Theoretical Biology
The mathematics of evolution, ecology, and interaction — from gene regulation and game theory to cultural transmission, with interactive simulations running in your browser.
This course develops the mathematical tools of theoretical biology from the ground up: dynamical systems, stochastic processes, evolutionary game theory, network models, and eco-evolutionary feedbacks. The emphasis throughout is on interaction-driven processes — how dynamics at one scale (genes, cells, individuals, populations) generate complexity at the next. It grew out of the research programme of the Dynamics of Living Systems group at the University of Würzburg.
Each chapter opens with a historical timeline, develops the theory with full derivations, and includes at least one in-browser interactive demonstration where you can adjust parameters and watch the dynamics unfold in real time. No installation, no licences, no data sent to any server.
What you will learn
By the end of this course you will be able to:
- Formulate, analyse, and interpret ODE and stochastic models of biological systems.
- Apply phase-plane methods, bifurcation theory, and linear stability analysis to biological problems.
- Model allele-frequency dynamics, genetic drift, and selection in finite populations.
- Construct and analyse evolutionary games — two-player, multiplayer, and on networks — and compute fixation probabilities.
- Build and critique models of epidemics, cancer, gene drives, mutualism, and coevolution.
- Use mathematical models of cultural transmission, cooperation, and cognitive bias to study human and animal behaviour.
- Translate between equations, code, and biological intuition by exploring interactive simulations.
Prerequisites
Calculus (differentiation, integration, basic ODEs), linear algebra (vectors, matrices, eigenvalues), and elementary probability theory. No prior exposure to mathematical biology is required — every concept is introduced in context.
How to navigate
The six parts below build progressively, and each chapter card lists its prerequisites. If a topic interests you, check the prerequisites first; if you have the background, jump straight in. Three suggested paths through the material:
- Biologists learning modelling: Start with Part 0 (Flows on the Line, Flows on the Plane), then work through Parts 1–2 before moving to game theory.
- Physicists or mathematicians entering biology: Skim Part 0 for notation, then jump to Part 3 (Evolutionary Game Theory) or Part 4 (Eco-Evolutionary Dynamics).
- Social scientists interested in cultural evolution: Read Evolutionary Game Theory (Part 3), then go directly to Part 5.
Companion reading
The course is self-contained, but pairs well with the following texts: Strogatz, Nonlinear Dynamics and Chaos (Parts 0–2); Nowak, Evolutionary Dynamics (Parts 1, 3–4); Hofbauer & Sigmund, Evolutionary Games and Population Dynamics (Part 3); Murray, Mathematical Biology (Parts 1–4); Boyd & Richerson, Culture and the Evolutionary Process (Part 5).
Mathematical Foundations
A compact introduction to the qualitative theory of ordinary differential equations — the language shared by every subsequent chapter.
Genes & Cells
Dynamics at the molecular and cellular scale — allele frequencies, gene circuits, and drug kinetics.
Population Genetics
Allele frequency dynamics — drift, selection, mutation, and the coalescent
Gene Regulatory Networks
From autoregulation to oscillatory circuits — the mathematics of gene expression
Pharmacokinetics
Mathematical models of drug absorption, distribution, and elimination
Populations & Ecosystems
The mathematics of populations, species interactions, and epidemics — from single-species growth to multi-trophic dynamics and disease.
Single-Species Population Dynamics
From exponential growth to chaos — the mathematics of populations
Interacting Species
Competition, predation, and mutualism — the ecology of multi-species systems
Infectious Disease Dynamics
Epidemics, endemic equilibria, and the coevolution of hosts and parasites
Evolutionary Game Theory
How strategic interactions shape the evolution of populations — from infinite-population dynamics to stochastic processes in finite groups.
Evolutionary Game Theory
From static equilibria to stochastic dynamics in finite populations
Multiplayer & Asymmetric Games
Beyond pairwise interactions — public goods, multiplayer dynamics, and bimatrix games
Games on Networks
How population structure shapes the evolution of cooperation and competition
Eco-Evolutionary Dynamics
When ecology and evolution operate on the same timescale — somatic evolution in cancer, the population genetics of gene drives, and the coevolutionary arms races between mutualists, hosts, and parasites.
Cancer & Cell Dynamics
Tumour growth, somatic evolution, and the ecology of cancer
Gene Drives & Evolutionary Interventions
When humans engineer evolution — the theory behind synthetic gene drives
Mutualism & Host-Parasite Coevolution
The eco-evolutionary logic of mutualisms, antagonistic coevolution, and the Red Queen
Cultural Evolution
When evolution meets culture — mathematical models of cultural transmission, the mechanisms of cooperation, and the cognitive architecture that shapes traditions.
Models of Cultural Evolution
Mathematical models of cultural transmission, biased learning, and gene-culture coevolution
The Evolution of Cooperation
From social dilemmas to the five mechanisms that sustain cooperation
Psycho-Cognitive Cultural Evolution of Traditions
How psychological mechanisms shape cultural transmission, intersubjectivity, and the evolution of traditions
Using the interactive elements
Each Explore section includes sliders, preset buttons, and canvas plots. Drag a slider to change a parameter and the trajectories redraw immediately. Presets load configurations that highlight particular regimes of behaviour (e.g. the bistable region of a toggle switch, or the limit cycle of a predator–prey system). Deliberately push parameters to extreme values to see where the assumptions of a model break down — this is often where the biology becomes most interesting.
Citing and contributing
The course is open-source. If you find an error, have a suggestion, or would like to contribute a new demonstration, please open an issue or pull request on the repository. If you use the material for your own teaching, please cite the Dynamics of Living Systems group. Feedback from students and instructors is welcomed and shapes the ongoing development of the course.