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MATH 572

Mathematical Modelling in Industry, Government, and Sciences

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Chapter 1��Introduction

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Course Objectives

  • Master Deterministic Modelling: To gain practical knowledge and hands-on experience in formulating mathematical models
  • Apply Theoretical and Numerical Methods: To learn essential analytical techniques and implement them using computational software
  • Explore Real-world Problems: To increase awareness of applied modelling by examining real-world systems
  • Execute an Independent Modelling Project: To understand and practice the iterative steps of the modelling process, from simplifying assumptions to computational validation
  • Develop Long-Term Skills: To bridge theoretical science with practical applications in industry and government, providing the foundational skills for continued learning in mathematical modelling beyond this course

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  • deterministic or stochastic
  • dynamic or static
  • discrete or continuous

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Why do we use mathematical models?

  • Advantages compared to empirical study?

​

  • Disadvantages compared to empirical study?

​

  • Integration with empirical study?

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Advantages of Mathematical Modelling (Compared to Empirical Study)

  • Cost and Speed: Building and running a computational or mathematical model is typically much faster and less expensive than setting up physical experiments, conducting long-term field observations, or building physical prototypes.
  • Absolute Control: Models allow you to perfectly isolate specific variables to observe their precise effects. In contrast, real-world empirical studies often struggle with "noise" and uncontrollable, confounding factors.
  • Safety and Feasibility: You can safely test extreme, dangerous, or physically impossible scenarios (such as global pandemics, nuclear reactions, or centuries of climate change) without ethical concerns or physical risks.
  • Predictive Exploration: Models allow you to project system behaviors far into the future or test hypothetical conditions that do not currently exist in the real world.

​

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Disadvantages of Mathematical Modelling (Compared to Empirical Study)

  • Oversimplification: To make the math solvable, models require simplifying assumptions. If critical real-world complexities or interacting variables are ignored (the "too little detail" side of the balance), the model's conclusions may be inaccurate.
  • Data Dependency: A model is only as good as the parameters and logic used to build it. If the foundational assumptions are flawed, the output will be functionally useless.
  • Lack of Tangible Proof: A model provides theoretical possibilities and likelihoods, not concrete reality. It cannot definitively prove how a natural system will act without real-world confirmation.

​

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Integration with Empirical Study

  • The Iterative Cycle: Mathematical modelling and empirical studies form a continuous, interdependent feedback loop. Models are used to generate specific, testable hypotheses, which then guide researchers on exactly what empirical data needs to be collected.
  • Parameterization: Empirical studies provide the raw, real-world measurements, rates, and initial conditions required to build, calibrate, and ground mathematical models in reality.
  • Validation and Refinement: Once a model makes a prediction, empirical experiments are used to test it. If the real-world data contradicts the model's output, the mathematical model must be revised, updated, or discarded.

​

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Modelling Process

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Rule of Thumb

We can always improve the model with more details, but meanwhile we want to keep the model as simple as possible such that we can obtain useful and in-depth results.

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In mechanics and physics, simple harmonic motion is a type of oscillation where the restoring force is proportional to the displacement and acts in the direction opposite to that of displacement. Ignoring the damping behavior, the restoring force (given by the product of mass and acceleration according to Newton's second law of motion for a constant mass) in a linear spring can be modelled by:

A solution of the equation                           (with the initial position                  ) is given by:

where A depends on the initial velocity

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Quick review

  • Read 1.2 Quick Review of the optional textbook “Mathematical Modelling: A Graduate Textbook” which includes 1.2.1 First‐order Differential Equations, 1.2.2 Second‐order Differential Equations, 1.2.3 Linear Algebra, 1.2.4 Scaling
  • After reading these basic techniques, please work on Exercises on page 14 at the end of Chapter 1. Basic Concepts and Quick Review

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Scaling

  • a useful technique in mathematical modelling to simplify the model for analysis
  • to make independent and dependent variables unitless
  • to normalize the variables
  • to reduce the number of parameters

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My expertise - STOICHIOMETRY

Stoichiometry in biology: Focuses on the balance of elements and energy within biological systems. This includes metabolism, growth, etc.

Stoichiometry in ecology: Focuses on the balance of elements and energy within ecological systems. This includes competition, commensalisms, etc.

Stoichiometry in chemistry: Focuses on the balance of elements within chemical equations.

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Low Light

High Light

(Extra)

High Light

Algal

P:C

Algal C

Daphnia

Daphnia

Daphnia

Extreme high light

🡪Daphnia extinction

Algal C

Algal C

Motivation for Stoichiometric Study

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What would happen to secondary production

if solar radiation was reduced?

Expectations from single-currency ecological theory

Solar Radiation

High

Low

Secondary Production,

Herbivore Biomass

Solar Radiation

Primary Production,

Autotroph Biomass

High

Low

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Terrestrial herbivore (Pieris)

Freshwater herbivore (Daphnia)

From: Elser, J.J., W.F. Fagan, R.F. Denno, D.R. Dobberfuhl, A. Folarin, A. Huberty, S. Interlandi, S.S. Kilham, E. McCauley, K.L. Schulz, E.H. Siemann, and R.W. Sterner. 2000. Nutritional constraints in terrestrial and freshwater food webs. Nature 408: 578-580.

Stoichiometric Imbalance Impairs Herbivores

In Freshwater and Terrestrial Ecosystems

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Solar Radiation

High

Low

Secondary Production,

Herbivore Biomass

Very High

Starvation

Junk food

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(CXNYPZ)prey + (CXNYPZ) predator -> Q (CXNYPZ) predator + (CXNYPZ)’ waste

From: Elser, J.J., and J. Urabe. 1999. The stoichiometry of consumer-driven nutrient recycling:

theory, observations, and consequences. Ecology 80: 735-751.

(CXNYPZ)inorganic + (CXNYPZ) autotroph + light -> Q (CXNYPZ)' autotroph + (CXNYPZ)’ inorganic

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Fundamental Reason: C supplies energy to cells, N is essential to build proteins, P is essential to build nucleic acids (DNA&RNA), … …

​

However, most predator-prey (=consumer-resource) models only consider the carbon flow (=biomass or population), which cannot explain the motivating experiment.

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S

0.11

S

0.14

C

0.18

C

20.2

Ca

3.22

Ca

2.5

Fe

4.18

Fe

0.01

Si

25.80

O

50.02

O

63.0

Al

7.30

Na

2.36

Na

0.10

K

2.28

K

0.11

H

0.25

H

9.9

P

0.11

P

1.14

N

0.03

N

2.5

Composition of Earth’s Crust

Composition of Human Body

Mismatch of C, N, P:

Living things make a very discriminating

selection of elements from the environment.

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Significant concentrations of C and N in Earth’s atmosphere?

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C and Si are in the same column of the periodic table, and are almost

equally abundant in the solar system, and Si is even more abundant

than C in the Earth’s crust. Why C, not Si to form living things?

(Possible answers: C has very high binding energy 🡪 store energy;

high degree of bonding flexibility of C 🡪 considerable architectural

flexibility)

​

Why N, P as main nutrient elements, not other elements?

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Laws and Hypotheses

  • Conservation Law of Matter
  • Liebig’s Minimum Law
  • Droop’s form
  • Lambert-Beer’s Law
  • Homeostasis

​

​

  • Light:Nutrient Hypothesis
  • Growth Rate Hypothesis

​

Wang, H.*, Sterner, R.W. and Elser, J.J., 2012. On the “strict homeostasis” assumption in ecological stoichiometry. Ecological Modelling, 243, pp.81-88.

​

Wang, H.*, et al., 2018. Weak dynamical threshold for the “strict homeostasis” assumption in ecological stoichiometry. Ecological Modelling, 384, pp.233-240.

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Wang, H., Garcia, P.V., Ahmed, S. and Heggerud, C.M., 2022. Mathematical comparison and empirical review of the Monod and Droop forms for resource-based population dynamics, Ecological Modelling, Vol. 466: 109887.

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Theoretical versus Empirical

Simple

Controlled

Replicated

Short

Small

Artificial

Complex

Uncontrolled

Unreplicated

Long

Large

Natural

Analytical

model

Simulation

model

Lab

flask

Small indoor

microcosms

Field

microcosms

Whole-ecosystem

manipulation

Field

sampling

Big indoor

microcosms

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Stoichiometric Modelling of Light:Nutrient Effects

Producer

Grazer

x' (t) = bx (1 -

min[K, (P - θy)/ q]

x

) - f (x)y

y' (t) = εmin (1,

θ

(P - θy) / x

) f (x)y - dy

From: Loladze, I, Y. Kuang, and J.J. Elser. 2000. Stoichiometry in producer-grazer systems: linking energy flow and element cycling. Bull. Math. Biol. 62: 1137-1162.

(carbon biomass)

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Logistic growth

(light-dependent)

Droop equation

(nutrient-dependent)

Plant P:C ratio

Herbivore P:C ratio

Assumption 1

Fixed total mass

of phosphorus, P,

in the entire system.

​

Assumption 2

Plant P:C varies

with a minimum q;

herbivore P:C is a

constant, θ.

​

Assumption 3

All phosphorus of

the system is either

in plants or in

herbivores.

Liebig’s Law

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light

light

light

From: Loladze, I, Y. Kuang, and J.J. Elser. 2000. Stoichiometry in producer-grazer systems: linking energy flow and element cycling. Bull. Math. Biol. 62: 1137-1162.

pplane10.m

Scientific interpretation

Stoichiometric Modelling of Light:Nutrient Effects

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light

Grazer

From: Loladze, I, Y. Kuang, and J.J. Elser. 2000. Stoichiometry in producer-grazer systems: linking energy flow and element cycling. Bull. Math. Biol. 62: 1137-1162.

bifurcation

Scientific interpretation

Stoichiometric Modelling of Light:Nutrient Effects

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  • Topics of final project will be posted on the course website (Final Project Topics) next week.
  • Project guidelines have been posted on the course website (Project Guidelines). Structures and rubrics of final report and final presentation are clarified in this document.
  • Each topic can be chosen by different students. However, each student needs to do his/her final project independently without sharing his/her work with other students.
  • You cannot seek any external help for the final report and the final presentation.

Course Project Information

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  • Labs will provide a hands-on experience with modelling during the course.
  • Lab materials such as existing programs will be posted on the course website timely.
  • Ensure that MATLAB is downloaded before mid-September. It is free for students: OnTheHub from the UofA website. Use your CCID and password to login for free download of Matlab software.
  • Ensure that XPPAUT is installed before mid-September as we will use this to run bifurcation diagrams in Chapter 3. It can be installed on your laptop (Windows, Mac), iPad/iPhone, Android system from University of Pittsburgh: https://sites.pitt.edu/~phase/bard/bardware/xpp/xpp.html
  • BRING YOUR LAPTOP to the classroom - on lab days

Labs and Software