Lotka-Volterra

sbisandbox.benchmarks.lotka_volterra.LotkaVolterraBenchmark

Bases: Benchmark

Lotka Volterra benchmark task.

The parameters \(\boldsymbol{\theta} = \{\alpha, \beta, \gamma, \delta\} \in \mathbb{R}^4\) are sampled from

\[ \alpha \sim \text{Lognormal}([-0.125, 0.5])\]

\[ \beta \sim \text{Lognormal}([-3, 0.5])\]

\[ \gamma \sim \text{Lognormal}([-0.125, 0.5])\]

\[ \delta \sim \text{Lognormal}([-3, 0.5])\]

The Lotka-Volterra equations characterize the evolution of two biological species: preys and predators, whose populations at a time \(t\) are denoted by \(x(t)\) and \(y(t)\) respectively. The two populations evolve as

\[ \frac{dx}{dt} = \alpha x - \beta xy \]

\[ \frac{dy}{dt} = -\gamma y + \delta xy \]

We define our data as the population levels of each species at 10 equally spaced time intervals, \(\boldsymbol{x} = \{(x(t_i), y(t_i))\}\) for \(t_i \in \{t_f / 10, 2t_f / 10, \ldots, t_f\}\), where the equations are integrated from \(t=0\) until \(t=t_f\).

Prey-predator populations for x(0) = 30 and y(0) = 1

Phase space plot for the same parameters and initial conditions

The model has interesting dynamics, displaying two different fixed points on its phase space. The wikipedia page is worth a read for a quick idea.

References

[1]: Lotka, Alfred James, and Vito Volterra. "Predator-prey model." Elements of Physical Biology (1925).