2.2. Lotka-Volterra equationsΒΆ
Here we will show how to generate forecasts for the (continuous) Lotka-Volterra equations, which describe the dynamics of biological systems in which two species interact (one predator, one prey).
\[\begin{split}\frac{dx}{dt} &= \alpha x - \beta xy \\
\frac{dy}{dt} &= \delta xy - \gamma y\end{split}\]
Symbol | Meaning |
---|---|
\(x(t)\) | The size of the prey population (1,000s). |
\(y(t)\) | The size of the predator population (1,000s). |
\(\alpha\) | Exponential prey growth rate in the absence of predators. |
\(\beta\) | The rate at which prey suffer from predation. |
\(\delta\) | The predator growth rate, driven by predation. |
\(\gamma\) | Exponential decay rate of the predator population. |
All of the state variables and parameters are stored in the state vector:
\[\mathbf{x_t} = [x, y, \alpha, \beta, \delta, \gamma]^T\]