# 有效種群大小

## 定義

### 方差有效群體大小

Wright-Fisher理想種群模型中，在給定上一代等位基因頻率${\displaystyle p}$ 時，等位基因頻率${\displaystyle p'}$ 條件方差爲：

${\displaystyle \operatorname {var} (p'\mid p)={p(1-p) \over 2N}.}$

${\displaystyle {\widehat {\operatorname {var} }}(p'|p)}$ 來表示該當前群體相同或通常更大的方差。方差有效群體大小${\displaystyle N_{e}^{(v)}}$ 定義爲具有相同方差的理想種群的大小。此時令${\displaystyle {\widehat {\operatorname {var} }}(p'|p)}$ ${\displaystyle \operatorname {var} (p'|p)}$ 相等並對${\displaystyle N}$ 求解，可得到

${\displaystyle N_{e}^{(v)}={p(1-p) \over 2{\widehat {\operatorname {var} }}(p)}.}$

### 近交有效群體大小

${\displaystyle F_{t}={\frac {1}{N}}\left({\frac {1+F_{t-2}}{2}}\right)+\left(1-{\frac {1}{N}}\right)F_{t-1}}$

${\displaystyle 1-F_{t}=P_{t}=P_{0}\left(1-{\frac {1}{2N}}\right)^{t}}$

${\displaystyle {\frac {P_{t+1}}{P_{t}}}=1-{\frac {1}{2N}}}$

${\displaystyle {\frac {P_{t+1}}{P_{t}}}=1-{\frac {1}{2N_{e}^{(F)}}}}$

${\displaystyle N_{e}^{(F)}={\frac {1}{2\left(1-{\frac {P_{t+1}}{P_{t}}}\right)}}}$

，儘管研究者很少直接使用這個方程式。

## 舉例

### 種群大小的變化

${\displaystyle {1 \over N_{e}}={1 \over t}\sum _{i=1}^{t}{1 \over N_{i}}}$

 ${\displaystyle {1 \over N_{e}}}$ ${\displaystyle ={{\begin{matrix}{\frac {1}{10}}\end{matrix}}+{\begin{matrix}{\frac {1}{100}}\end{matrix}}+{\begin{matrix}{\frac {1}{50}}\end{matrix}}+{\begin{matrix}{\frac {1}{80}}\end{matrix}}+{\begin{matrix}{\frac {1}{20}}\end{matrix}}+{\begin{matrix}{\frac {1}{500}}\end{matrix}} \over 6}}$ ${\displaystyle ={0.1945 \over 6}}$ ${\displaystyle =0.032416667}$ ${\displaystyle N_{e}}$ ${\displaystyle =30.8}$

### 雌雄異體

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {1}{2}}\end{matrix}}}$

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {D}{2}}\end{matrix}}}$

N很大時，Ne近似等於N，因此這通常不重要，可以忽略，即：

${\displaystyle N_{e}=N+{\begin{matrix}{\frac {1}{2}}\approx N\end{matrix}}}$

### 非費雪式性別比

${\displaystyle N_{e}^{(v)}=N_{e}^{(F)}={4N_{m}N_{f} \over N_{m}+N_{f}}}$

 ${\displaystyle N_{e}}$ ${\displaystyle ={4\times 80\times 20 \over 80+20}}$ ${\displaystyle ={6400 \over 100}}$ ${\displaystyle =64}$

### 對下一代的不均等貢獻

${\displaystyle \operatorname {var} (k)={\bar {k}}=2.}$

${\displaystyle \operatorname {var} (k)>2.}$

${\displaystyle N_{e}^{(v)}={4N-2D \over 2+\operatorname {var} (k)}}$

### 世代重疊及種群和年齡結構的關係

#### 單倍體

${\displaystyle v_{i}=}$  對年齡${\displaystyle i}$ 的費雪氏繁殖值（Fisher's reproductive value），
${\displaystyle \ell _{i}=}$  一個個體存活到年齡${\displaystyle i}$ 的幾率，以及
${\displaystyle N_{0}=}$  每個繁殖季節的新生個體數。

${\displaystyle T=\sum _{i=0}^{\infty }\ell _{i}v_{i}}$

${\displaystyle N_{e}^{(F)}={\frac {N_{0}T}{1+\sum _{i}\ell _{i+1}^{2}v_{i+1}^{2}({\frac {1}{\ell _{i+1}}}-{\frac {1}{\ell _{i}}})}}}$  （Felsenstein 1971）

#### 二倍體

{\displaystyle {\begin{aligned}{\frac {1}{N_{e}^{(F)}}}={\frac {1}{4T}}\left\{{\frac {1}{N_{0}^{f}}}+{\frac {1}{N_{0}^{m}}}+\sum _{i}\left(\ell _{i+1}^{f}\right)^{2}\left(v_{i+1}^{f}\right)^{2}\left({\frac {1}{\ell _{i+1}^{f}}}-{\frac {1}{\ell _{i}^{f}}}\right)\right.\,\,\,\,\,\,\,\,&\\\left.{}+\sum _{i}\left(\ell _{i+1}^{m}\right)^{2}\left(v_{i+1}^{m}\right)^{2}\left({\frac {1}{\ell _{i+1}^{m}}}-{\frac {1}{\ell _{i}^{m}}}\right)\right\}.&\end{aligned}}}

## 參考文獻

• Emigh TH, Pollak E. Fixation probabilities and effective population numbers in diploid populations with overlapping generations. Theoretical Population Biology. 1979, 15: 86–107.
• Felsenstein J. Inbreeding and variance effective numbers in populations with overlapping generations. Genetics. 1971, 68: 581–597.
• Johnson DL. Inbreeding in populations with overlapping generations. Genetics. 1977, 87: 581–591.
• Kempthorne O. An Introduction to Genetic Statistics. Iowa State University Press. 1957, [1969].
• Wright S. Evolution in Mendelian populations. Genetics. 1931, 16: 97–159. reprint
• Wright S. Size of population and breeding structure in relation to evolution. Science. 1938, 87: 430–431.