# A Tour of the Generalized Lotka-Volterra Model

*2020-05-19*

# 1 Introduction

These lectures were prepared for the ICTP-SAIFR/IFT-UNESP **“School on Community Ecology: from patterns to principles”**, held in São Paulo, Brazil, January 20-25, 2020.

The four 90-minute lectures provide a semi-rigorous introduction to the Generalized Lotka-Volterra model, along with its applications to community ecology. Throughout the text, I have included code that can be used to simulate the processes that are introduced from a mathematical point of view.

The emphasis is squarely on models for large communities, composed of \(n \gg 1\) species. In particular, I often refer to “random” communities, i.e., dynamical systems with random parameters. This allows me to derive interesting results that are meant to describe the “typical” or “expected” community.

At the end of each lecture, I refer to my own work in the area, highlighting some of the strongest contributions from my laboratory at the University of Chicago.

## 1.1 Prerequisites

The lectures are fairly self-contained, but some working knowledge of calculus, dynamical systems, linear algebra, and game theory would be beneficial.

The code is written in `R`

, and meant to be run using a fairly recent version of `R`

along with the packages `tidyverse`

, `deSolve`

, `MASS`

, `lpSolve`

and `igraph`

, which need to be installed.

### References

Baigent, Stephen A. 2016. “Lotka-Volterra Dynamics: An Introduction.” In. World Scientific.

Hadeler, Karl Peter, Michael C Mackey, and Angela Stevens. 2017. *Topics in Mathematical Biology*. Springer.

Hofbauer, Josef, and Karl Sigmund. 1998. *Evolutionary Games and Population Dynamics*. Cambridge university press.