## APP MTH 7035 - Modelling with Ordinary Differential Equations III

### North Terrace Campus - Semester 1 - 2014

Differential equation models describe a wide range of complex problems in biology, engineering, physical sciences, economics and finance. This course focuses on ordinary differential equations (ODEs) and develops students' skills in the formulation, solution, understanding and interpretation of coupled ODE models. A range of important biological problems, from areas such as resource management, population dynamics, and public health, drives the study of analytical and numerical techniques for systems of nonlinear ODEs. A key aim of the course is building practical skills that can be applied in a wide range of scientific, business and research settings. Topics covered are: analytical methods for systems of ODEs, including vector fields, fixed points, phase-plane analysis, linearisation of nonlinear systems, bifurcations; general theory on existence and approximation of ODE solutions; biological modelling; explicit and implicit numerical methods for ODE initial value problems, computational error, consistency, convergence, stability of a numerical method, ill-conditioned and stiff problems.

• General Course Information
##### Course Details
Course Code APP MTH 7035 Modelling with Ordinary Differential Equations III Applied Mathematics Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week MATHS 1012 APP MTH 3013, APP MTH 3004 MATHS 2102 Ongoing assessment 30%, Exam 70%
##### Course Staff

Course Coordinator: Professor Yvonne Stokes

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
Students who successfully complete the course should:
1. understand how to model time-varying systems using ordinary differential equations
2. be able to identify and analyse stability of equilibrium solutions
3. be able to numerically solve ordinary differential equations
4. be able to analyse how the structure of solutions can change depending on a parameter
5. understand the analytical solution theory for linear systems of ordinary differential equations
6. appreciate the necessity of numerical and qualitative methods for analysing solutions for nonlinear systems
7. have a detailed understanding of several ordinary differential equations models arising in physics, biology and chemistry, namely oscillator models, Lotka-Volterra competition and predator-prey models, Michaelis-Menton kinetics and SIR epidemic models

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 2,3,4,5,6,7
A proficiency in the appropriate use of contemporary technologies. 3,7
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. all
• Learning Resources
None.
##### Recommended Resources
1. Butcher, John. Numerical Methods for Ordinary Differential Equations (Wiley, 2008)
2. Chicone, Carmen. Ordinary Differential Equations with Applications (Springer, 2006)
3. Dahlquist, Germund and Bjorck, Ake. Numerical Methods (Dover, 2003)
4. de Vries, Gerda et al. A Course in Mathematical Biology (SIAM, 2006)
5. Edelstein-Keshet, Leah. Mathematical Models in Biology (SIAM, 2005)
6. Strogatz, Steven. Nonlinear Dynamics and Chaos (Perseus, 2001)
##### Online Learning
This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, sample solutions, discussion boards, etc. It is recommended that the students make appropriate use of these resources. Link to MyUni login page: https://myuni.adelaide.edu.au/webapps/login/
• Learning & Teaching Activities
##### Learning & Teaching Modes
This course relies on lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
 Activity Quantity Workload hours Lectures 30 90 Tutorials 6 18 Assignments 5 40 Total 148
##### Learning Activities Summary
Lecture Outline
1. Modelling examples, necessity for theory and computation
2. Logistic growth model, fixed points
3. Phase line, stability criteria
4. Euler’s method for numerical solutions. Saddle-node bifurcation
5. Spruce budworm model
6. Transcritical bifurcation. Pitchfork bifurcation
7. Supercritical and subcritical pitchfork bifurcations. Hysteresis
8. All ODEs are first-order. Phase space for a system of ODEs
9. Existence and uniqueness theorem
10. Continuity in initial conditions theorem
11. Implications to phase space. Examples
12. Numerical schemes for initial value problems
13. Error of numerical methods
14. Ill-conditioned problems. Stability. Euler and backwards Euler methods
15. Stiff problems. Predictor-corrector and Runge-Kutta schemes
16. Exercises in numerical solutions to ODEs using Matlab
17. Numerical schemes for boundary value problems
18. Linear systems in two dimensions. The phase plane
19. Nonlinear systems in two dimensions.
20. Linearisation and the Hartman-Grobman theorem
21. Models for nonlinear systems: nonlinear oscillators
22. Lotka-Volterra predator-prey equations
23. Limit cycles. Periodic orbits.
24. Hopf bifurcation. Oscillating chemical reactions
25. Lotka-Volterra competition models for species
26. Linear nonautonomous systems in higher dimensions
27. Linear nonautonomous systems continued
28. Michaelis-Menton chemical kinetic model
29. SIR epidemic spreading model. Lorenz model for the atmosphere
30. Course summary and revision
Tutorial outline
1. One-dimensional models
2. Bifurcations and existence/uniqueness
3. Numerical schemes
4. Two-dimensional models
5. Nonautonomous linear systems
6. Higher-dimensonal models
##### Specific Course Requirements
None.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Component Weighting Objective assessed Assigments 30% all Exam 70% all
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail
 Assessment Item Distributed Due Date Weighting Assignment 1 Week 3 Week 4 6% Assignment 2 Week 5 Week 6 6% Assignment 3 Week 7 Week 8 6% Assignment 4 Week 9 Week 10 6% Assignment 5 Week 11 Week 12 6%
##### Submission
1. All written assignments are to be submitted to the designated hand-in boxes within the School of Mathematical Sciences with a signed cover sheet attached.
2. Late assignments will not be accepted.
3. Assignments will have a two week turn-around time for feedback to students.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
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