6pt] \end{array} \right\} t= (n+1)\tau, n= 1, 2, \cdots. \end{array} \right. \end{equation} $

(3.19)

容易得到系统$(3.19)$的比较方程

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \frac{dx_{1}(t)}{dt}=-(c+d_{1})x_{1}(t)-\beta(\overline{c_{e}(t)}+\varepsilon)x_{1}(t), \\ \displaystyle \frac{dy_{1}(t)}{dt}= cx_{1}(t)-d_{1}y_{1}(t)-\beta(\overline{c_{e}(t)}+\varepsilon)y_{1}(t), \\ \end{array} \right\} t\in( n\tau, (n+l)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x_{1}(t)=y_{1}(t)(a-by_{1}(t)), \\ \displaystyle \triangle y(t)=0, \\[6pt] \end{array} \right\} t= (n+l)\tau, n= 1, 2, \cdots, \\ \left. \begin{array}{llc} \displaystyle \frac{dx_{1}(t)}{dt}=-d_{2}x_{1}(t), \\[6pt] \displaystyle \frac{dy_{1}(t)}{dt}=-d_{2}y_{1}(t), \\[6pt] \end{array} \right\} t\in( (n+l)\tau, (n+1)\tau], \\ \left. \begin{array}{llc} \displaystyle \triangle x_{1}(t)=0, \\ \displaystyle \triangle y_{1}(t)=0, \\ \end{array} \right\} t= (n+1)\tau, n= 1, 2, \cdots. \end{array} \right. \end{equation} $

(3.20)

并考虑到$(2.4)$式的子系统$(3.10).$

从而容易得到与$(3.20)$式的频闪映射

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x_{1}((n+1)\tau^{+})=A_{1}x_{1}(n\tau^{+})+B_{1}x_{1}(n\tau^{+})-b[C_{1}x_{1}(n\tau^{+})+D_{1}y_{1}(n\tau^{+})]^{2}e^{-d_{2}(1-l)\tau}, \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle y_{1}((n+1)\tau^{+})=E_{1}x_{1}(n\tau^{+})+F_{1}y_{1}(n\tau^{+}), \end{array} \right. \end{array} \right. \end{equation} $

(3.21)

其中

$ \begin{eqnarray*}&& A_{1}=[e^{-[c+(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})]l\tau}+a(1-e^{cl\tau})e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l\tau}]e^{-d_{2}(1-l)\tau}, \\ && B_{1}=ae^{-[(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l+d_{2}(1-l)]\tau}, \\ && C_{1}=(1-e^{-cl\tau})e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l\tau}<1, \\ && D_{1}=e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l\tau}<1, \\ && E_{1}=(1-e^{-cl\tau})e^{-[(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l+d_{2}(1-l)]\tau}<1, \\ && F_{1}=e^{-[(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l+d_{2}(1-l)]\tau}<1.\end{eqnarray*} $

计算$(3.21)$式可得到两个不动点$Q_{1}(0, 0)$与$Q_{2}(x^{\ast}, y^{\ast})$, 其中

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x_{1}^{\ast}=\frac{(1-F_{1})[(A_{1}-1)(1-F_{1})+B_{1}E_{1}]}{b[C_{1}(1-F_{1})+D_{1}E_{1}]^{2}}\times e^{-d_{2}(1-l)\tau}, (A_{1}-1)(1-F_{1})+B_{1}E_{1}>0, \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle y_{1}^{\ast}=\frac{E_{1}[(A_{1}-1)(1-F_{1})+B_{1}E_{1}]}{b[C_{1}(1-F_{1})+D_{1}E_{1}]^{2}}\times e^{-d_{2}(1-l)\tau}, (A_{1}-1)(1-F_{1})+B_{1}E_{1}>0. \end{array} \right. \end{array} \right. \end{equation} $

(3.22)

于是类似定理1与定理3可得

定理4 (ⅰ)如果$(A_{1}-1)(1-F_{1})+B_{1}E_{1} < 0, $那么不动点$Q_{1}(0, 0)$是全局渐近稳定的;

(ⅱ)如果$(A_{1}-1)(1-F_{1})+B_{1}E_{1}>0, $那么不动点$Q_{2}(x_{1}^{\ast}, y_{1}^{\ast})$是全局渐近稳定的, 其中$x_{1}^{\ast}$和$y_{1}^{\ast}$按照$(3.22)$式定义.

定理5 (ⅰ)如果$(A_{1}-1)(1-F_{1})+B_{1}E_{1} < 0, $那么系统$(3.20)$的平凡周期解$(0, 0)$是全局渐近稳定的;

(ⅱ)如果$(A_{1}-1)(1-F_{1})+B_{1}E_{1}>0, $那么系统$(3.20)$的正周期解$(\widetilde{x_{1}(t)}, \widetilde{y_{1}(t)})$是全局渐近稳定的, 其中

$ \begin{equation} \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \widetilde{x_{1}(t)}= \end{array} \right. \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle x_{1}^{\ast}e^{-[c+(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast}))](t-n\tau)}, t\in(n\tau, (n+l)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle x_{1}^{\ast\ast}e^{-d_{2}(t-(n+l)\tau)}, t\in((n+l)\tau, (n+1)\tau], \\ \end{array} \right. \\ \left. \begin{array}{llc} \displaystyle \widetilde{y_{1}(t)}= \end{array} \right. \left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle (1-e^{-c(t-n\tau)})x^{\ast}e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})(t-n\tau)}-y^{\ast}e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})(t-n\tau)}, \\ ~~~~~t\in(n\tau, (n+l)\tau], \\ \left. \begin{array}{llc} \displaystyle \end{array} \right. \displaystyle y^{\ast\ast}e^{-d_{2}(t-(n+l)\tau)}, t\in((n+l)\tau, (n+1)\tau], \end{array} \right. \end{array} \right. \end{equation} $

(3.23)

而

$ \begin{eqnarray*}&& x_{1}^{\ast\ast}=(x_{1}^{\ast}+y_{1}^{\ast})e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l\tau}+x_{1}^{\ast} e^{-[c+(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast}))]l\tau}, \\ && y_{1}^{\ast\ast}=(x_{1}^{\ast})(1-e^{-cl\tau})e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l\tau}+y^{\ast})e^{-(d_{1}+\beta(c_{e}^{\ast}+c_{e}^{\ast\ast})l\tau}, \end{eqnarray*} $

且$x_{1}^{\ast}$与$y_{1}^{\ast}$如$(3.22)$式所定义.

由定理3与定理5容易得

定理6 (ⅰ)如果$(A-1)(1-F)+BE < 0, $那么系统$(2.4)$种群灭绝;

(ⅱ)如果$(A_{1}-1)(1-F_{1})+B_{1}E_{1}>0, $系统$(2.4)$种群持久.

本文考虑污染环境、生物冬眠与种群阶段结构, 建立毒素脉冲输入与种群脉冲出生的阶段结构切换单种群动力学模型, 根据定理6的结论可以知道脉冲输入的毒素浓度存在一个阈值$\mu^{\ast}$, 当$\mu>\mu^{\ast}$时, 系统$(2.4)$种群灭绝; 当$\mu < \mu^{\ast}$时, 系统$(2.4)$种群持久.所得结论告诉我们控制治理污染的力度能够保护环境生物的多样性, 特别是环境污染控制的阈值为污染环境下生物资源管理提供决策支持.