Module 2 – Mathematical Modelling

“As time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.”

Paul Dirac

Nobel Prize Physics 1933

The creation of mathematical models is fundamental to all aspects of our lives. Whether it is sending a man to the moon, predicting the stock market or deciding on the best measures in a global pandemic; mathematical modeling plays a crucial role.

The process of mathematical modelling is extremely varied and also often uses technology. This module is designed to look at four different modelling scenarios, and in all cases to make the most of technology that we all have available to us.

We will begin with a task looking at population growth of fish in a pond and how this leads to the Logistic Mapping. We will make a deep dive on this mapping which will allow us to understand the importance of parameters in a model, as well as how a simple model can lead to complex results. The mapping will allow us to see how chaos can come from a simple form and how other mappings are related to the Logistic Mapping. Students will learn how to use Geogebra to a high level of skill.

The second part of the module will look at the Predator – Prey module and here students will discover how they can still examine a dynamic module without the use of Calculus. Students will learn more about technology and how we can use difference equations to solve dynamic systems and plot phase portraits.

The third part of the module will change tack and look at a physical system, in this case the bicycle. Using some simple physics and our basic knowledge of bicycles, students will learn to understand how the power of a bike rider relates to their speed. The goal of the module is to find the correct gearing when riding up hills of different gradients.

The final model to be looked at will be the classic SIR model for pandemics. Again, with the use of technology, students will discover how to chart the evolution of a pandemic given certain parameters. The work will look at the real challenge of many mathematical models, the understanding of the parameters that define the evolution of a model.