Mathematical modelling
WHAT IS MATHEMATICAL
MODELLING?
Dr. Gerda de Vries
Assistant Professor
Department of Mathematical Sciences
University of Alberta
1
Mathematical modelling is the use of
mathematics to
• describe real-world phenomena
• investigate important questions about the observed world
• explain real-world phenomena
• test ideas
• make predictions about the real world
2
The real world refers to
• engineering
• physics
• physiology
• ecology
• wildlife management
• chemistry
• economics
• sports
• ...
3
EXAMPLES of real-world questions that
can be investigated with mathematical models
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.
In the management of a fishery, it may be important to determine the optimal sustainable
yield of a harvest and the sensitivity of the
species to population fluctuations caused by
harvesting.
4
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 . . . .
Instead of undertaking experiments in the real
world, a modeller undertakes experiments
on mathematical representations of the real
world.
5
Process of mathematical modelling
Formulation
Real-world
data
✲
Model
✻
Test
Analysis
❄
Predictions/
explanations
✛
Interpretation
Mathematical
conclusions
There is no best model, only better models.
6
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.”
Howard Emmons
7
Two hands-on modelling activities
• Modelling short-track running races
• How should a bird select worms?
8
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.
Who is the better runner?
9
Distance and velocity profiles of Maurice
Green’s 100-metre race at the 1997 World
Championships in Athens, Greece
100
distance (m)
80
60
40
20
0
0
2
4
6
8
10
0
2
4
6
8
10
velocity (m/s)
15
10
5
0
time (s)
10
Simulated distance and velocity profiles
(A = 12.2 m/s2 and τ = 0.892 s)
100
distance (m)
80
60
40
20
0
0
2
4
6
8
10
12
0
2
4
6
8
10
12
velocity (m/s)
15
10
5
0
time (s)
11
Effect of drag term and headwind on
simulated race times
A
12.2
12.2
12.2
τ
0.892
0.892
0.892
D
0
0.00166
0.00166
w
0
0
−2
Race Time
10.08 s
10.21 s
10.26 s
12
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)
Which worms should the bird eat?
13
Afterword
Experimental scientists are very good at taking
apart the real world and studying small components.
Since the real world is nonlinear, fitting the
components together is a much harder puzzle.
Mathematical modelling allows us to do just
that.
Ideally, the combination of science and modelling leads to a complete understanding of the
phenomenon being studied.
14
Contact information
Email:
devries@math.ualberta.ca
Webpage:
http://www.math.ualberta.ca/˜devries
Download slide presentation, modelling activities, answer keys:
http://www.math.ualberta.ca/˜devries/erc2001
15
MODELLING?
Dr. Gerda de Vries
Assistant Professor
Department of Mathematical Sciences
University of Alberta
1
Mathematical modelling is the use of
mathematics to
• describe real-world phenomena
• investigate important questions about the observed world
• explain real-world phenomena
• test ideas
• make predictions about the real world
2
The real world refers to
• engineering
• physics
• physiology
• ecology
• wildlife management
• chemistry
• economics
• sports
• ...
3
EXAMPLES of real-world questions that
can be investigated with mathematical models
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.
In the management of a fishery, it may be important to determine the optimal sustainable
yield of a harvest and the sensitivity of the
species to population fluctuations caused by
harvesting.
4
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 . . . .
Instead of undertaking experiments in the real
world, a modeller undertakes experiments
on mathematical representations of the real
world.
5
Process of mathematical modelling
Formulation
Real-world
data
✲
Model
✻
Test
Analysis
❄
Predictions/
explanations
✛
Interpretation
Mathematical
conclusions
There is no best model, only better models.
6
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.”
Howard Emmons
7
Two hands-on modelling activities
• Modelling short-track running races
• How should a bird select worms?
8
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.
Who is the better runner?
9
Distance and velocity profiles of Maurice
Green’s 100-metre race at the 1997 World
Championships in Athens, Greece
100
distance (m)
80
60
40
20
0
0
2
4
6
8
10
0
2
4
6
8
10
velocity (m/s)
15
10
5
0
time (s)
10
Simulated distance and velocity profiles
(A = 12.2 m/s2 and τ = 0.892 s)
100
distance (m)
80
60
40
20
0
0
2
4
6
8
10
12
0
2
4
6
8
10
12
velocity (m/s)
15
10
5
0
time (s)
11
Effect of drag term and headwind on
simulated race times
A
12.2
12.2
12.2
τ
0.892
0.892
0.892
D
0
0.00166
0.00166
w
0
0
−2
Race Time
10.08 s
10.21 s
10.26 s
12
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)
Which worms should the bird eat?
13
Afterword
Experimental scientists are very good at taking
apart the real world and studying small components.
Since the real world is nonlinear, fitting the
components together is a much harder puzzle.
Mathematical modelling allows us to do just
that.
Ideally, the combination of science and modelling leads to a complete understanding of the
phenomenon being studied.
14
Contact information
Email:
devries@math.ualberta.ca
Webpage:
http://www.math.ualberta.ca/˜devries
Download slide presentation, modelling activities, answer keys:
http://www.math.ualberta.ca/˜devries/erc2001
15