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2013/14 Undergraduate Module Catalogue

MATH3567 Evolutionary Modelling

15 creditsClass Size: 61

Module manager: Dr M. Mobilia
Email: m.mobilia@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2013/14

Pre-requisite qualifications

(MATH2391 or MATH2375) and MATH1715, or equivalent. Some knowledge of Stochastic Processes, as in MATH2750, is useful but not required.

This module is mutually exclusive with

MATH5567M

Advanced 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 sciences and have also inspired research in the mathematical and physical sciences. In the last decades, mathematical and modelling analysis has thus 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. In this context, the students of this module will familiarize themselves with fundamental evolutionary ideas that will be introduced through influential models and paradigmatic examples. These will be analysed by combining methods of nonlinear and stochastic dynamics. In this module, the students 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 evolutionary dynamics.

Objectives

This module consists of four parts:
- deterministic non-spatial models with applications to population genetics and evolutionary games;
- deterministic spatial models with applications to biological movement and pattern formation;
- stochastic biological modelling with discrete and continuous Markov chains;
- modelling evolutionary games and populations genetics with Markov chains.

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. These will be described by combining:
- difference equations (Part I);
- ordinary differential equations (Part I);
- partial differential equations (Part II);
- discrete and continuous Markov chains (Part IV);
- principles of evolutionary dynamics in game theory and population genetics (Parts I and IV).

Syllabus

- Difference equations: linearization, stability, bifurcations, applications to logistic map and parasitoid models;
- Ordinary differential equations: basic techniques for scalar and coupled equations, elements of bifurcation , two-species interacting population models, law of mass action and epidemic model;
- Deterministic approach to evolution: basic notions of population genetics (selection, fitness, allele, ...) and evolutionary game theory (equilibrium, stability, replicator dynamics,...);
- Modelling biological motion: macroscopic theory, advection-diffusion and reaction-diffusion equation, directed motion and chemotaxis, travelling waves;
- Pattern formation and morphogensis: Turing instability and bifurcation, applications to activatorinhibitor systems;
- Review of probability theory and discrete-time Markov chains: random variables (distributions, moments, ...), stochastic processes, Chapman-Kolmogorov equation, first-passage properties, random walks;
- Continuous-time Markov chains: Poisson process, master equations; waiting times, birth-death processes, extinction;
- Evolutionary games in finite population: 2x2 games, Moran model, fixation probability, weak selection and evolutionary stability in finite population;
- Diffusion processes and applications to population genetics: diffusion processes and Kolmogorov differential equations, first-passage properties, Wright-Fisher model, genetic drift, fixation properties.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	33	1.00	33.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Private study

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

Opportunities for Formative Feedback

Examples sheet and practice exam near the end of the semester.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr 30 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: 17/02/2014

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