WHAT IS MATHEMATICAL MODELLING?
Dr. Gerda de Vries Assistant Professor Department of Mathematical Sciences University of Alberta
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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
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The real world refers to • engineering • physics • physiology • ecology • wildlife management • chemistry • economics • sports • ...
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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.
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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.
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Process of mathematical modelling
Formulation
Real-world data
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Model
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Analysis
Test
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Predictions/ explanations
Interpretation
Mathematical conclusions
There is no best model, only better models.
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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
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Two hands-on modelling activities
• Modelling short-track running races • How should a bird select worms?
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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?
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Distance and velocity profiles of Maurice Green’s 100-metre race at the 1997 World Championships in Athens, Greece 100
distance (m)
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velocity (m/s)
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time (s)
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Simulated distance and velocity profiles (A = 12.2 m/s2 and Ď„ = 0.892 s) 100
distance (m)
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velocity (m/s)
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Effect of drag term and headwind on simulated race times A 12.2 12.2 12.2
Ď„ 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
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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?
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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.
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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
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