ロトカ‐ヴォルテラ方程式
Lotka-Volterra equation
2種競争系
$ \frac{dx}{dt} = r_1x \left( 1 - \frac{x + ay}{K_1} \right),
$ \frac{dy}{dt} = r_2y \left( 1 - \frac{y + bx}{K_2} \right)
2種捕食系
$ \frac{dx}{dt} = ax - bxy,
$ \frac{dy}{dt} = -cy + dxy
2種競争系の平衡解の導出:
$ \frac{dx}{dt}= r_1x \left( 1 - \frac{x + ay}{K_1} \right)=0,
$ \frac{dy}{dt}= r_2y \left( 1 - \frac{y + bx}{K_2} \right)=0
$ (x,y)=(0,0), (K_1,0), (0,K_2),\left(\frac{K_2-bx}{1-ab},\frac{K_1-ax}{1-ab}\right) $ \square
2種捕食系の平衡解の導出:
$ \frac{dx}{dt} = ax - bxy=0,
$ \frac{dy}{dt} = -cy + dxy=0
$ (x,y)=\left(\frac{a}{b},\frac{c}{d}\right),(0,0) $ \square