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
    Course Modelling with Ordinary Differential Equations III
    Coordinating Unit Applied Mathematics
    Term Semester 1
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 hours per week
    Prerequisites MATHS 1012
    Incompatible APP MTH 3013, APP MTH 3004
    Assumed Knowledge MATHS 2102
    Assessment 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
    University Graduate Attributes

    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
    Required Resources
    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:
  • 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
  • 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%
    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.
    Course Grading

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

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    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 ( 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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