2023/24 Taught Postgraduate Module Catalogue

MATH5567M Advanced Evolutionary Modelling

20 creditsClass Size: 40

Module manager: Professor Mauro Mobilia
Email: m.mobilia@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2023/24

Pre-requisite qualifications

(MATH1012 or MATH1400) and MATH1710, or equivalent. Some knowledge of Stochastic Processes, as in MATH2750, is useful but not required.

This module is mutually exclusive with

MATH3567

Evolutionary Modelling

This module is approved as an Elective

Module summary

Darwin's natural selection paradigm is a cornerstone of modern evolutionary biology and ecology. Darwinian ideas have applications in social and behavioural science, and have also inspired research in the mathematical and physical sciences. In the last decades, mathematical analysis and theoretical modelling have led to tremendous progress in the quantitative understanding of evolutionary phenomena. Yet, many questions of paramount importance, like the "origin of cooperative behaviour" or "what determines biodiversity", are subjects of intense research and their investigation requires advanced mathematical and computational tools. The students of the co-taught MATH3567/MATH5567M modules will be exposed to fundamental ideas of evolutionary modelling. These will be introduced through influential models and paradigmatic examples that will be analysed by a combination of methods drawn from the theory of nonlinear dynamics and stochastic processes. The students of these modules will thus be introduced to some areas of applied mathematics that currently give rise to exciting new developments and prominent challenges in mathematical biology and in evolutionary dynamics.

Objectives

In this module, various tools and approaches will be used:
- Evolutionary modelling with maps and ordinary differential equations;
- Modelling evolutionary dynamics with random and stochastic processes in discrete and continuous time;
- Evolutionary dynamics with continuous-time Markov chains, birth-and-death processes, and diffusion processes.

Learning outcomes
On the completion of this module, students should have become familiar with a set of paradigmatic models and mathematical methods describing an important class of biological and evolutionary phenomena. In particular, students will have been exposed to many fundamental concepts underlying evolutionary game theory, Mendelian and population genetics.
In MATH5567M the introductory examples seen in MATH3567 will be completed by more advanced and specific topics, such as (i) some applications of two-dimensional maps; (ii) model of ordinary differential equations exhibiting bifurcations (Holling response); (iii) mutation-selection balance; (iv) asymmetric and iterated games; (v) recurrent, transient and absorbing states; (vi) detailed balance and Gillespie algorithm; (vii) notion of risk-dominance; (viii) selection in the Wright-Fisher and Moran models.

Syllabus

1. Introduction to evolutionary modelling;
2. Modelling with difference equations (incl. (i));
3. Modelling with ordinary differential equations (incl. (ii));
4. Mendelian genetics (incl. (iii));
5. Evolutionary game theory and social behaviour, (incl. (iv));
6. Random and stochastic processes (incl. (v) and (vi));
7. Evolutionary game theory in finite populations (incl. (vii));
8. Diffusion theory (Fokker-Planck) and population genetics (incl. (viii)).

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	44	1.00	44.00
Private study hours			156.00
Total Contact hours			44.00
Total hours (100hr per 10 credits)			200.00

Private study

Studying and revising of course material.
Reading as directed.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Examples sheets with detailed solutions.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	3 hr 00 mins	100.00
Total percentage (Assessment Exams)		100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 17/05/2023

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