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:

\[\mathbf{x_t} = [x, y, \alpha, \beta, \delta, \gamma]^T\]