APP MTH 7035 - Modelling with Ordinary Differential Equations III

North Terrace Campus - Semester 1 - 2015

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, linearization 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
    Available for Study Abroad and Exchange Y
    Prerequisites (MATHS 2101 and MATHS 2102) or (MATHS 2201 and MATHS 2202)
    Incompatible APP MTH 3013, APP MTH 3004
    Assumed Knowledge MATHS 2104
    Course Description 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, linearization 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.
    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
    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.
    Workload

    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 35
    Mid-semester test 1 5
    Total 148
    Learning Activities Summary
    Lecture Outline
    1. Modelling examples, necessity for theory and computation.
    2. Autonomous scalar equations, fixed points, phase line analysis, stability criteria, fisheries management model.
    3. Nondimensionalisation, constant yield fishing model, bifurcation, saddle-node bifurcation.
    4. Constant effort fishing model, transcritical bifurcaiton.
    5. Supercritical pitchfork bifurcation.
    6. Subcritical pitchfork bifurcation, hysteresis.
    7. The spruce-budworm model.
    8. A bifurcation diagram in a two parameter space.
    9. Exact solution of 1D models, existence and uniqueness.
    10. Existence and uniqueness theorem.
    11. Numerical error, well and ill-conditioned problems.
    12. Stable and unstable algorithms, conditioning and stability.
    13. Numerical solution of 1st order initial value problems (IVPs), finite difference approximation.
    14. Explicit and implicit methos, forward and backward Euler methods, consistence, convergence.
    15. Stability of forward and backward Euler methods, the stability region of a numerical method.
    16. Optimal step size, stiff problems.
    17. More numerical solution methods, Matlab ODE solvers, non-linear IVPs.
    18. Predictor-corrector and Runge-Kutta schemes.
    19. Systems of IVPs, second-order IVPs.
    20. Numerical solution of boundary value problems
    21. Linear autonomous systems in two dimensions: some models.
    22. The Kermack-McKendrick epidemic model, existence and uniqueness of solutions, the phase plane.
    23. Analysis of linear systems in two dimensions.
    24. Nonlinear systems and linearisation, the Hartman-Grobman theorem, general population interaction model, Lotka-Volterra predator-prey equations.
    25. Mutualism and competition population models, analysis of the Kermack-McKendrick epidemic model.
    26. Limit cycles. Bifurcations in 2D systems, Hopf bifurcation.
    27. Chemical kinetics, Michaelis-Menten kinetics.
    28. Linear nonautonomous systems in higher dimensions: homogeneous systems.
    29. Linear nonautonomous systems: nonhomogeneous systems.
    30. Course summary and revision.

    Tutorial outline

    1. One-dimensional models: scaling, equilibria and their stability, bifurcation, benefits of numerical and analytic solution.
    2. Phase-line analysis, bifurcation diagrams, classification of bifurcations.
    3. Understanding a model, bifurcation analysis and interpretation, hysteresis.
    4. Ill- and well-conditioned problems, stable and unstable algorithms, numerical error.
    5. Stiff problems, Matlab ODE solvers, regions of stability for explicit and implicit Euler methods, numerical solution of vector IVPs.
    6. Two-dimensonal models, linearisation of non-linear models, phase portraits.
    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 20% all
    Mid-semester test 10% 1,2,3,4
    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 4%
    Assignment 2 Week 5 Week 6 4%
    Assignment 3 Week 7 Week 8 4%
    Assignment 4 Week 9 Week 10 4%
    Assignment 5 Week 11 Week 12 4%
    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.
    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 (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.

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