This module considers the study of pairs of differential equations. Theoretical analysis, complemented by results obtained using computer simulation, will be used to study models drawn from a variety of disciplines.

• Appreciate the structure of a range of mathematical models.
• Analyse the behaviour of the models in both quantitative and qualitative terms.
• Appreciate the use of computer simulations to illustrate and extend the results obtained from qualitative analysis.
• Modify a standard model to incorporate some additional feature and assess the changes in behaviour induced.

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills:

Problem formulation and modelling. Solution and qualitative analysis of ordinary differential equations using mathematical techniques. 

Transferable Skills:

Mathematical modelling of continuous time systems. Visualisation and interpretation of the results obtained from a mathematical model.

Formative assessment is carried out by means of regular tutorial exercises.  These might include some exercises that involve the use of simple computer simulations to illustrate and complement theoretical results. Feedback to students on their solutions and their progress towards learning outcomes will be provided in lectures.  

Summative assessment is by written examination at the end of the module. 

The examination gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions

• Review of single variable models, equilibrium points and stability

• Introduction to suitable computer simulation software.

• Systems with two variables.
  • Phaseplane analysis.
  • Classification of equilibrium points and their stability for the linear case.
  • Linearisation of non-linear models.

• Study of a range of examples.
  • These would be drawn from physical problems as well as from biological and environmental topics.  In each case the study would include a justification of the model and a theoretical analysis. Where appropriate, results obtained from computer simulations will be used to illustrate and confirm the analysis.

Not applicable.

Differential Equations, Lomen, D., & Lovelock, L., Wiley