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

### North Terrace Campus - Semester 1 - 2017

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 Modelling with Ordinary Differential Equations III Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week Y (MATHS 2101 and MATHS 2102) or (MATHS 2201 and MATHS 2202) APP MTH 3013, APP MTH 3004 MATHS 2104 Ongoing assessment 30%, Exam 70%
##### Course Staff

Course Coordinator: Dr Luke Bennetts

##### 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)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
all
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
1,2,4,6
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
1,4,6
• technology savvy
• professional and, where relevant, fully accredited
• forward thinking and well informed
• tested and validated by work based experiences
all
Self-awareness and emotional intelligence
• a capacity for self-reflection and a willingness to engage in self-appraisal
• open to objective and constructive feedback from supervisors and peers
• able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
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 (Canvas) 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/

This course also uses the online Matlab coding tool "Cody Coursework" which allows automated marking of computer codes. You will be given instructions on how to access this tool.
• 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.  Some computer programming in Matlab will also be required to promote understanding of computational methods.

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 Mid-semester test 1 8 Total 156
##### 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
Understanding of and ability to use analytic solution methods for first-order and second-order differential equations as taught in Differential Equations II or Engineering Mathematics IIA.

Ability to write a simple Matlab code from scratch, for example, to solve a first-order initial value problem using Euler's method. Knowledge of numerical methods to the level taught in Numerical Methods II is assumed.
• 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.

Some computer programs will need to be written that will form part of the assignment assessment.
##### 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. Where required computer codes are to be submitted via "Cody Coursework".
3. Late assignments will not be accepted.
4. 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
• Fraud Awareness

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