Mathematical Biology involves using mathematical techniques and computational tools to answer problems that arise in Biology.
Overview
The BSc in Mathematical Biology is the only dedicated undergraduate mathematical biology degree offered in the UK. We also offer a four or five year MSci.
Our course offers a unique blend of modern mathematics applied to important biological processes. For example, mathematics can be used to understand how cells in a tumour grow.
The BSc (Hons) Mathematical Biology is a degree programme certified by the Institute of Mathematics and its Applications.
Researchled teaching
We believe that undergraduates are best served by studying in a division that is active in research to the highest level, and we are fortunate in having an excellent international reputation for our research work, with a dedicated Mathematical Biology Research Group. Our courses are taught by those who have great enthusiasm for their subject, a dedication to it, and an appreciation of the needs of students.
We are a relatively small division and operate with an excellent staff/student ratio. One advantage of this is that we can get to know each student personally, and so can offer a friendly and supportive learning experience. Staff are ready and willing to help at all levels, and in addition, our StudentStaff Committee meets regularly to discuss matters of importance to our students.
Computing facilities
Computing power is used extensively in modern mathematics. You will have access to wellequipped PC suites, and be able to use specialist Mathematics software packages.
Student society
You will be able to join DUMaS (Dundee University Maths Society), an active society open to all students studying Mathematics or Mathematical Biology.
Entry Requirements
The following are the minimum, uptodate entry requirements.
Courses starting 2018 and 2019  

Qualification  Level 1 Entry  Advanced Entry to Level 2 
SQA Higher/Advanced Higher  BBBB (minimum)  AABB (typical) at Higher including mathematics at B plus either biology or physics at B, and chemistry at C  AB at Advanced Higher including mathematics at A, plus AB at Higher in different subjects, including chemistry and biology at B (with at least one of them at Advanced Higher and the other at Higher) 
GCE ALevel  BCC (minimum)  BBB (typical) including ALevel mathematics at B, plus either ALevel biology or physics at B, and AS Level chemistry at C  BBC  ABB at ALevel including ALevel mathematics at A, and chemistry and biology at B (with at least one of them at ALevel and the other at AS Level) 
BTEC  A relevant BTEC Level 3 Extended Diploma with DDM  A relevant BTEC Level 3 Extended Diploma with DDD. 
International Baccalaureate (IB) Diploma  30 points at Higher Level grades 5, 5, 5 to include mathematics plus either biology or physics, and chemistry. A combination of IB Certificate plus other qualifications, such as ALevels, Advanced Placement Tests or the International Baccalaureate Careerrelated Programme (IBCP), will also be considered. 
34 points at Higher Level grades 6, 6, 5 to include mathematics at grade 6, and chemistry and biology at grade 5 
Irish Leaving Certificate (ILC)  H2H2H3H3 at Higher Level including Higher Level mathematics plus either biology or physics at H3, and chemistry at H4  Level 2 entry is not possible with this qualification 
Graduate Entry  
SQA Higher National (HNC/HND)  A relevant HNC with B in the Graded Unit including Mathematics for Engineering 1  A relevant HNC with A in the Graded Unit including Mathematics for Engineering 2 and 120 SCQF points. A relevant HND with BB in the Graded Units including Mathematics for Engineering 2 
Scottish Baccalaureate  Pass with BC at Advanced Higher in Mathematics and a Science/Engineering subject  Distinction with AB at Advanced Higher in Mathematics and a Science/Engineering subject 
SWAP Access  Relevant science subjects with ABB grades including Mathematics and Physics Units at SCQF Level 6  Level 2 entry is not possible with this qualification 
Advanced Diploma  Grade B with ASLA Levels at AB in Mathematics and a Science/Engineering subject  Grade B with ASLA Level at AA in Mathematics and a Science/Engineering subject 
Welsh Baccalaureate  Pass with A level at AB in Mathematics and a Science/Engineering subject  Pass with A level at AA in Mathematics and a Science/Engineering subject 
European Baccalaureate  70% overall with 7 in Mathematics  75% overall with 7.5 in Mathematics 
Other Qualifications  
Notes 
EU and International qualifications
English Language Requirement
For non EU students
IELTS Overall  6.0 

Listening  5.5 
Reading  5.5 
Writing  6.0 
Speaking  5.5 
Equivalent grades from other test providers
English Language Programmes
We offer PreSessional and Foundation Programme(s) throughout the year. These are designed to prepare you for university study in the UK when you have not yet met the language requirements for direct entry onto a degree programme.
Teaching Excellence Framework (TEF)
The University of Dundee has been given a Gold award – the highest possible rating – in the 2017 Teaching Excellence Framework (TEF).
Teaching & Assessment
How you will be taught
You will learn by traditional methods such as lectures, tutorials, and workshops as well as via computer assisted learning. From Level 2 onwards we teach the use of professional mathematical software packages in order to allow you to explore mathematics far beyond the limits of traditional teaching. This also prepares you for the way in which Mathematicians work in finance, industry and research.
Our excellent staff student ratio allows for close contact between students and lecturers and provides a lively learning environment. For example, we run "Maths Base'', a dropin facility in the Division, which provides another source of help with your studies over and above what you can expect from your lectures and tutorials.
How you will be assessed
Assessment for most modules involves a final examination and takes into account varying amounts of continuous assessment, including assignments and class tests.
What you will study
MSci Honours Degree
An MSci honours degree normally takes five years, full time, you study levels 15, as described below.
BSc Honours Degree
An honours degree normally takes four years, full time, you study levels 14, as described below.
Advanced Entry Honours Degree
It is possible to study for the MSci or BSc honours degree in one year less if you have the required grades and subjects as listed in the Entry Requirements section. You study levels 24 for the BSc or 25 for the MSci. There are definite advantages to considering this route as the time needed to study is reduced by one year which enables you to start working and earning earlier.
Typical Degree Programme
Level 1
The Mathematics modules at Level 1 provide you with an introduction to university mathematics, following on from material that you will have studied at school. Two thirds of your time is spent studying Life Science modules to give you the Biological background required.
Semester 1
Number of credits: 10
This module will broaden knowledge and understanding of atomic and molecular structure, enzyme function, energy and thermodynamics that can be applied to the Life Sciences.
Topics include:
 molecular structure and biochemical properties (to include organisation and charge/polarity) of nucleotides, lipids and phospholipids, carbohydrates and amino acids
 protein structure (primary to quaternary)
 biological enzymes and their importance in the catalysis of biochemical reactions
 energy and thermodynamics relating to biochemical reactions
Credits: 10
Semester: 1
Natural selection has favoured structures whose shape and composition contribute to their function. This module aims to broaden knowledge and understanding of the evolution of animal, plant and microbial cell structure and function, including organelles and membranes.
Topics include:
 cellular evolution
 interpretation of phylogenetic analysis
 evolution of organelles
 membrane structure and transport
 key features of life and death of animal, plant and microbial cells
Semester: 1
Number of credits: 10
This module will start with a mandatory introduction to health and safety, basic lab skills and equipment. There will be one field excursion (Botanic Gardens, Perth Road, Dundee) and a series of practical classes that will cover techniques of isolation and culture of microorganisms and gram staining, isolation and identification of terpenes from plant material, enzyme kinetics, thermodynamics and PCR.
Semester: 1
Number of credits: 10
This module will extend and develop the generic skills introduced in BS11003 with specific emphasis on health and safety and basic laboratory practice. The ability to work effectively as part of a group will form a significant part of this module. You will extend your information literacy skills by locating and accessing scientific resources to support your learning.
To support the group lab project, you will receive guidance on lab book / record keeping, experimental design and project planning, in addition to interpretation of data and presentation of project results. You'll also be encouraged to reflect on and evaluate your own learning throughout the semester, identifying areas for development and consolidation.
Semester: 2
Number of credits: 10
This module aims to broaden knowledge and understanding of the flow of genetic information (DNA to mRNA to protein), cell division (to include mitosis, meiosis and binary fission) and the cell cycle.
Topics include:
 cellular reproduction (DNA replication, mitosis, meiosis and binary fission)
 an introduction to the cell cycle
 an introduction to genes and gene expression
 an introduction to ribosomes and their function
 multigene, polygene inheritance of traits
 an introduction to population genetics
 genetical theory of natural selection
Semester: 2
Number of credits: 10
Physiological functions are often compartmentalised into different cells, tissues, organs and systems which have structures that support specialised activities. This module aims to broaden knowledge and understanding of the development of systems (e.g. neural, respiratory, cardiovascular, musculoskeletal and excretory) with reference to model organisms (e.g. nematodes, Drosophila, zebra fish, chick, mouse).
Topics include:
 an introduction to the development of an organism, with reference to the germ layers (ectoderm, mesoderm and endoderm)
 an overview of the key features of the body plan
 model organisms commonly used in scientific research (e.g. nematodes, Drosophila, zebra fish, chick, mouse)
 developmental origins of the following systems (with reference to the selected model organisms covered previously): neural, cardiovascular, respiratory, musculoskeletal, dermal, gastrointestinal and excretory
Semester: 2
Number of credits: 10
This module will extend and develop laboratory and research skills introduced in semester 1 of Level 1
 Optical techniques – Students will use a spectrophotometer to produce a standard absorbance spectrum, apply the Beer Lambert Law to derive unknown concentrations from known values of absorbance and perform a Bradford Assay
 Protein purification – Students will experience two techniques that are commonly used to separate mixtures of proteins: size exclusion chromatography (SEC) and SDS polyacrylamide gel electrophoresis (SDS PAGE)
 PCR  In conjunction with its associated workshop, this laboratory exercise aims to give students a basic understanding of the practical application of the polymerase chain reaction (PCR) and the use of agarose gel electrophoresis for the analysis of DNA samples.
 Enzyme kinetics – Students will gain practical experience of a typical enzyme assay procedure.
 Digital skills – online scientific literature searches
 Protein expression – Students will learn how to purify and analyse a recombinant protein.
Semester: 2
Number of credits: 10
This module will extend and develop the generic skills introduced in BS12003 with specific emphasis on data presentation, interpretation and analysis. The ability to work effectively as part of a group and the application of peer support and peerassessment will form a significant part of this module. Students will extend their information literacy and scientific writing skills by researching and presenting an area of current research in poster format, giving due attention to scientific writing protocols. Students will be encouraged to reflect on and evaluate their own learning throughout the semester, identifying areas for development and consolidation and setting appropriate targets.
About the module
This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 1 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on nonmathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor.
Prerequisites
Students taking this module should typically have at least a B in Mathematics in Scottish Highers, AS, or ALevel, or an equivalent qualification.
Indicative Content

Functions
Number systems (N, Z, Q, R), open and closed intervals, elementary functions, domain, range, composition, inverse. Inequalities. Idea of a limit for functions and for sequences.

Differential Calculus
Derivatives, tangents and rates of change. Simple derivatives by first principles. Treatment of (f + g)′ , (fg)′, (f/g)′, (f ? g)′ and inverse functions. Higher order derivatives. Implicit Differentiation. Revision of index laws and log to base a. Definitions and elementary properties of exp and ln. Solution of equations involving exponential and logarithmic functions. Differentiation of functions involving exponential and logarithmic functions. Logarithmic differentiation. Tangents and Normals to curves. Increasing and decreasing functions. Critical points. Curve sketching (including asymptotes).

Polynomials
Quadratic polynomials. Algebra and geometric representation of complex numbers. Division algorithm, Remainder theorem. Roots of polynomials. Techniques of partial fraction decomposition.

Trigonometry
Definitions and properties of the six trigonometric functions, including formulae for sin(A + B), sin A sin B, sin A + sin B, etc. Solution of trigonometric equations (including a cos(x) + b sin(x) = c).

Series
Series as sequences of partial sums. Summation of series and sigma notation. Convergence of series, geometric series. Examples of finite and infinite series. Binomial theorem.

Conics
Classification, standard forms, parametric representations.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via coursework (100%) consisting of homeworks, projects and tests.
Credit Rating
This module is a Scottish Higher Education Level 1 or SCQF level 7 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.
About the module
This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 1 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on nonmathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor.
Prerequisites
Students taking this module must have taken the module MA11001, or equivalent.
Indicative Content
Calculus Component

Integral Calculus
Idea of integral, including elementary treatment of the definite integral as a limit using rectangles. Fundamental theorem of calculus. Methods of integration including integration by substitution, by parts and with partial fractions. Relation of integrals with areas. Trapezium and Simpson′s rules for numerical integration.

Differential Equations
First order ordinary differential equations by (a) separation and (b) integrating factor. Second order ordinary differential equations with constant coefficients and simple right hand sides. (Complex roots included, but no resonance problems.)
Algebra Component

Vectors
Vector geometry in R² and R³ vector properties and manipulation. Unit vectors, position vectors, Cartesian coordinates. Scalar product and vector product.

Matrices and linear equations
Matrix properties, addition, multiplication. Inverse matrices, determinants. Linear mappings in R² (rotation, reflection). Systems of linear equations, Gaussian elimination and row operations.

(Further) complex numbers
Polar form, exponential notation. Multiplication, de Moivre′s Theorem, powers and roots.

Lines, planes and spheres
Implicit and parametric equations of lines. Implicit equations of planes. Intersections, distances between points, lines and planes. Equations of spheres, tangent planes. Linear dependence and independence, colinear and coplanar vectors.
Delivery and Assessment
This module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (50%) and coursework (50%) consisting of homeworks and tests and project work.
Credit Rating
This module is a Scottish Higher Education Level 1 or SCQF level 7 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.
Level 2
At Level 2 students taking a degree fully within Mathematics take either 4 or 5 modules in Mathematics, with the remaining 1 or 2 modules being taken in a subject of their choice. Joint degrees typically involve a 5050 split between Mathematics and the other subject.
Semester 1
Number of credits: 10
An introduction to statistics for the biosciences, bringing you up to speed on the structure of datasets and how one infers differences. This will include an introduction to the design of experiments, concepts of randomisation and blocking. Regression and Multiple regression; ANOVA (including posthoc testing); principle tests of signal versus noise. You will also be introduced to appropriate graphing and representation of data. You will be expected to complete a substantial piece of coursework related to a component of the indicative content.
Semester 1
Number of credits: 10
To introduce and broaden knowledge and understanding of key inter and intracellular communication mechanisms with reference to selected examples of relevance to both biological and biomedical contexts. This module will provide you with an overview of inter and intracellular communication mechanisms (e.g. paracrine, endocrine, hormonal and neural) and subsequent cell responses.
On successful completion of the module you will be able to demonstrate knowledge and understanding of:
 key extracellular signalling molecules, receptor types and cell response (e.g. changes in gene expression and protein modifications)
 the general modes of action of steroid and peptide hormones
 neurotransmission (e.g. compare and contrast with hormones)
 nitric oxide signalling in the cardiovascular system
 plant cell signalling
 regulation of the cell cycle
Skills:
 access and use a range of defined and self selected (with guidance) learning resources to further your studies
 evaluate own learning, identifying strengths and weaknesses within criteria largely set by others
 use the knowledge gained to solve a set of typical problems in the biological sciences
 work individually and in groups to extract pertinent information from the scientific literature
Number of credits: 20
The aim of this module is to give you a sound foundation in biomolecular mechanisms and processes. This module will study the main mammalian metabolic pathways and their control including the molecular processes involved. The module will also look at current topics in microbiology including disease and resistance and introduce immunology and virology.
About the module
This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 2 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on nonmathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, or equivalents.
Indicative Content
Calculus Component

Differential Equations
Revision of linear differential equations of second order with constant coefficients using undetermined coefficients. General solutions and solutions satisfying initial conditions. Resonance. Equations of higher order.

Hyperbolic Functions
Hyperbolic Functions; Solution of simple equations, inverse functions. Revision of standard methods of definite integration, including hyperbolic substitutions.

Fundamentals of Calculus
Limits, Continuity and Differentiability. Rolle′s Theorem, Mean Value Theorem. Definition and properties of the Riemann integral, Fundamental Theorem of Calculus. L′Hôpital′s Rule and Indeterminate Forms. Infinite and improper integrals. Taylor & Maclaurin series.
Algebra Component

Vectors and vector spaces
Definition of a vector space, R^{n}. Vectors, lines and planes in R^{n}. Span, linear independence. Basis and dimension. Subspaces.

Inner product
Scalar product, length. Projection. Normal form of hyperplanes in R^{n}. Orthogonality.

Linear equations and matrices
Systems of linear equations. Gaussian elimination. Matrices and matrix operations, transposes and inverses. Matrix equations. LU factorisation. Determinants.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.
Credit Rating
This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.
About the module
The aim of this module is to make the students familiar with a Computer Algebra software package and to use this software to solve a number of problems from the area of Dynamical Systems. This module is mandatory for all Level 2 students on Mathematics (including Mathematics combined) degrees except for those taking the BSc or MSci in Mathematics and Physics, for whom it is optional. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, or equivalents.
Indicative Content

An introduction to Maple
The Maple front end and syntax. Plotting. Integration/differentiation. Differential equations.

An introduction to Dynamical Systems
Vector fields, the gradient field, integrals of motion, fixed points and their classification. Examples of dynamical systems, including mass on a spring, pendulum, Van Der Pol oscillator, nonlinear oscillator. Conservation laws for a system of interacting bodies. Orbits in a gravitational field. Nonautonomous systems.
Delivery and Assessment
The module is delivered in the form of lectures and computer lab workshops. Assessment is entirely computerbased and is via an exam (60%) and coursework (40%) consisting of homeworks and tests.
Credit Rating
This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.
About the module
This module provides an introduction for Level 2 students to various topics in Discrete Mathematics. This module is mandatory for Level 2 students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematical Biology. This module is optional for students taking the BSc in Mathematics combined with any of Accountancy, Economics, Financial Economics or Psychology. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, or in EG11003 and EG12003, or equivalents.
Indicative Content

Sets and Graphs
Sets and subsets: definitions, examples, Set operations, basic identities, power of a set, Cartesian product of sets, relations on sets, Basic graph terminology.

Recurrence relations (Difference Equations)
Definition of a recurrence relation (difference equations), Homogeneous and inhomogeneous difference equations, Nonlinear difference equations: x_{n+1} = g(x_{n}), Fixed points, linearisation, stability of fixed points. Applications: the Newton and Secant Methods to solve nonlinear equations f(x) = 0, Programming: Short introduction to Matlab, Numerical algorithms for difference equations: Newton′s method, Fibonacci sequences, Recursion.

Markov Chains
Definition of Markov chains, probability vectors, and stochastic matrices, Connection between a Markov chain and a second order difference equation, Long time behaviour of a process described by a Markov chain, Random walk as a Markov chain, Absorbing and irreducible Markov chains.

Combinatorics
Permutations and combinations, Binomial coefficients and their properties, Binomial theorem, Principle of inclusion and exclusion. Derangements, Partitions and Stirling numbers, Transpositions and Cycles, Multinomial Theorem, Newton′s Binomial Theorem.

Game Theory
Strategic form games, Dominated Strategies, Nash Equilibrium, Prisoner′s Dilema, Twoperson zerosum games, The minimax Theorem, Extensive form games with perfect information.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.
Credit Rating
This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.
About the module
This module consists of a Calculus and Algebra component. It is part of a series of four modules, Mathematics 1A, 1B, 2A, 2B, which are the core Mathematics modules in years 1 and 2, and provide the foundations in Calculus, Algebra and Geometry for all mathematics modules in higher levels. This module is mandatory for all Level 2 students on Mathematics (including Mathematics combined) degrees. The module is also suitable for students on nonmathematics degrees and recommended for students on physics and computing degrees. If you have questions about this module please contact our Undergraduate Admissions Tutor or your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA11001 and MA12001, and have taken MA21001, or equivalents.
Indicative Content
Calculus Component

Multivariable Calculus
Limits and continuity of functions of two variables. Partial Derivatives. Method of Lagrange Multipliers. Taylor series in two variables. Stationary points for functions of two variables. Double integrals, Jacobian of a change of variables.

Series
Tests for convergence of series of numbers. Convergence of power series, radius of convergence. Application to Taylor and Maclaurin series (mention of Taylor's Theorem).
Algebra Component

General vector spaces and subspaces
P_{n}, C^{n} and other vector spaces. Span, linear dependence/independence, bases. Reduction to rowecehelon form, relation to linear independence Intersections, unions and direct sums of subspaces. Range and nullspace of a matrix.

Inner products
Definition of inner products and inner product spaces. GramSchmidt orthogonalisation.

Eigenvalues and eigenvectors
Definitions and examples. Complex and repeated eigenvalues, algebraic and geometric multiplicity. Diagonalization of matrices. The CayleyHamilton theorem.

Linear mappings
Definitions and matrix representations. Composition of linear mappings. Kernel and image.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%).
Credit Rating
This module is a Scottish Higher Education Level 2 or SCQF level 8 module and is rated as 20 SCOTCAT credits or 10 ECTS credits.
Levels 3 and 4
Third and fourth year modules are designed to cover both Mathematical and Biological topics related to Mathematical Biology. Mathematics topics will include:
About the module
This module provides an indepth study of Differential Equations aimed at Level 3 students. This module is mandatory for all Level 3 students on Mathematics (including Mathematics combined) degrees. If you have questions about this module please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.
Indicative Content

First Order Differential Equations
Separable equations, Linear equations with constant coefficients, Linear equations with variable coefficients, integrating factors, Homogeneous equations, Exact equations and integrating factors.

Second Order Differential Equations
Homogeneous equations with constant coefficients, Fundamental solutions of linear homogeneous equations, Linear independence and the Wronskian (including Abel′s formula), Reduction of order and reduction to the normal form, Nonhomogeneous equations, Method of undetermined coefficients, Initial conditions.

Systems of First Order Linear Equations
Transformation of an nth order equation to a system of n first order equations, Homogeneous linear systems with constant coefficients, Fundamental sets of solutions and fundamental matrices, the Wronskian and Abel′s formula, The exponential of a matrix, Nonhomogeneous linear systems, Variation of parameters, Homogeneous linear systems of two first order equations with constant coefficients, Stability and the phase plane.

Partial Differential Equations and Fourier Series
Fourier series of functions of one variable, Dirichlet′s Conditions, Technique for determining Fourier coefficients (even/odd functions). Gibbs′ phenomena. Introduction to Partial Differential Equations, Technique of separation of variables with application to initial and boundary value problems.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
The aim of this module is to provide the Level 3 student with a variety of basic mathematical techniques with which to analyse a wide class of mathematical models arising in science and engineering. This module is mandatory for all Level 3 students on Mathematics (including Mathematics combined) degrees. If you have questions about this module please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.
Indicative Content

Revision of vector products and scalar functions of three variables.

Orthogonal coordinates.

Curves in space, parameterization and arc length

Surfaces in space, parameterization, normal vectors and tangent planes.

The operators grad, div, curl.

Line integrals, surface integrals, and volume integrals.

Divergence Theorem and Stokes Theorem.

Scalar and vector potentials.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
The aim of this course is to introduce you to some biological phenomena and their formulation in terms of mathematical models, which lead to difference equations and ordinary differential equations, and to investigate the solutions of these equations. This module is mandatory for students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematical Biology, and may optionally be taken in combination with other modules by students on any of the other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents, and must have taken modules MA31002 and either MA31007 or MA32002.
Indicative Content

Single Species Population dynamics
Difference equations: graphical analysis, fixed points and linear stability analysis. First order systems of ordinary differential equations: logistic equation, steady states, linearisation, and stability. Harvesting and fisheries.

Interacting Species
Systems of difference equations (hostparasitoid systems). Systems of ordinary differential equation (predatorprey and competition models).

Molecular Dynamics
Biochemical kinetics: MichaelisMenten kinetics. Metabolic pathways: activation and inhibition.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module is a personal research project which runs over both semesters. During the first semester there will be classes in computer labs where students will develop skills that will be useful in completing a project. The module consists of a substantial project together with a project report. This module is mandatory for Level 4 students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematical Biology. If you have questions about this module please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA31002 and either MA31007 or MA32002, or equivalents.
Indicative Content

MATLAB
Become familiar with the basic concepts of MATLAB and its use in solving differential equations.

Communicating Mathematics
Learn how to typeset mathematics using LaTeX. Understand how to structure, set out, and typeset a mathematical report.

Presentations
Prepare and deliver presentations on general topics in mathematics, and on your chosen project.

Project
Carry out a substantial project in an area of mathematics and document the work in a project report.
Delivery and Assessment
The project will require substantial independent work as well as regular meetings with the project supervisor(s), as well as the inclass contact hours in semester 1. Assessment will be based on coursework (100%) consisting of the project report and presentations.
Credit Rating
This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 30 SCOTCAT credits or 15 ECTS credits.
About the module
Ordinary Differential Equations (ODEs) are an important modelling tool in Science and Engineering. These can rarely be solved exactly and so techniques have been developed to derive approximate solutions that may, in principle, be made as accurate as desired. This module, aimed at Level 4 students, will investigate these techniques. This module is mandatory for Level 4 students taking a BSc or MSci in Mathematical Biology, or an MSci in Mathematics and Physics. This module may be taken in combination with other Level 3 or 4 modules by other Level 4 students. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA31002 and either MA31007 or MA32002, or equivalents.
Indicative Content

Numerical methods for initial value problems for ODEs
Taylor Series Methods; Linear multistep methods: onestep methods (Euler, Trapezoidal and Backward Euler methods) and twostep methods; Consistency, zerostability, weak stability theory and Astability; Provision of the extra starting values and the potential for instability; RungeKutta methods: construction and weak stability theory; Application to systems.

Boundary value problems for ODEs
BVPs for second order ODEs; eigenvalues and eigenfunctions; orthogonality; Green′s functions and maximum principles; Finite difference methods: 2nd order methods; Treatment of boundary conditions; Discrete maximum principles; Convergence.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
The aim of this module is to introduce Level 4 students to some biological phenomena and their formulation in terms of mathematical models, building on the work in MA32009/MA41002 (Mathematical Biology I). This module is mandatory for Level 4 students taking the BSc or MSci in Mathematical Biology, and may optionally be taken in combination with other modules by students taking the BSc or MMath in Mathematics or any of the other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in the module MA32009, or in MA41002, or equivalent.
Indicative Content

Modelling of Biological Systems using Partial Differential Equations
Derivation of conservation equations. Different models for movement (e.g. diffusion, convection, directed movement). Connection between diffusion and probability.

Linear reactiondiffusion equations
Fundamental solution for linear diffusion equations. Speed of a wave of invasion.

Nonlinear reactiondiffusion equations
Travelling wave solutions for monostable equations (e.g. Fisher equation). Travelling wave solutions for bistable equations.

Systems of reactiondiffusion equations
Travelling wave solutions for systems of reactiondiffusion equations. Pattern formations in systems of reactiondiffusion equations. Pattern formations in chemotaxis equations.

Mathematical modelling of infection diseases (SIR)
Derivation of a simple SIR model. Travelling wave solutions for the simple SIR model. Generalisation of the simple SIR model. Stochastic SIR model.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module gives a broad introduction to PDEs that includes classification into different types, classical solution methods, qualitative properties and, for the majority of problems that cannot be solved exactly, provides techniques for constructing approximate solutions. This module is mandatory for Level 4 students taking the BSc or MSci in Mathematical Biology, and may optionally be taken in combination with other modules by students taking the BSc or MMath in Mathematics or any of the other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have taken the module MA41003, or equivalent.
Indicative Content

First and Second Order PDEs
Basic Theory; examples of fundamental solutions Second order linear PDEs; classi fication, characteristics; dAlemberts solution of the onedimensional wave equation.

Boundary Value Problems for PDEs
Finitedifference methods for second order problems (Poisson's equation): the treatment of boundary conditions and curved boundaries in two dimensions.

Initial Value Problems for PDEs
Parabolic and Hyperbolic equations: Fundamental solutions. General discussion of basic qualitative properties such as dissipation (energy inequalities) and characteristics. Construction of numerical methods: twolevel methods and brief reference to threelevel methods (if time permits). Local truncation errors. Stability and choice of norm: Maximum norm, L2 norm via von Neumann's method. Application to hyperbolic systems. The Method of Lines.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
Further 2 modules
Two further maths modules can be chosen from the following topics:
About the module
This module provides an indepth study of Analysis aimed at Level 3 or 4 students in which the concepts are defined precisely and the results are proved rigorously. This module is mandatory for Level 3 students taking a BSc or MMath in Mathematics. This module may be taken in combination with other Level 3 or 4 modules by Level 4 students on Mathematics combined degrees other than those taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.
Indicative Content

Normed and Metric Spaces
Supremum, Completeness Axiom. Definitions and properties of normed and metric spaces, convergence of sequences, continuity, closed sets (in terms of limit points)

Connectedness and Completeness
Connected sets: definition in metric spaces; relation to the concept of continuity. Cauchy sequences, completeness and relation to closed sets, Banach's contraction mapping theorem

Compact Sets
General definition with open sets of the notion of compactness, its sequences characterisations on metric spaces, and its connection with closed subsets. "Closed and bounded" characterisation of the compact sets in R^{n} (i.e., the HeineBorel Theorem), connection with limit points, Weierstrass Theorem. Connection between continuity and compactness. The concept of uniform continuity and its connection with compactness. Urysohn's Lemma [1 lecture]. Partition of Unity on R^{n}

Convergence and Equicontinuity
Uniform convergence of sequences of functions. The concept of equicontinuity of a family of functions and ArzelaAscoli Theorem.

Series
Ratio test, comparison test, Weierstrass Mtest. Power series, Taylor series (mention of Taylor's theorem).
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module provides an indepth study of Operational Research aimed at Level 3 or 4 students. This module is mandatory for Level 3 students taking a BSc or MMath in Mathematics. This module may be taken in combination with other Level 3 or 4 modules by Level 4 students on Mathematics combined degrees other than those taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.
Indicative Content

Linear Programming
Structure of Linear Programming problems, Graphical solutions, Simplex method, Duality, Integer linear programming, Branch and Bound method.

Transportation Problems
Transportation and Assignment problems, initial basic feasible solution, Hungarian method.

Network Programming
Graphs, Minimum cost flow, Maxflow/Mincut theorem, FordFulkerson method.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
Matrix algebra is a fundamental and widely used resource for modelling a wide variety of problems in science, technology, industry and commerce. The aim of this course is to use computers to implement algorithms and to solve a number of problems that can be stated in terms of matrixrelated equations, and to understand the relevant matrix theory that underpins these algorithms. This module is mandatory for students taking the BSc or MMath in Mathematics and may optionally be taken in combination with other modules by students on any of the Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.
Indicative Content

Direct Methods for Solving Linear Systems of Equations
Basic properties of matrices, Gaussian elimination, partial pivoting. LUfactorization. Tridiagonal systems.

Iterative Methods
A general iterative method and convergence, Jacobi method, GaussSeidel method, SOR (successive overrelaxation).

Iterative Methods for Solving Eigenvalue Problems
Review of eigenvalue problems, QR factorizations

Using MATLAB to solve problems in linear algebra
Introduction to MATLAB, Application of MATLAB to algorithms for LU factorization, iterative methods and QR factorizations.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and computer labs, and assessed via an exam (70%) and computer homeworks (30%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module introduces the notions of differentiation and integration for functions of a complex variable. It develops the theory with important applications such as evaluation of path integrals via residue calculus, the fundamental theorem of algebra and conformal mappings. This module is mandatory for students taking the BSc or MMath in Mathematics and may optionally be taken in combination with other modules by students on any of the Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.
Indicative Content

Algebraic properties of complex numbers

Definition of the derivative; CauchyRiemann equations

Power series; radius of convergence

Logarithmic, exponential and trigonometrical functions; branch points

Line integrals. The Cauchy integral theorem and integral formula

The Cauchy formula for derivatives; Taylor series

Liouville's theorem; fundamental theorem of algebra

Laurent's theorem; poles and the residue theorem; zeros of analytic functions

Evaluation of integrals

Conformal mappings
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
The basic ideas at the foundations of many physical theories, such as continuum mechanics, fluid dynamics, electromagnetism, thermodynamics, general relativity and gauge theories, are geometrical. This course develops some of the geometrical concepts and tools that are essential for understanding classical and modern physics and engineering. This module is mandatory for students taking the BSc or MMath in Mathematics and may optionally be taken in combination with other modules by students on any of the Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents, and have taken MA31007 or MA32002.
Indicative Content

Manifolds
Submanifolds in R^{n}. Implicit Function theorem. Examples

Vector fields on manifolds
The tangent space. Vectors as differential operators. Vector fields and flows.

Covectors and exterior forms
Linear functionals and the dual space. Differential of a function. The pullback of a covector.

The exterior algebra
The geometric meaning of forms in R^n. Exterior product. Inner product. Exterior differential. Relation to vector analysis.

Integration of forms
Line and surface integrals. Independence of parameterisation. Integrals and pullbacks. Stoke′s Theorem.

Liederivative
The Liederivative of forms. Relation to equations in physics.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module provides an introduction to the Mathematics of Fluids and Plasmas, focusing on Fluid Dynamics. This module is mandatory for students taking the BSc or MMath in Mathematics or the BSc or MSci in Mathematics and Physics, and is optional for students taking any other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA31002 and either MA31007 or MA32002, or equivalents.
Indicative Content

Fundamentals
Fields, flux, potentials. Representation of fields: fieldlines/streamlines, contours, flux surfaces. Gauss' and Stokes' theorems.

Conservation laws
Conservation of mass, conservation of momentum, Euler's Eq., energy equation, equation of state.

Common approximations
Incompressible, irrotational, potential flows, Bernoulli’s the orem. Laplace’s equation, boundary conditions, uniqueness theorem, separable solutions.

Vorticity
Vorticity and circulation, Kelvin's circulation theorem, vorticity evolution.

The solar wind
Introduction to solar features, Parker's solar wind solution.

Waves
Sound waves, linearisation, dispersion relations, wave properties.

Viscous flow
Stress tensor, viscous stresses, viscosity, energy dissipation, the Reynolds number.

Turbulence and Chaos
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module aimed at Level 4 students, covers the theory and application of graphs, including both theoretical work and the use of algorithms. This module is mandatory for students taking the BSc or MMath in Mathematics, and is optional for students taking any other Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents.
Indicative Content

Introduction to Graph Theory
Fundamental definitions of graph theory. Introduction to some special types of graphs. Degree sequences and corresponding graphs.

Connectedness
Sufficient conditions to ensure connectedness. Connectivity and edge connectivity. Tarry's algorithm.

Eulerian and Hamiltonian Graphs
Necessary and sufficient conditions for graphs to be Eulerian. Fleury's algorithm. Necessary conditions for graphs to be Hamiltonian.

Trees
Properties of trees. Spanning trees and labelled spanning trees. Finding minimum weight spanning trees.

Planar and Nonplanar Graphs
Necessary conditions for graphs to be Planar. Toroidal graphs. Genus of graphs.

Graph Colourings
Vertex and edge colourings Chromatic polynomials. The 4colour theorem.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).
Credit Rating
This module is a Scottish Higher Education Level 4 or SCQF level 10 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
Level 5
This year is only followed by the MSci Honours students. Students undertake a yearlong personal project as well as studying the following modules:
About the module
This module, aimed at the Level 5 student, takes an advanced look at dynamical systems. The time evolution of many biological, chemical, or physical processes, as well as systems considered in engineering or economics, can be described by difference or differential equations. Dynamical systems theory allows us to study these systems of equations and inver information about the behaviour of the corresponding biological, chemical or physical systems. It addresses questions like the existence and stability of solutions, how the behaviour of solutions changes depending on the system parameters, or determines the existence of strange attractors or chaos in the system.
This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in each of the modules MA31002 and MA32001, or equivalent.
Indicative Content

OneDimensional Maps
Definition, Cobweb Plot: Graphical Representation of an Orbit, Stability of Fixed Points, Periodic Points, Chaos: Lyapunov Exponents.

Ordinary Differential Equations
Background, Examples of main Physical and Biological Processes described by Ordinary Differential Equations (ODEs), Existence and uniqueness of solutions of ODEs, Linearised Stability Analysis, Twodimensional Systems: Hamiltonian and Gradient systems, Periodic solutions: Floquet theory, Poincare Map and Stability of Periodic Orbits, Bifurcation and Chaos.

Partial Differential Equations
Definitions, Background, Wellposedness, Maximum Principles, Spectral Theorem for Laplace Equation, Semigroups for Evolution Equations in Banach Spaces, Nonlinear Evolution Equations: Linearised Stability Analysis for ReactionDiffusion Equations.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/presentation classes and assessed via an exam (60%) and coursework (40%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
In this module the Level 5 student will learn to write their own code and to apply builtin "black box" solvers in MATLAB and COMSOL to mathematical modelling problems. This module is mandatory for Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology. This module may be taken in combination with another at Level 5 by students taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in each of the modules MA32005 and MA42003, or equivalent.
Indicative Content

MATLAB fundamentals
Students will learn basic operations in MATLAB, and implement various finite difference schemes to solve ODEs (primarily initial value problems) originating in celestial mechanics, population dynamics, and cell biomechanics.

MATLAB ODE solvers for initial value problems
Students will learn to use standard builtin solvers with MATLAB, particularly ode45 and ode23s, and possibly dde23. We will apply these solvers to initial value problems (and possibly delay differential equations) stemming from celestial mechanics, cell biomechanics, and population dynamics.

MATLAB random variables, stochastic processes, and SDEs
After a brief introduction to stochastic differential equations (SDEs), students will learn MATLAB solution techniques, with applications to Brownian motion and related physical processes. We will also learn to simulate discrete and continuous stochastic processes, and generate samples from random variables with arbitrary distributions.

MATLAB ODE solvers for boundary value problems
Students will implement a standard "shooting" method to solve a BVP from heat transfer. We will learn to use the standard builtin solvers, particularly bvp4c. We explore alternate solution techniques, such as by formulating the discretised equation as a linear algebraic system, and as the steady state solution to a PDE; these approaches help drive us towards PDE solution methods. The class will apply these solvers to boundary value problems stemming from heat transfer and fluid mechanics.

MATLAB for PDEs
Students will implement explicit finite difference methods in MATLAB, with a focus on reactiondiffusion problems. The overall goal will be to solve coupled reactiondiffusion problems (with heterogeneous coefficients) and cell growth.

Weak formulations for partial differential equations; introduction to FEMs
We repose PDEs using a weak formulation, using the context of function spaces. Using this framework, we develop an understanding of finite element methods (FEMs).

FEMs and COMSOL fundamentals
Students will learn to solve reactiondiffusion equations using the builtin FEMs in COMSOL.
Delivery and Assessment
Delivery of this module will take a handson, interactive approach, where lectures are integrated with guided computer lab time. Assessment will be based on computational coursework (100%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This Level 5 module will undertake analysis and computational simulation of mathematical models of cancer growth and treatment. In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.
Indicative Content

Overview of Cancer Growth and Spread
The biology of cancer.

Avascular Solid Tumour Growth
ODE models of solid tumour growth Greenspan′s model. Reactiondiffusion moving boundary models.

Tumourinduced Angiogenesis
Continuum PDE models. Hybrid discretecontinuum models.

Cancer Invasion
Continuum PDE models. Hybrid discretecontinuum models.

Immune Response to Cancer
ODE models of the immune response to cancer. PDE models of the immune response to cancer.

Chemotherapy and Radiotherapy Treatment
Chemotherapy drug scheduling models. The linearquadratic radiobiological model.
Delivery and Assessment
The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
In this module we will undertake analysis and computational simulation of mathematical models of ecological and epidemiological systems including predatorprey systems, hostparasitoid systems, plantherbivore systems, spread and transmission of disease (e.g. AIDS, SARS, measles, rabies). In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.
Indicative Content

Nonspatial models
Difference equation models. Ordinary differential equation models. Delay difference/differential equation models.

Spatial models
Integrodifference equation models. Partial differential equation models. Integrodifferential equation models.
Delivery and Assessment
The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module, aimed at Level 5 students, studies mathematical models of cellular physiology in metabolism, genetic networks, and electrophysiology. This module is mandatory for Level 5 students taking the MSci in Mathematical Biology and may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA32009, or MA41002, or equivalent.
Indicative Content

Biochemical Reactions
Enzyme kinetics. Glycolysis

Genetic Networks
Central Dogma of biology, review of bifurcation theory. Feedback loops and Oscillation. Circadian rhythms. Cell cycle model.

Electrophysiology
The HodgkinHuxley model. The FitzHughNagumo equations. Small network dynamics of coupled neurons.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/presentation classes and assessed via exam (70%) and coursework (30%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
Plus one of
About the module
This module, aimed at Level 5 students, gives a nonmeasure theoretic introduction to stochastic processes, considering the theory and some applications and going on to introduce stochastic differential equations and their solutions. This module may optionally be taken in combination with others by Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology or Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module would find it beneficial to have taken each of the modules MA32001 and MA51007, or equivalent.
Indicative Content

Probability fundamentals
Elementary probability concepts such as random variables, expected value, moment generating and characteristic functions, conditional exceptions, probability inequalities and limit theorems, etc.

The Poisson process
(Homogeneous) Poisson process and related examples such as interarrival and waiting time distributions and conditional distribution of the arrival times. Some practical examples such as the busy period of the M/G/1 queueing system. Introduction to the nonhomogeneous Poisson process.

Markov chains
(Discretetime) Markov chains and some related examples. ChapmanKolmogorov equations and classification of states.

Continuoustime Markov chains
Continuoustime Markov chains, birth and death processes, and the Kolmogorov differential equations.

Brownian motion and stochastic differential equations
Basics of Brownian motion, Ito^ integral and Ito^ formula, and then stochastic differential equations (SDEs). A number of commonly used SDEs and their solutions will be discussed.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/presentation classes and assessed via coursework (100%) consisting of homeworks and a presentation.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This is a Level 5 course that offers a robust understanding of the inverse problems theoretical framework and methods suitable for medical and financial applications. The aim is to achieve comprehensive knowledge in the theoretical fundaments and general methodology for inverse problems in various heterogeneous media, including medical applications and finance. This module may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA32001, or equivalent.
Indicative Content

Examples of Inverse Problems
Examples from medical applications and finance.

Inverse Methodology Preliminary Foundation
Necessary Basic Definitions and Theorems in Measure Theory and Function Spaces

General Regularisation Theory
Tikhonov′s regularization method. Landweber Iteration. The Discrepancy Principle of Morozov. Conjugate gradient method.

Galerkin Methods
Galerkin General formulation. The Least Squares Method. The Dual Least Squares Method.

The Truncated Singular Value Decomposition Method

Stable inversion via the Mollification Method

Inverse Problems in General Heterogeneous Media and Medical Applications
Backward heat conduction problem. Inverse problems in reactiondiffusion equations.

Inverse problems in finance
Formulation of forward model: BlackScholes and Dupire's Formula. Inverse Problem formulation of market volatility. Reconstruction of time and pricedependent volatilities.
Delivery and Assessment
The module is delivered in the form of lectures and assessed via coursework (100%) consisting of tests and homeworks.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
Careers
Mathematics is central to the sciences, and to the development of a prosperous, modern society. The demand for people with mathematical qualifications is considerable, and a degree in mathematics is a highly marketable asset.
Mathematics graduates are consistently amongst those attracting the highest graduate salaries and can choose from an everwidening range of careers in research, industry, science, engineering, commerce, finance and education.
Exciting new applications of mathematical biology are opening up yet more career options in the biotech industries. Here you could be involved in designing new anticancer drugs or new treatment regimes for patients with diabetes.
Even if you do not take your mathematics any further than university, employers know that mathematics graduates are intelligent, logical problem solvers. With this training behind you, the career options become almost limitless.
Fees & Funding
The fees you pay will depend on your fee status. Your fee status is determined by us using the information you provide on your application.
Find out more about fee status
Fees for students starting 201920
Fee category  Fees for students starting 201920 

Scottish and EU students  £1,820 per year of study 
Rest of UK students  £9,250 per year, for a maximum of 3 years, even if you are studying a four year degree. See our scholarships for rest of UK applicants. 
Overseas students (nonEU)  £20,950 per year of study 
Scottish and EU students can apply to the Students Award Agency for Scotland (SAAS) to have tuition fees paid by the Scottish Government.
Rest of the UK students can apply for financial assistance, including a loan to cover the full cost of the tuition fees, from the Student Loans Company.
Tuition fees for Overseas (nonEU) students are guaranteed not to increase by more than 3% per year, for the length of your course.
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Graduation fee  Studio fee  Field trips 
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Your Application
Unistats data set  formerly the Key Information Set (KIS)
Degree  UCAS Code  Unistats Data  

Apply Now  Mathematical Biology BSc (Hons)  CG11  
Apply Now  Mathematical Biology MSci  GC11 