<< Lotka-Volterra_model

Lotka-Volterra Competition Model

\[ \newcommand{\d}{\text d} \newcommand{\dt}{\d t} \newcommand{\dN}{\d N} \]

# Modeling

For the two competition species with population number \(N_1, N_2\), the environment environmental capacity and instant growth rate are \(K_1, r_1, K_2, r_2\) respectively. For the species 1, based on Logistic equation, we have \[ \frac{\dN_1}{\dt} = r_1N_1\left( 1-\frac{N_1}{K_1}-\alpha\frac{N_2}{K_1} \right) \] where \(\alpha\) is the competition coefficient from species 2 to species 1, the physical meaning is “one animal of species 2 is the same as \(\alpha\) animals of species 1”. Similarly, for species 2 we have \[ \frac{\dN_2}{\dt} = r_2N_2\left( 1-\frac{N_2}{K_2}-\alpha\frac{N_1}{K_2} \right) \] where \(\alpha\) is the competition coefficient of species 1 to species 2.

The two equations give the Lotka-Volterra competition model.

# Conclusion¹

We could use the reciprocal of capacity \(K^{-1}\) as the indicator of the intensity of intraspecific competition, and \(\alpha K_1^{-1}\) as the intensity of interspecific competition from species 2 to species 1, symmetrically, \(\beta K_2^{-1}\) as the intensity of interspecific completition from species 1 to species 2.

One species win in the interspecific competition, if and only if the intensity of interspecific completition larger than the intensity of intraspecific competition, that is

1. \(K_1^{-1} < \beta K_2^{-1}, K_2^{-1} > \alpha K_1^{-1}\), species 1 win, species 2 is excluded.
2. \(K_1^{-1} > \beta K_2^{-1}, K_2^{-1} < \alpha K_1^{-1}\) , species 2 win, species 1 is excluded.
3. \(K_1^{-1} < \beta K_2^{-1}, K_2^{-1} < \alpha K_1^{-1}\), both species might win, unstable equilibrium.
4. \(K_1^{-1} > \beta K_2^{-1}, K_2^{-1} > \alpha K_1^{-1}\), both species exist.