The Lotka–Volterra Equations

Sunday May 31, 2020

•  mathematics •  differential equations •  ecology •  environmental science •  CMNA • 

My favorite set of differential equations are the Lotka-Volterra equations. These equations describe predator-prey relationships. What impresses me the most about them are their complete simplicity:

[latex]\begin{align} \frac{dx}{dt} &= \alpha x - \beta x \text{, and} \\ \frac{dy}{dt} &= \delta x y - \gamma y \end{align}[/latex]

In these equations, [latex]x[/latex] and [latex]y[/latex] are the population counts of some prey species and some predator species. [latex]\frac{dx}{dt}[/latex] and [latex]\frac{dy}{dt}[/latex] are their respective growth rates. The Greek describes the relationship between the two. Obviously, this is a bit simpler than we might imagine the reality, but it is, after all, only a model. But I think it is a powerful demonstration of how differential equations can model dynamic systems with elegance and beauty.

The Python agent-based modelling platform, Mesa, includes a dynamic system that models the Lotka-Volterra equations in a different way, but show the relationship through a population of sheep and wolves. Here’s a video of a typical run: