ICTS QSB 2025

We have four lectures and a tutorial. We are going to concentrate on the Generalized Lotka-Volterra model—which turns 100 years old in 2026.

• Lecture 1. The Generalized Lotka-Volterra model for \(n\) interacting populations.
• Lecture 2. GLV with random parameters.
• Lecture 3. GLV to model experimental data.
• Tutorial. Applications of GLV models for experimental microbial communities.
• Lecture 4. GLV and metapopulation dynamics.

Computing

The GitHub repository associated with the lecture notes contains the code used to generate the lecture notes, including all the figures and simulations. Crucially, the code depends on the following R packages:

library(tidyverse) # plotting, data organization
library(deSolve) # integrate differential equations

Notation

• We write \(x(t)\) for the density of population \(x\) at time \(t\). In many cases, we will write simply \(x\), as the dependency on time is always assumed. For discrete-time models, we write \(x_t\) instead. Typically, \(x(t)\) is measured in either \([\text{mass}]/[\text{area/volume}]\) or \([\text{number of individuals}]/[\text{area/volume}]\)

• The density of a population typically changes in time, and the change is modulated by several parameters. Unless specified, we use Greek letters for scalars, lower-case Latin letters for vectors, and upper-case Latin letters for matrices.

• Other useful notation:

  • \(i\) is the imaginary unit, such that \(i^2 = -1\)
  • \(0_n\) is a vector of zeros of lenght \(n\)
  • \(1_n\) is a vector of ones of lenght \(n\)
  • \(I\) is the identity matrix (i.e., a matrix with \(1_n\) on the diagonal, and zeros elsewhere)
  • \(D(a)\) is a diagonal matrix with vector \(a\) on the diagonal
  • \(H(A)\) is the symmetric part of matrix \(A\): \(H(A) = \frac{1}{2} (A + A^T)\)
  • \(\circ\) is the Hadamard (element by element) product (i.e., \(a \circ b = c\) with \(c_i = a_i b_i\), the same holds for matrices)
  • \(\dfrac{d x(t)}{dt}\) is sometimes written as \(\dfrac{d x}{dt}\) or \(\dot{x}\)