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Module summary: Darwin’s natural selection theory is a cornerstone of modern science. In the last decades, mathematical and computational modelling has led to significant advances in our understanding of evolutionary puzzles, such as what determines biodiversity or the origin of cooperative behaviour. Students of this module will be exposed to fundamental ideas of evolutionary modelling, and to the mathematical tools for their study. These will be illustrated by numerous paradigmatic examples motivated by exciting developments and challenges in mathematical biology.
Learning outcomes: On the completion of the module, students will be familiar with a range of mathematical tools, ideas and paradigmatic models allowing them to understand an important class of evolutionary phenomena. In particular, students will have been exposed to fundamental concepts of evolutionary game theory, Mendelian and population genetics.
Pre-requisites for MATH3567/MATH5567M: Ordinary Differential Equations as in (MATH1013 from 2024 / MATH1012 until 2024) or MATH1400, or equivalent; in addition to Probability and Statistics as in MATH1710, or equivalent. Some knowledge of Stochastic Processes, as in MATH2750, is useful but not strictly required.
Topics to be covered in MATH3567/5567M:
1. Introduction to evolutionary modelling
2. Modelling with difference equations
3. Modelling with ordinary differential equations
4. Introduction to Mendelian genetics
5. Introduction to game theory
6. Evolutionary game theory
7. Random processes: Discrete-time Markov chains
8. Random processes: Continuous-time Markov chains & birth-and-death processes
9. Evolutionary games in finite populations
10. Diffusion theory, Fokker-Planck equations & applications
11. Applications of diffusion processes to population genetics
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