WHAT IS MATHEMATICAL MODELLING?
Dr. Gerda de Vries Assistant Professor
Mathematical modelling is the use of mathematics to
• describe real-world phenomena
• investigate important questions about the
ob-served world
• explain real-world phenomena • test ideas
The real world refers to
• engineering • physics
• physiology • ecology
• wildlife management • chemistry
• economics • sports
EXAMPLES of real-world questions that can be investigated with mathematical mod-els
Suppose there is a baseball strike. We might be interested in predicting the effects of higher players’ salaries on the long-term health of the baseball industry.
One can think of mathematical modelling as an activity or process that allows a mathematician to be a chemist, an ecologist, an economist, a physiologist . . . .
Process of mathematical modelling Predictions/ explanations Real-world data Mathematical conclusions Model ✛ ✻ ✲ ❄ Formulation Analysis Interpretation Test
Challenge in mathematical modelling
“. . . not to produce the most comprehensive descriptive model
but
to produce the simplest possible model that incorporates the major features of the
phenomenon of interest.”
Two hands-on modelling activities
Modelling short-track running races
Consider the following two situations:
Situation 1:
Donovan Bailey runs the 100-metre dash at sea-level against a headwind of 2 m/s. His time is 9.93 seconds.
Situation 2:
Maurice Green runs the 100-metre dash at an altitude of 500 metres in windless conditions. His time is 9.92 seconds.
Distance and velocity profiles of Maurice Green’s 100-metre race at the 1997 World Championships in Athens, Greece
0 2 4 6 8 10
time (s) 0 5 10 15 velocity (m/s)
0 2 4 6 8 10
Simulated distance and velocity profiles (A = 12.2 m/s2 and τ = 0.892 s)
5 10 15
velocity (m/s)
0 2 4 6 8 10 12
0 20 40 60 80 100
Effect of drag term and headwind on simulated race times
A τ D w Race Time
How should a bird select worms?
Consider a bird searching a patch of lawn for worms, and suppose that there are two types of worms living in the lawn:
big, fat, juicy ones (highly nutritious)
and
long, thin, skinny ones (less nutritious)
Afterword
Experimental scientists are very good at taking apart the real world and studying small com-ponents.
Since the real world is nonlinear, fitting the components together is a much harder puzzle.
Mathematical modelling allows us to do just that.
Contact information Email:
devries@math.ualberta.ca
Webpage:
http://www.math.ualberta.ca/˜devries
Download slide presentation, modelling ac-tivities, answer keys:
