# 洛特卡－沃爾泰拉方程

$\frac{dx}{dt} = x(\alpha - \beta y)$
$\frac{dy}{dt} = - y(\gamma - \delta x)$
• 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

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

### 掠食者族群的增值速度

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

## 方程式的解

### 族群規模的平衡

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

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

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

### 不動點的穩定性

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

#### 第一不動點

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

$\lambda_1 = \alpha,\quad \lambda_2 = -\gamma$

#### 第二不動點

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

$\lambda_1 = i \sqrt{\alpha \gamma},\quad \lambda_2 = -i \sqrt{\alpha \gamma}$

## 饱和沃尔泰拉方程

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

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

## 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.