Mathematics in Action - Modelling the Real World Using Mathematics

With 91 Spreadsheet Based Models to Accompany the Text

By Richard Beare

Chartwell-Bratt/Studentlitteratur 1997 (530 pages). ISBN 0-86238-492-3.

Contents of book

Note on levels of mathematical sophistication

• Topics without an asterisk require only basic mathematics (i.e. simple algebra, the idea of rates of change, the awareness of functions such as exp x and ln x)

• Topics with one asterisk require further mathematics (as would be covered in British A-level mathematics courses)

• A few sections marked with two asterisks are advanced material included to provide more complete coverage for more advanced students

Preface

Chapter 1 What is mathematical modelling?

Types of models and reasons for modelling

1.1 The difference between a model and a problem

1.2 Types of model

1.3 Example one: petrol station pricing strategies

1.3 Example two: Olympic track records

1.4 Example three: Hospital corners

1.5 Example four: Animal fur

1.7 Reasons for Modelling

1.8 The modelling process

Chapter 2 Getting started with graphs, data and simple algebra

Models based on simple algebraic relationships and graphs, empirical models, curve fitting and dimensional analysis 33

2.1 Models based on simple algebraic functions

Example one: maximising traffic flow - more haste less speed?

Example two: the best place to kick from to convert a try in rugby

2.2 Empirical models

Example one: Search for the missing planet

Example two: The population of England and Wales since 1751

2.3 Using transformations to simplify relationships

Example one: the growth of a population of birds (log-linear transformation)

Example two: The orbital periods of the planets and their distances from the Sun (log-log transformation)

Example three: Growth of a yeast culture - a transformation for the logistic curve

2.4 Qualitative models using graphs

Analysis of the supply and demand for a product

2.5 Dimensional analysis

Example one: Adding up a shopping list

Example two: Forces on an aircraft

Example three Air resistance on balls in different sports

Exercises

Modelling problems

More structured problems

Chapter 3 Modelling Step by Step Processes

Dynamic models involving difference equations (recurrence relations)

3.1 Introduction to step by step models

3.2 Example: Growth of savings with interest paid at fixed intervals

3.3 Words or symbols?

3.4 A model for the growth of a population of annual plants

3.5 Difference equation models in general

3.6 Making the annual plants model more realistic by including limited resources - a non-linear model

* 3.7 Limit cycles and chaos in the discrete logistic model

Observing limit cycles and chaos

Bifurcation or Feigenbaum diagrams

The significance of chaos for mathematical modelling and our understanding of the world we live in

** 3.8 Analysing the non-linear annual plants model using cobweb diagrams

3.9 Modelling a nation's economy - using a second order difference equation

** 3.10 Some background mathematics: linear difference equations with constant coefficients

Linear first order difference equations with constant coefficients

Linear second order difference equations with constant coefficients

Application to the national economy model in Section 3.9

3.11 A time-lag model of population growth using a second order difference equation

3.12 Supply and demand for a product - using a cobweb diagram to illustrate a process involving a first order difference equation

Exercises

Modelling problems

More structured problems

Chapter 4 Modelling Continuous Processes

Dynamic models involving ordinary differential equations

4.1 Introduction to dynamic models involving continuous change

4.2 A model of a savings account

4.3 Models involving first order ordinary differential equations

4.4 Exponential growth and decay

The equation for exponential growth or decay and its solution

Properties of the exponential function

Instantaneous rates of change and average changes over finite intervals of time

Example one: World population

Example two: radiocarbon dating of archaeological finds

4.5 Exponential growth and decay relative to a "baseline"

Example When it rains and when it doesn't

4.6 The simple Euler numerical method for solving differential equations

Numerical methods in general

The Euler method

Example one: Growth of savings with interest added on a daily basis

A brief look at accuracy of numerical methods

Example two: Drug half lives and dosage intervals

* Extending the drug dose model to include gradual release of a drug

4.7 Non-linear first order differential equation models

Example one The logistic differential equation for population growth

* Analytic Solution to the logistic equation

Phase plane analysis of the logistic equation

Example two A leaky pond

* 4.8 More accurate numerical methods

* The modified Euler method

** The fourth order Runge-Kutta method

** 4.9 Understanding local error and global error in terms of Taylor's series

** Accuracy of the modified Euler method

Exercises

Modelling problems

Structured problems

Chapter 5 Flow Models and Compartment Models

Dynamic models using systems of linear first order differential equations

5.1 Introduction

5.2 Example one: a cascade model involving containers of water

Applying the simple Euler method to find a numerical solution

5.3 Example two: carbon flow through leaf litter and soil in ecosystems

* Analytic Solution

5.4 Example three: pollution in lakes - a simple a carrier-tracer model

Dimensional analysis of the lake pollution model

5.5 Example four: cholesterol in the body - a two-way carrier-tracer model

** 5.6 Solution of the general two compartment problem

** 5.7 Example five: the kinetics of a pain-killing drug

* 5.8 Example six: Modelling the decline in numbers of loggerhead sea turtles using a Leslie matrix approach

Exercises

Modelling problems

Structured problems

Chapter 6 Population Interactions

Numerical and graphical methods for dynamic models involving systems of non-linear first order differential equations

6.1 Population Interactions

The Hudson's Bay Company records for snowshoe hare and Canada lynx pelts

6.2 Population cycles and the basic Lotka-Volterra equations

Assumptions made in population interaction models

Direction flows and direction fields for predator-prey models

Phase plane plots (or state space diagrams) for predator-prey models

6.3 Developing the basic Lotka-Volterra model

Finite carrying capacity for the prey

Refuges for the prey

Varying appetite for the predators

Combining the three refinements in one model

6.4 Competition models

Phase plane analysis of competition models

** 6.5 The stability of equilibrium solutions

Introduction

Linearising the equations near a point of equilibrium

Possible types of behaviour

Example: The stability of equilibrium solutions in the competition model of Section 6.4

Exercises

Modelling problems

Structured problem

Chapter 7 Case Study - Epidemics

The progressive development and refinement of a model using analytic, numerical and graphical methods

7.1 Introduction

7.2 The simplest possible model - no recovery or immunity

7.3 A model with recovery but no immunity

7.4 Phase plane analysis

7.5 Model with immunity and isolation

Some simple conclusions from examining the model equations

Application of the model to the Great Plague of 1666 in England

7.6 Model with immunity, isolation and replenishment

Exercises

Modelling problem

Structured problem

Chapter 8 Models in Mechanics

Models using second order ordinary differential equations

8.1 Free fall with and without air resistance

Introduction

Free fall without air resistance

Including air resistance

Numerical solution

* Analytic solution

8.2 Car suspensions, sleeping policemen, and roads in the Outback

Introduction

Numerical solution using the leapfrog method

** Analytic solution for the damped oscillator

Sleeping policemen

** Analytic solution for the corrugations model

8.3 Orbits of the satellites of Jupiter

8.4 Projectile motion with air resistance

Introduction

Exercises

Modelling problems

Chapter 9 Modelling Random Processes

Monte Carlo models using random distributions models that calculate probabilities, differential equation models that predict average behaviour

9.1 Modelling using random distributions

Introduction

A random walk model of diffusion in one dimension (example of a discrete distribution)

Two-dimensional random walk models of diffusion

9.2 Various types of random distribution

Bernoulli or alternative distribution (discrete)

Uniform distribution (continuous)

Binomial distribution (discrete)

Poisson processes: the Poisson (discrete) and negative exponential (continuous) distributions

Normal distribution (continuous)

9.3 The probability of extinction in small populations with random births and deaths - an event driven model involving a Poisson process

The difference between event driven and time driven Monte Carlo models (or event sequenced and time slicing models)

9.4 Queues for showers on a campsite - two examples of an event driven model involving a Poisson process

A basic model involving just one shower

Extending the model: several showers, peak periods, varying times in the shower

9.6 How birds can migrate over long distances using only minimal directional clues - an example of a Monte Carlo model using a

discrete distribution

9.7 Hedgehogs on roads: comparing five different mathematical approaches to modelling random death

The various approaches to modelling random phenomena

A random death model for hedgehogs

A deterministic model to predict average behaviour

A time driven or time-slicing Monte Carlo model

An event driven or event-sequenced Monte Carlo model

* A discrete time model that calculates probabilities and how they change with time

** A continuous time model that calculates probabilities and how they change with time

Exercises

Modelling problems

Chapter 10 Spatial and Diffusion Models

Models using partial differential equations

10.1 Introduction to spatial models

Introduction

A river flowing parallel to the coastline

Examining the assumptions involved

** Extending the water table model to two dimensions

** Numerical solution in two dimensions

** Water from a well or borehole - a circularly symmetric model

** 10.3 Temperature variations below the soil surface

** Introduction

** Numerical solution in two dimensions

** 10.4 Diffusion in two dimensions

** Varying water table levels

** Numerical solution

** Diffusion of species - muskrats in Europe

** 10.5 Diffusion as a microscopic and a macroscopic process - comparing approaches

** The alternatives of a microscopic probabilistic model and a macroscopic continuous model

** Approximating a random walk by using the normal distribution

Comparison of random walk model with continuous model using a partial differential equation

**10.6 Appendix: a few formulae for use in three dimensional spatial models

Exercises

Modelling problems

Appendices