# 洛特卡-沃爾泰拉方程

${\displaystyle {\frac {dx}{dt}}=x(\alpha -\beta y)}$
${\displaystyle {\frac {dy}{dt}}=-y(\gamma -\delta x)}$
• y掠食者（如）的數量；
• x獵物（如兔子）的數量；
• dy/dtdx/dt 表示上述兩族群相互對抗的時間之變化；
• t 表示時間；
• α, β, γδ 表示與兩物種互動有關的係數，皆為正實數

## 生物學上的意義

### 獵物族群的增值速度

${\displaystyle {\frac {dx}{dt}}=\alpha x-\beta xy}$

### 掠食者族群的增值速度

${\displaystyle {\frac {dy}{dt}}=\delta xy-\gamma y}$

## 方程式的解

### 族群規模的平衡

${\displaystyle x(\alpha -\beta y)=0}$
${\displaystyle -y(\gamma -\delta x)=0}$

${\displaystyle \left\{y=0,x=0\right\}}$

${\displaystyle \left\{y={\frac {\alpha }{\beta }},x={\frac {\gamma }{\delta }}\right\},}$

### 不動點的穩定性

${\displaystyle J(x,y)={\begin{bmatrix}\alpha -\beta y&-\beta x\\\delta y&\delta x-\gamma \\\end{bmatrix}}}$

#### 第一不動點

${\displaystyle J(0,0)={\begin{bmatrix}\alpha &0\\0&-\gamma \\\end{bmatrix}}}$

${\displaystyle \lambda _{1}=\alpha ,\quad \lambda _{2}=-\gamma }$

#### 第二不動點

${\displaystyle J\left({\frac {\gamma }{\delta }},{\frac {\alpha }{\beta }}\right)={\begin{bmatrix}0&-{\frac {\beta \gamma }{\delta }}\\{\frac {\alpha \delta }{\beta }}&0\\\end{bmatrix}}}$

${\displaystyle \lambda _{1}=i{\sqrt {\alpha \gamma }},\quad \lambda _{2}=-i{\sqrt {\alpha \gamma }}}$

## 饱和沃尔泰拉方程

${\displaystyle {\frac {dr}{dt}}=2*r(t)-{\frac {\alpha *r(t)*f(t)}{1+s*r(t)}}}$;[1]

${\displaystyle {\frac {df}{dt}}=-f(t)+{\frac {\alpha *r(t)*f(t)}{1+s*r(t)}}}$

## 参考文献

1. ^ Richard H. Enns George C. McCGuire, Nonlinear Physics, p25, Birkhauser,1997
• E. R. Leigh (1968) The ecological role of Volterra's equations, in Some Mathematical Problems in Biology - a modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903.
• Understanding Nonlinear Dynamics. Daniel Kaplan and Leon Glass.
• Vito Volterra. Variations and fluctuations of the number of individuals in animal species living together. In Animal Ecology. McGraw-Hill, 1931. Translated from 1928 edition by R. N. Chapman.