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\end_header

\begin_body

\begin_layout Section
离散动态系统模型
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
引言
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
要理解并预测由差分方程 
\begin_inset Formula $x_{n+1}=Ax_{n}$
\end_inset

 所描述的动态系统的长期行为或演化, 关键在于掌握矩阵 
\begin_inset Formula $A$
\end_inset

 的特征值与特征向量.
 在本节中, 我们将通过应用实例来介绍矩阵对角化在离散动态系统模型中的应用.
 这些应用实例主要针对生态问题,是因为相对于物理问题或工程问题,它们更容易说明和解释,但实际上动态系统在许多科学领域中都会出现.
\end_layout

\end_deeper
\begin_layout Subsection
区域人口迁移预测问题
\end_layout

\begin_layout Frame
\begin_inset Argument 3
status open

\begin_layout Plain Layout
allowframebreaks
\end_layout

\end_inset


\begin_inset Argument 4
status open

\begin_layout Plain Layout
区域人口迁移预测问题
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
设定一个初始的年份, 比如说 2008 年, 用 
\begin_inset Formula $r_{0},s_{0}$
\end_inset

 分别表示这一年城市和农村的人口.
 设 
\begin_inset Formula $x_{0}$
\end_inset

 为初始人口向量, 即 
\begin_inset Formula $\boldsymbol{x}_{0}=\begin{bmatrix}r_{0}\\
s_{0}
\end{bmatrix}$
\end_inset

, 对 2009 年以及后面的年份, 我们用向量 
\begin_inset Formula 
\[
\boldsymbol{x}_{1}=\begin{bmatrix}r_{1}\\
s_{1}
\end{bmatrix},\quad\boldsymbol{x}_{2}=\begin{bmatrix}r_{2}\\
s_{2}
\end{bmatrix},\quad\boldsymbol{x}_{3}=\begin{bmatrix}r_{3}\\
s_{3}
\end{bmatrix},\cdots
\]

\end_inset


\end_layout

\begin_layout Standard
表示每一年城市和农村的人口.
 我们的目标是
\series bold
用数学公式表示出这些向量之间的关系
\series default
.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
newpage
\end_layout

\end_inset


\end_layout

\begin_layout Standard
假设每年大约有 
\begin_inset Formula $5\%$
\end_inset

 的城市人口迁移到农村 (
\begin_inset Formula $95\%$
\end_inset

 仍然留在城市), 有 
\begin_inset Formula $12\%$
\end_inset

 的农村人口迁移到城市 (
\begin_inset Formula $88\%$
\end_inset

 仍然留在农村), 如图所示,
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{center}
\end_layout

\begin_layout Plain Layout


\backslash
begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto]
\end_layout

\begin_layout Plain Layout

			
\backslash
node[state] (ct) {城市};
\end_layout

\begin_layout Plain Layout

			
\backslash
node[state,accepting] (cnt) [right=of ct] {农村};
\end_layout

\begin_layout Plain Layout

			
\backslash
path[->] (ct) edge [loop left]  node {0.95} ()
\end_layout

\begin_layout Plain Layout

							edge [bend left]  node [swap] {0.05} (cnt)
\end_layout

\begin_layout Plain Layout

			 		(cnt) edge [loop right] node {0.88} ()
\end_layout

\begin_layout Plain Layout

							edge [bend left]  node {0.12} (ct);
\end_layout

\begin_layout Plain Layout


\backslash
end{tikzpicture}
\end_layout

\begin_layout Plain Layout


\backslash
end{center}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
忽略其他因素对人口规模的影响, 则一年之后, 城市与农村人口的分布分别为
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

$$
\end_layout

\begin_layout Plain Layout

r_0
\backslash
begin{bNiceMatrix}[last-col]
\end_layout

\begin_layout Plain Layout

0.95 & 
\backslash
text{留在城市}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

0.05 & 
\backslash
text{移居农村}
\end_layout

\begin_layout Plain Layout


\backslash
end{bNiceMatrix}
\backslash
 ,
\backslash
quad
\end_layout

\begin_layout Plain Layout

s_0
\backslash
begin{bNiceMatrix}[last-col]
\end_layout

\begin_layout Plain Layout

0.12 & 
\backslash
text{移居城市}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

0.88 & 
\backslash
text{留在农村}
\end_layout

\begin_layout Plain Layout


\backslash
end{bNiceMatrix}
\end_layout

\begin_layout Plain Layout

$$
\end_layout

\end_inset


\end_layout

\begin_layout Standard
因此, 2009 年全部人口的分布为
\begin_inset Formula 
\[
\begin{bmatrix}r_{1}\\
s_{1}
\end{bmatrix}=r_{0}\begin{bmatrix}0.95\\
0.05
\end{bmatrix}+s_{0}\begin{bmatrix}0.12\\
0.88
\end{bmatrix}=\begin{bmatrix}0.95 & 0.12\\
0.05 & 0.88
\end{bmatrix}\begin{bmatrix}r_{0}\\
s_{0}
\end{bmatrix},
\]

\end_inset

即
\begin_inset Formula 
\[
\boldsymbol{x}_{1}=\boldsymbol{M}\boldsymbol{x}_{0},
\]

\end_inset

其中 
\begin_inset Formula $\boldsymbol{M}=\begin{bmatrix}0.95 & 0.12\\
0.05 & 0.88
\end{bmatrix}$
\end_inset

 称为
\series bold
迁移矩阵
\series default
.
 如果人口迁移的百分比保持不变, 则可以继续得到 2010 年, 2011 年, 
\begin_inset Formula $\cdots$
\end_inset

 的人口分布公式: 
\begin_inset Formula 
\[
\boldsymbol{x}_{2}=\boldsymbol{M}\boldsymbol{x}_{1},\ \boldsymbol{x}_{3}=\boldsymbol{M}\boldsymbol{x}_{2},\cdots,
\]

\end_inset

一般地, 有 
\begin_inset Formula 
\[
\boldsymbol{x}_{n+1}=\boldsymbol{A}\boldsymbol{x}_{n},\quad(n=0,1,2,\cdots).
\]

\end_inset


\end_layout

\begin_layout Standard
这里, 向量序列 
\begin_inset Formula $\left\{ x_{0},x_{1},x_{2},\cdots\right\} $
\end_inset

 描述了城市与农村人口在若干年内的分布变化.
 
\end_layout

\begin_layout Example
已知某城市 
\begin_inset Formula $2002$
\end_inset

 年的城市人口为 
\begin_inset Formula $5000000$
\end_inset

, 农村人口为 
\begin_inset Formula $7800000$
\end_inset

, 忽略其它因素对人口规模的影响, 计算 
\begin_inset Formula $2022$
\end_inset

 年的人口分布.
\end_layout

\begin_layout Solution*
迁移矩阵 
\begin_inset Formula $\boldsymbol{M}=\begin{bmatrix}0.95 & 0.12\\
0.05 & 0.88
\end{bmatrix}$
\end_inset

 的全部特征值是 
\begin_inset Formula $\lambda_{1}=1$
\end_inset

, 
\begin_inset Formula $\lambda_{2}=0.83$
\end_inset

, 其对应的特征向量分别是
\begin_inset Formula 
\[
p_{1}=\begin{bmatrix}2.4\\
1
\end{bmatrix},\ p_{2}=\begin{bmatrix}1\\
-1
\end{bmatrix}.
\]

\end_inset

因为 
\begin_inset Formula $\lambda_{1}\neq\lambda_{2}$
\end_inset

, 故 
\begin_inset Formula $\boldsymbol{M}$
\end_inset

 可对角化.
\end_layout

\begin_layout Solution*
令 
\begin_inset Formula $P=\left(p_{1},p_{2}\right)=\begin{bmatrix}2.4 & 1\\
1 & -1
\end{bmatrix}$
\end_inset

, 有 
\begin_inset Formula $P^{-1}\boldsymbol{M}P=\begin{bmatrix}1 & 0\\
0 & 0.83
\end{bmatrix}$
\end_inset

, 则 
\begin_inset Formula $\boldsymbol{M}=P\begin{bmatrix}1 & 0\\
0 & 0.83
\end{bmatrix}P^{-1}\eqqcolon P\Lambda P^{-1}$
\end_inset

.
\end_layout

\begin_layout Solution*
因 
\begin_inset Formula $2002$
\end_inset

 年的初始人口为 
\begin_inset Formula $x_{0}=\begin{bmatrix}5000000\\
7800000
\end{bmatrix}$
\end_inset

, 故对 
\begin_inset Formula $2022$
\end_inset

 年, 有
\end_layout

\begin_layout Solution*
\begin_inset Formula 
\[
\begin{aligned}x_{20} & =Mx_{19}=\cdots=M^{20}x_{0}=P\Lambda^{20}P^{-1}x_{0}\\
 & =\begin{bmatrix}2.4 & 1\\
1 & -1
\end{bmatrix}\begin{bmatrix}1 & 0\\
0 & 0.83^{20}
\end{bmatrix}\begin{bmatrix}2.4 & 1\\
1 & -1
\end{bmatrix}^{-1}\begin{bmatrix}5000000\\
7800000
\end{bmatrix}\approx\begin{bmatrix}8938145\\
3861855
\end{bmatrix}.
\end{aligned}
\]

\end_inset

即 
\begin_inset Formula $2022$
\end_inset

 年中国的城市人口约为 
\begin_inset Formula $8938145$
\end_inset

, 农村人口为 
\begin_inset Formula $3861855$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Subsection
捕食者与被捕食者系统
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 3
status open

\begin_layout Plain Layout
allowframebreaks
\end_layout

\end_inset


\begin_inset Argument 4
status open

\begin_layout Plain Layout
捕食者与被捕食者系统
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Example
某森林中, 猫头鹰以鼠为食.
 记猫头鹰和鼠在时间 
\begin_inset Formula $n$
\end_inset

 的数量为 
\begin_inset Formula $X_{n}=\begin{bmatrix}O_{n}\\
M_{n}
\end{bmatrix}$
\end_inset

, 其中 
\begin_inset Formula $n$
\end_inset

 是以月份为单位的时间指标, 
\begin_inset Formula $O_{n}$
\end_inset

 是研究区域中的猫头鹰, 
\begin_inset Formula $M_{n}$
\end_inset

 是鼠的数量 (单位: 千).
 假定生态学家已建立了猫头鹰与鼠的自然系统模型:
\end_layout

\begin_layout Example
\begin_inset Formula 
\begin{equation}
\begin{cases}
O_{n+1}=0.4O_{n}+0.3M_{n},\\
M_{n+1}=-pO_{n}+1.2M_{n},
\end{cases}\label{eq:4.5-1}
\end{equation}

\end_inset

其中 
\begin_inset Formula $p$
\end_inset

 是一个待定的正参数.
 第一个方程中的 
\begin_inset Formula $0.4O_{n}$
\end_inset

 表明, 如果没有鼠做食物, 每个月只有 
\begin_inset Formula $40\%$
\end_inset

 的猫头鹰可以存活, 第二个方程中的 
\begin_inset Formula $1.2M_{n}$
\end_inset

 表明, 如果没有猫头鹰捕食, 鼠的数量每个月会增加 
\begin_inset Formula $20\%$
\end_inset

.
 如果鼠充足, 猫头鹰的数量将会增加 
\begin_inset Formula $0.3M_{n}$
\end_inset

, 负项 
\begin_inset Formula $-pO_{n}$
\end_inset

 用以表示猫头鹰的捕食所导致野鼠的死亡数 (事实上, 平均每个月一只猫头鹰吃掉约 
\begin_inset Formula $1000p$
\end_inset

 只鼠).
 当捕食参数 
\begin_inset Formula $p=0.325$
\end_inset

 时, 则两个种群都会增长, 估计这个长期增长率及猫头鹰与鼠的最终比值.
\end_layout

\begin_layout Solution*
当 
\begin_inset Formula $p=0.325$
\end_inset

 时, (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:4.5-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

) 的系数矩阵 
\begin_inset Formula $A=\begin{bmatrix}0.4 & 0.3\\
-0.325 & 1.2
\end{bmatrix}$
\end_inset

, 求得 
\begin_inset Formula $A$
\end_inset

 的全部特征值 
\begin_inset Formula $\lambda_{1}=0.55$
\end_inset

, 
\begin_inset Formula $\lambda_{2}=1.05$
\end_inset

, 其对应的特征向量分别是 
\begin_inset Formula $p_{1}=\begin{bmatrix}2\\
1
\end{bmatrix}$
\end_inset

, 
\begin_inset Formula $p_{2}=\begin{bmatrix}6\\
13
\end{bmatrix}$
\end_inset

.
\end_layout

\begin_layout Solution*
初始向量 
\begin_inset Formula $x_{0}=c_{1}p_{1}+c_{2}p_{2}$
\end_inset

.
 令 
\begin_inset Formula $P=\left(p_{1},p_{2}\right)=\begin{bmatrix}2 & 6\\
1 & 13
\end{bmatrix}$
\end_inset

, 当 
\begin_inset Formula $n\geq0$
\end_inset

 时, 则
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
vspace{-4mm}{
\backslash
scriptsize{
\end_layout

\end_inset


\begin_inset Formula 
\begin{align*}
\begin{bmatrix}2 & 6\\
1 & 13
\end{bmatrix}\begin{bmatrix}0.55^{n} & 0\\
0 & 1.05^{n}
\end{bmatrix}\begin{bmatrix}2 & 6\\
1 & 13
\end{bmatrix}^{-1}x_{0} & =\begin{bmatrix}\begin{bmatrix}2\\
1
\end{bmatrix} & \begin{bmatrix}6\\
13
\end{bmatrix}\end{bmatrix}\begin{bmatrix}0.55^{n} & 0\\
0 & 1.05^{n}
\end{bmatrix}\begin{bmatrix}2 & 6\\
1 & 13
\end{bmatrix}^{-1}\begin{bmatrix}p_{1} & p_{2}\end{bmatrix}\begin{bmatrix}c_{1}\\
c_{2}
\end{bmatrix}\\
 & =\begin{bmatrix}p_{1} & p_{2}\end{bmatrix}\begin{bmatrix}0.55^{n} & 0\\
0 & 1.05^{n}
\end{bmatrix}\begin{bmatrix}c_{1}\\
c_{2}
\end{bmatrix}\\
 & =\begin{bmatrix}p_{1} & p_{2}\end{bmatrix}\begin{bmatrix}0.55^{n}c_{1}\\
1.05^{n}c_{2}
\end{bmatrix}=0.55^{n}c_{1}p_{1}+1.05^{n}c_{2}p_{2}.
\end{align*}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}}
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
vspace{-4mm}
\end_layout

\end_inset


\begin_inset Formula 
\[
x_{n}=PA^{n}P^{-1}x_{0}=\begin{bmatrix}2 & 6\\
1 & 13
\end{bmatrix}\begin{bmatrix}0.55^{n} & 0\\
0 & 1.05^{n}
\end{bmatrix}\begin{bmatrix}2 & 6\\
1 & 13
\end{bmatrix}^{-1}x_{0}=0.55^{n}c_{1}\begin{bmatrix}2\\
1
\end{bmatrix}+1.05^{n}c_{2}\begin{bmatrix}6\\
13
\end{bmatrix}.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Solution*
假定 
\begin_inset Formula $c_{2}>0$
\end_inset

, 则对总够大的 
\begin_inset Formula $n$
\end_inset

, 
\begin_inset Formula $0.55^{n}$
\end_inset

 趋于 
\begin_inset Formula $0$
\end_inset

, 进而
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
vspace{-4mm}
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
x_{n}\approx c_{2}1.05^{n}\begin{bmatrix}6\\
13
\end{bmatrix},\label{eq:4.5-2}
\end{equation}

\end_inset


\begin_inset Formula $n$
\end_inset

 越大 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:4.5-2"
plural "false"
caps "false"
noprefix "false"

\end_inset

) 式的近似程度越高, 故对于充分大的 
\begin_inset Formula $n$
\end_inset

, 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
vspace{-4mm}
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
x_{n+1}\approx c_{2}1.05^{n+1}\begin{bmatrix}6\\
13
\end{bmatrix}=1.05x_{n},\label{eq:4.5-3}
\end{equation}

\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:4.5-3"
plural "false"
caps "false"
noprefix "false"

\end_inset

) 式的近似表明, 最后 
\begin_inset Formula $x_{n}$
\end_inset

 的每个元素 (猫头鹰和鼠的数量) 几乎每个月都近似地增长了 
\begin_inset Formula $0.05$
\end_inset

 倍, 即有 
\begin_inset Formula $5\%$
\end_inset

 的月增长率.
 由 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:4.5-2"
plural "false"
caps "false"
noprefix "false"

\end_inset

) 式知, 
\begin_inset Formula $x_{n}$
\end_inset

 约为 
\begin_inset Formula $(6,13)^{T}$
\end_inset

 的倍数, 所以 
\begin_inset Formula $x_{n}$
\end_inset

 中元素的比值约为 
\begin_inset Formula $6:13$
\end_inset

, 即每 
\begin_inset Formula $6$
\end_inset

 只猫头鹰对应着约 
\begin_inset Formula $13000$
\end_inset

 只鼠.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
作业
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Problem
设 
\begin_inset Formula $A=\begin{bmatrix}0.5 & 0.2 & 0.3\\
0.3 & 0.8 & 0.3\\
0.2 & 0 & 0.4
\end{bmatrix}$
\end_inset

, 
\begin_inset Formula $x_{0}=\begin{bmatrix}1\\
0\\
0
\end{bmatrix}$
\end_inset

, 考虑一个由 
\begin_inset Formula $x_{n+1}=Ax_{n}$
\end_inset

, 
\begin_inset Formula $n=1,2,3,\cdots$
\end_inset

 描述的系统.
 随时间的变化, 这个系统将如何变化? 通过计算状态向量 
\begin_inset Formula $x_{1},\cdots,x_{15}$
\end_inset

 来求解.
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\end_body
\end_document