## Table of contents for Mathematical modelling with case studies : a differential equations approach using Maple and MATLAB / Belinda Barnes, Glenn Robert Fulford.

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1 Introduction to mathematical modelling                                   1
1.1  Mathematical models  . ....................          .      .   .  .. .  2
1.2 An overview of the book  . ................... ...             . .. .  3
1.3  Some modelling approaches  . .................      ....  .   . .   5
1.4  Modelling for decision-making  . ................     .      .  .  ..  7
2 Compartmental models                                                     9
2.1  Introduction   .  ...................     .  . .  ....  ...  .  .  . 10
2.2 Exponential decay and radioactivity . ............       . . .. . . 10
2.3  Case Study: Detecting art forgeries  . ..............  .   ..  . . . 17
2.4  Case Study: Pacific rats colonise New Zealand . ..  . . .  . . . .. ..19
2.5  Lake pollution  models  . ..................     ........... ..    20
2.6  Case Study: Lake Burley Griffin .. . ........    . . .  . . . . . .26
2.7 Drug assimilation into the blood ...............        .    .  . . ..28
2.8  Case Study: Dull, dizzy  or dead?  . . . . .. ...........  .   .   .  33
2.9 Cascades of compartments  . . . . .  ...................         . .. 37
2.10  First-order linear DEs  .  ....................         .    .  .  ...  .  38
2.11 Equilibrium points and stability  . . . ................ .    . .  40
2.12 Case Study: Money, money, money makes the world go around . . . .  .  41
2.13  Exercises for Chapter 2  .... .... .. . . . ...... ..... . . . .. . 44
3 Models of single populations                                            51
3.1  Exponential growth  .. .  ... .................      ..  .    .  .   .  . 52
3.2  Density  dependent growth  . .... .  .................      ..  . . 56
3.3 Limited growth with harvesting  .............. . .        . . . . .63
3.4  Case Study: Anchovy  wipe-out  . ..................          .  .   . .  65
3.5  Case Study: How can 2 x 106 birds mean rare? . . . . . .  .     . . .  66
3.6 Discrete population growth and chaos.. .  . . . .  . . . ..... ..67
3.7  Time-delayed  regulation  . ................ ...      .      .  .  .. .  74
3.8  Case Study: Australian blowflies........     .  . . . .  . . . . . ..76
3.9 Exercises for Chapter 3 ............. ................ .            77
4 Numerical solution of differential equations                            83
4.1  Introduction  . ............. .................         .     .  . .   84
4.2 Basic numerical schemes  .. . . . . .  .  . . .  ............. . .  84
4.3 Computer implementation using Maple and MATLAB . . . . . .   .  .   .  87
4.4  Instability  . .............. ................               .  .  .  .. 89
4.5  D iscussion  .  .  .  . . . . . . . . . . . . . . . . . . . .... . . .  . .  .  .   .  .  . 90
4.6  Exercises for  Chapter 4  ...................     ...   .   ......  .  .  91
5 Interacting population models                                          95
5.1  Introduction  . . . .  . ..................  ........ .    ...... .  96
5.2  An epidemic model for influenza  . ..................      ...  .  97
5.3  Predators  and  prey  .  .  .......................    .   .   .   .  . 106
5.4  Case Study: Nile Perch  catastrophe  . ..............  ...  .   .  .. 113
5.5  Competing species  ...  . . . . .  .  .... ................ . .  .. 114
5.6  Case Study: Aggressive protection of lerps and nymphs  . . . . . . . ....  119
5.7 Model of a battle  . ............. .................. ..         120
5.8  Case Study: Rise and fall of civilisations  . .................. .. 125
5.9 Exercises for Chapter 5 ..... .  ................. .       .  .   . ..128
6 Phase-plane analysis                                                  135
6.1  Introduction  . . . . . .  ......  . ......... ............ .  .. 136
6.2 Phase-plane analysis of epidemic model . . . . ... .. .... . .. ..139
6.3 Analysis of a battle model  . .....   .................. ..      143
6.4  Analysis of a predator-prey model  . ................  .  ..... .. 148
6.5  Analysis of competing species models  . ................ . ... . .. 152
6.6 Closed trajectories for the predator-prey  . .................. .. 158
6.7  Case Study: Bacteria battle in  the gut  . .............. .. .  .  .   . .. 160
6.8  Exercises for Chapter 6  . .............. .............. ..     162
7 Linearisation analysis                                                169
7.1  Introduction  . ........... ..............       . . .  .  .  .  . .  .. 170
7.2 Linear theory . ........... ....................... ..           170
7.3  Applications of linear theory  . ...............  ......... ..  179
7.4  Nonlinear theory...  . . . ..... ......................   .. . .   181
7.5  Applications of nonlinear theory  ...  .  .......... .. .  .. . .  . . .. 184
7.6 Exercises for Chapter 7 ..... ..........      . . . .  . . . . ... .. 187
8 Some extended population models                                       191
8.1 Introduction... . ................................ ..            192
8.2  Case Study: Competition, predation and diversity  . . . . .  .  .  .  .....  192
8.3  Extended  predator-prey model  ...................    .  .   . ..  .. 193
8.4  Case Study: Lemming mass suicides? .... . .........  . . . . . ..198
8.5  Case Study: Prickly-pear meets its moth  . . . . . ....... .  .  .  . .  200
8.6  Case Study: Geese defy mathematical convention . . . . . . .... .....  202
8.7  Case Study: Possums threaten New Zealand cows  . . . . . .  .  .  .  .....  206
8.8  Exercises for Chapter 8  . ............ ...............       .  .. 212
9 Formulating basic heat models                                        217
9.1 Introduction  .... .  . ............................. ..218
9.2  Some basic physical laws  . ........................... ..      219
9.3  Model for a  hot water heater  . .................. .    .  .  . . .. 222
9.4 Heat conduction and Fourier's law ........... . . . . .  .  .  . .....  226
9.5 Heat conduction through a wall . ....................... ..      228
9.6  Radial heat conduction  . ...............        .....   .. .  .. 232
9.7  H eat  fins  .  .  .  .  .  .   . . . . . .. . . . . . . . . . . . . . . . . . . . . .   .  . . 234
9.8  Exercises for Chapter 9  . ............... ............. ..     238
10 Solving time dependent heat problems                                241
10.1  The cooling coffee problem  revisited  . .................. . . . .  242
10.2 The hot water heater problem revisited  . .............. . ....  244
10.3  Case Study: It's hot and stuffy in the attic  . ................. .. 248
10.4 Spontaneous combustion  . .................. ......... ..       250
10.5 Case Study: Fish and chips explode  . .............. ........ . 257
10.6 Exercises for Chapter 10  . .................. ......... ..     258
11 Solving heat conduction problems                                    263
11.1  Boundary value problems  . ................. ...    .   .. . ... .  264
11.2  Heat loss through a wall  ............  . . . . . . .  .  .  .  .... .   266
11.3 Case Study: Double glazing: What's it worth?  . ............... .. 271
11.4 Insulating a water pipe . .......... .................. ..      274
11.5 Cooling a computer chip  . ............. ........ . . . ..279
11.6 Exercises for Chapter 11  . ............. ..............        284
12 Introduction to partial differential equations                      289
12.1  The heat conduction equation  ............  . . . .  .  .   .  .  .  . 290
12.2  Oscillating soil temperatures  . .................. .. .... .. .. 292
12.3 Case Study: Detecting land mines ......  ............... . ..   296
12.4 Lake pollution revisited .. . . . .................       . . . . .  299
12.5  Exercises for Chapter 12  .  .................. ......... ..   306
A Differential equations                                               309
A.1 Properties of differential equations . .............. . . . ... . 309
A.2  Solution  by  inspection  . ..................  .........      .. 310
A.3  First-order separable equations  . ................ .  . .  .  .   .  311
A.4 First-order linear equations and integrating factors . . . . . .  . . .....  312
A.5  Homogeneous equations  . .................. .......... ..       313
A.6 Inhornogeneous equations . . . ................. . ...... ..314
B Further mathematics                                                  317
B.1  Linear algebra  . . . . . ...........   ..............      .  .. 317
B.2  Partial derivatives and Taylor expansions  . ..................  320
B.3  Review  of complex numbers  . ..................     .   .....   .  .. 323
B.4  Hyperbolic functions  . ............... .............. ..       323
B.5  Integration using partial fractions  . . .  .. ................ .  .. 325
C Notes on Maple and MATLAB                                            327
C.1  Brief introduction  to  Maple  . .................. ........ .. 327
C.2 Solving differential equations with Maple .... . . . .  . .  . . .  327
C.3 Brief introduction to MATLAB  .......... . . . . .  .     .  .  ......  329
C.4 Solving differential equations with MATLAB  . ................ .. 330

Library of Congress subject headings for this publication: Differential equations Mathematical models, Maple (Computer file)