To study, by example, how mathematical models are hypothesised, modified and elaborated. To introduce basic approaches to the analysis of differential-equation/difference-equation models.
After taking this unit, students should be able to:
* Construct an initial mathematical model for a real world process.
* Assess this model critically, and to suggest alterations or elaborations of proposed model in light of discrepancies between model predictions and observed data or failures of the model to exhibit correct qualitative behaviour.
* Take first steps into the study of the behaviour of solutions of nonlinear ordinary differential equations/difference equations, in particular, stability of equilibrium solutions and the phase plane method.
* Write the relevant mathematical arguments in a precise and lucid fashion.
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).
Modelling and the scientific method: Objectives of mathematical modelling; the iterative nature of modelling; falsifiability and predictive accuracy; Occam's razor, paradigms and model components; self-consistency and structural stability. The three stages of modelling:(a) model formulation, including the use of empirical information, (b) model fitting and (c) model validation.
Difference equations: Solving linear and affine equations, change of variables, equilibria, stability, systems, nonlinear equations (e.g. logistic equation, Ricker), chaos, linear stability for nonlinear systems.
Ordinary differential equations: (1) Revisit: scalar equations, systems, formulate higher order equations as systems, behaviour near equilibria for linear equations in the plane. (2) Nonlinear equations: modelling with nonlinear rates, Michaelis-Menten-kinetics, unique existence of solutions, linearisation at equilibria, stability, preservation of positivity and funnel theorems, the phase plane method, simple examples of bifurcation scenarios.
Application of these methods to the analysis of distinguished model cases. Possible case studies: dynamics of measles epidemics; population growth; dynamics of genetic selection; prey-predator and competition models; modelling water pollution; chemical reactions and the chemostat.