一類生物種群模型及其穩(wěn)定性數(shù)學(xué)畢業(yè)論文

1、一類生物種群模型及其穩(wěn)定性摘 要：本文討論一類種群發(fā)展方程，建立了年齡依賴種群系統(tǒng)的連續(xù)模型，半離散模型和離散化模型，并由特征值簡要討論了它們的穩(wěn)定性。關(guān)鍵詞：種群模型，半離散，穩(wěn)定性，特征值 引 言： 在自然界中生存的各種生物種群的發(fā)展受到各種影響，本文就年齡結(jié)構(gòu)變化對(duì)單一生物種群的發(fā)展影響進(jìn)行分析建模，結(jié)合1文，為討論方便，我們假設(shè)在一穩(wěn)定狀態(tài)環(huán)境中生物的生存條件僅受年齡結(jié)構(gòu)變化限制，由此得出以下幾種模型。1、線性種群發(fā)展方程線性種群發(fā)展方程是分析、預(yù)測(cè)和定量控制的基礎(chǔ)。在一穩(wěn)定的狀態(tài)環(huán)境中，用r表示年齡，t表示時(shí)間，r, t皆為連續(xù)變化量，用表示t時(shí)刻年齡小于r的種群總數(shù)。顯然且當(dāng)時(shí)，

2、，即對(duì)于固定的t為r的單調(diào)增函數(shù)，稱為種群函數(shù)。表示t時(shí)刻種群函數(shù)，m記為種群所能達(dá)到的最高年齡，則有的定義，易知=。當(dāng)r, t都連續(xù)變化時(shí)，是r, t 的連續(xù)函數(shù)，假設(shè)的一階偏導(dǎo)數(shù)，都是一元連續(xù)函數(shù)。設(shè)=，稱為種群按年齡分布函數(shù)簡稱種群密度函數(shù)，由的單調(diào)性知且。設(shè)為充分小的年齡空間，0時(shí)，則t時(shí)刻年齡在r和+r之間的種群總數(shù)為，另外有=，=t時(shí)刻年齡在和（）之間的種群總數(shù)為設(shè)t時(shí)刻年齡在內(nèi)平均單位時(shí)間內(nèi)消亡總數(shù)為，為同一時(shí)刻年齡在內(nèi)活著的種群數(shù)。定義 (1.2)稱為相對(duì)消亡率函數(shù)，對(duì)于充分小的及，由t到,年齡在中消亡總數(shù)為即 =設(shè)為充分小的時(shí)間區(qū)間，t時(shí)刻在之間的種群總數(shù)為，過了時(shí)間到達(dá)時(shí)，

3、在此期間消亡數(shù)為，而在此期間沒消亡的種群到了時(shí)變成了年齡在中的種群，其總數(shù)為，用表示年齡在中的種群在時(shí)間內(nèi)增長或消亡的種群總數(shù)，規(guī)定增生為正，消亡為負(fù)，稱為t時(shí)刻r歲種群的增消率，由于r和t具有相同的量綱，所以，于是有下式成立 (1.3) 變化為等式兩邊同除以得到由于，令得到 (1.4) 這就是所求種群連續(xù)發(fā)展方程，這是一階線性偏微分方程。取可得初始條件，可由統(tǒng)計(jì)數(shù)據(jù)給出。設(shè)邊界條件為，若設(shè)為t時(shí)刻消亡與增殖數(shù)之比，稱為更新率為種群增殖成活率，則在t時(shí)刻在內(nèi)消亡數(shù)為所以有 (1.5) 由此可得 (1.6)這即為種群發(fā)展方程的連續(xù)模型，這是一階線性偏微分方程系統(tǒng)。2、半離散種群發(fā)展方程當(dāng)t連續(xù)r

4、離散時(shí)的種群發(fā)展方程稱為半離散模型。下面用半離散逼近法求半離散模型。給定區(qū)間的一個(gè)分劃，記，用表示t年代滿 歲但不滿歲的種群總數(shù)，則 (2.1)由于這里，從而 (2.2)其中,對(duì)（1）中第一個(gè)方程兩邊從 到 積分得=即 由（2）有 這里,舍掉高階項(xiàng)有 (2.3)當(dāng)取年齡間隔為1，即時(shí) ，為 (2.4) 即對(duì)初始條件做離散化處理記 則有 (2.5) 對(duì)于外界條件 有對(duì)右端應(yīng)用積分中值定理有所以有 即有 (2.6)因此我們有半離散模型: (2.7) 引進(jìn)向量和矩陣記號(hào)有XGXAB則（2.7）即半離散模型可表示為 (2.8)這是一階線性常微分方程組。下面考慮半離散模型（2.8）在定常情況下的穩(wěn)定性。

5、（定常情形指消亡率、成活率、增消率都不隨時(shí)間變化）。在一個(gè)相對(duì)安定的環(huán)境下，方程（2.8）可變?yōu)椋?(2.9)其中 A=B 稱為種群的增生率.A、B都是m-1階常數(shù)方陣，容易得出A+的特征多項(xiàng)式= (2.10)對(duì)于 的增生率 稱為種群臨界增生率由（2.10）易推得=由文2的方法可證明下述結(jié)論引理2.1: 0是A+的代數(shù)單特征值引理2.2:當(dāng)時(shí),A+有且只有一個(gè)正特征值,且此特征值的代數(shù)重?cái)?shù)為1引理2.3當(dāng) 時(shí)，A+ 的每個(gè)特征值都有負(fù)實(shí)部；且A+ 的每個(gè)非零特征值也具有負(fù)實(shí)部。從引理2.1、引理2.2、引理2.3易得定理2.1：對(duì)于系統(tǒng)（2.9），如果 ， 那么系統(tǒng)是不穩(wěn)定的；如果 ，那么系統(tǒng)

6、是漸近穩(wěn)定的，即對(duì)任意初始值，系統(tǒng)（2.9）的解 隨時(shí)間t的增加指數(shù)衰減到零的；如果 ，那么系統(tǒng)穩(wěn)定。上述結(jié)果與文3中連續(xù)型方程的穩(wěn)定性一致。3、離散種群發(fā)展方程為便于數(shù)值計(jì)算以利于統(tǒng)計(jì)分析，在定量計(jì)算中，為了用計(jì)算機(jī)求解種群發(fā)展方程，必須把r和t同時(shí)離散化。離散后的r和t我們 取整數(shù)值以年度為單位，將連續(xù)種群方程變成一個(gè)差分方程組，這即為種群發(fā)展過程的離散模型。離散模型不但適合于計(jì)算機(jī)計(jì)算、模擬和數(shù)據(jù)處理，而且又與傳統(tǒng)的統(tǒng)計(jì)方法相一致。下面我們?cè)诎腚x散模型的基礎(chǔ)上建立離散模型。對(duì)r離散，由半離散模型有，記為種群狀態(tài)向量。再對(duì)t離散，單位取年，由(1.3)有p（r+）消去 ，令 ，上式兩邊對(duì)r

7、 從i 到 積分，得到 (3.1)對(duì)等式右邊第一項(xiàng)應(yīng)用積分中值定理，有這里滿足定義 為 t年代i歲按年齡消亡率，則（3.1）為或 (3.2)這里對(duì)于初始條件p(r,0)=作離散化處理，記 , 有 (3.3)對(duì)于邊界條件，p(0,t)表示t時(shí)刻單位時(shí)間內(nèi)種群的新增生數(shù)，取，則p(0,t)就是 t-1 年到t年新增生種種群數(shù)，由p(0,t)= 有 (3.4)（3.4）的實(shí)際意義是這樣的，表示t年代I歲種群的消亡數(shù)， 表示消亡后I歲種群的消亡數(shù)，表示消亡后I歲種群的增殖更新數(shù)，為成活率，則 表示t年代I歲種群的增生數(shù)。因此即 為t年代各年齡種群增生數(shù)。如對(duì)于森林系統(tǒng)，表示t年i齡級(jí)林木采消率， 為t

8、年代林木更新率即林木更新棵數(shù)與采消棵數(shù)之比，表示成活率，即 。因此表示t年代i齡級(jí)林木采伐棵數(shù)，則表示t年代i 齡級(jí)林木更新增殖數(shù)，表示t年代i齡級(jí)林木增值成活數(shù)，表示t年代各齡級(jí)林木增植成活總棵數(shù)。于是得到離散種群方程組（時(shí)間與林齡同步純林離散模型）： (3.5) （3.5）是一個(gè)以年度為時(shí)間間隔的查分方程組，引進(jìn)向量和矩陣符號(hào)：G(t)= HB則（3.5）可表示成 這里H(t)稱為種群狀態(tài)轉(zhuǎn)移陣，B(t)稱為種群消亡陣,G(t)稱為干擾向量，加上初始條件可得完整的種群發(fā)展離散模型 (3.6)（3.6）是一個(gè)離散的雙線性系統(tǒng)，是控制量，通過改變 來達(dá)到控制種群狀態(tài)的目的。對(duì)于離散系統(tǒng)（3.6

9、），由文23，可得到與半離散情形一致的穩(wěn)定性結(jié)果。 參考文獻(xiàn)： 1. 姜啟元. 數(shù)學(xué)建模 2. 宋 健. 于景元, 人口控制論. 1985 ,190-2013. Wang Dingjiang , The Stability of Forest evolution systems and the Critieal Proliferation rate of forest ,Applied Functional Analysis. 1995.Vol2.235-2384. Song jian etal.Scientia sinica (seriesA) (2)1996.113-123 5. Pazy

10、 A. Semigroups of Linear operators and applications to partial differential equations. Spinger-verlay New Yor ,1983 A Class of Bio -population Model and Its stabilityAbstract: In this paper , We study a class of poplution evlution equations, the continuous model and semidiscrete models and discrete

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23、2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx

24、2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGp

25、P$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#Qc

26、UE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&M

27、uWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVku

28、m&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqvadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89A

29、mYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTX

30、Rm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*

31、3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*

32、3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpaza

33、dNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%MadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpaz

34、adNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*adNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA

35、5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv

36、*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tn

37、GK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazad

38、Nu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7Jn

39、D6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8

40、!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEw

41、Z#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN

42、&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcv

43、R9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjq

44、v$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%MzadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT

45、#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazad

46、Nu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazad

47、Nu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6

48、YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz84!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z

49、89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z

50、89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN&M

51、uWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNuGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zV

52、kum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmY

53、WpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqvUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$U*3tnGK8!z89AmYWpazadNu#KN

54、&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89Amv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5uxY7JnD6YWRrWwcvR9CpbK!zn%Mz849GxGjqv$UE9wEwZ#QcUE%&qYpEh5pDx2zVkum&gTXRm6X4NGpP$vSTT#&ksv*3tnGK8!z89AmYWv*3tnGK8!z89AmYWpazadNu#KN&MuWFA5u