Student Records
Programme & Unit Catalogues

## MA20221: Modelling and dynamical systems

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Intermediate (FHEQ level 5)
Period: Semester 2
Assessment: CW 25%, EX 75%
Supplementary Assessment: MA20221 Mandatory extra work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA10207 and take MA10209 and take MA10210 and take MA10208 and take MA20220 and you must have used MATLAB or an equivalent package to solve ODEs.
Description: Aims:
To study, by example, how mathematical models are hypothesised, modified and elaborated. To introduce basic approaches to the analysis of differential-equation/difference-equation models.

Learning Outcomes:
After taking this unit, students should be able to:
* Construct an initial mathematical model for a real world process.
* Assess this model critically, and to suggest alterations or elaborations of proposed model in light of discrepancies between model predictions and observed data or failures of the model to exhibit correct qualitative behaviour.
* Take first steps into the study of the behaviour of solutions of nonlinear ordinary differential equations/difference equations, in particular, stability of equilibrium solutions and the phase plane method.
* Write the relevant mathematical arguments in a precise and lucid fashion.

Skills:
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).

Content:
Modelling and the scientific method: Objectives of mathematical modelling; the iterative nature of modelling; falsifiability and predictive accuracy; Occam's razor, paradigms and model components; self-consistency and structural stability. The three stages of modelling:(a) model formulation, including the use of empirical information, (b) model fitting and (c) model validation.
Difference equations: Solving linear and affine equations, change of variables, equilibria, stability, systems, nonlinear equations (e.g. logistic equation, Ricker), chaos, linear stability for nonlinear systems.
Ordinary differential equations: (1) Revisit: scalar equations, systems, formulate higher order equations as systems, behaviour near equilibria for linear equations in the plane. (2) Nonlinear equations: modelling with nonlinear rates, Michaelis-Menten-kinetics, unique existence of solutions, linearisation at equilibria, stability, preservation of positivity and funnel theorems, the phase plane method, simple examples of bifurcation scenarios.
Application of these methods to the analysis of distinguished model cases. Possible case studies: dynamics of measles epidemics; population growth; dynamics of genetic selection; prey-predator and competition models; modelling water pollution; chemical reactions and the chemostat.
Programme availability:

#### MA20221 is Compulsory on the following programmes:

Department of Mathematical Sciences
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 2
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2

#### MA20221 is Optional on the following programmes:

Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 2
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 2
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 2
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
Department of Physics
• USXX-AFB03 : BSc (hons) Mathematics and Physics (Full-time) - Year 3
• USXX-AKB04 : BSc (hons) Mathematics and Physics with Placement (Full-time with Thick Sandwich Placement) - Year 4
• USXX-AAB04 : BSc (hons) Mathematics and Physics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 3

Notes:
* This unit catalogue is applicable for the 2013/4 academic year only. Students continuing their studies into 2014/15 and beyond should not assume that this unit will be available in future years in the format displayed here for 2013/14.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.