This self-contained introduction to the fast-growing field of Mathematical Biology is written for students with a mathematical background. It sets the subject in a historical context and guides the reader towards questions of current research interest. A broad range of topics is covered including: Population dynamics, Infectious diseases, Population genetics and evolution, Dispersal, Molecular and cellular biology, Pattern formation, and Cancer modelling. Particular attention is paid to situations where the simple assumptions of homogenity made in early models break down and the process of mathematical modelling is seen in action.
Author(s): Nicholas F. Britton
Series: Springer Undergraduate Mathematics Series
Publisher: Springer
Year: 2004
Language: English
Commentary: +OCR; 8 colors scan; no frontmatter
Pages: 370
List of Figures......Page 8
1.1 Introduction......Page 11
1.2 Linear and Nonlinear First Order Discrete Time Models......Page 12
1.2.1 The Biology of Insect Population Dynamics......Page 13
B.1 First-order Ordinary Differential Equations 271......Page
1.3 Differential Equation Models......Page 21
1.4 Evolutionary Aspects......Page 26
1.5 Harvesting and Fisheries......Page 27
1.6 Metapopulations......Page 31
1.7 Delay Effects......Page 34
1.8 Fibonacci's Rabbits......Page 37
1.9 Leslie Matrices: Age-structured Populations in Discrete Time......Page 40
1.10.1 Discrete Time......Page 44
1.10.2 Continuous Time......Page 48
1.11 The McKendrick Approach to Age Structure......Page 51
1.12 Conclusions......Page 54
2. Population Dynamics of Interacting Species......Page 57
2.2 Host-parasitoid Interactions......Page 58
2.3 The Loth-Volterra Prey-predator Equations......Page 64
2.4 Modelling the Predator Functional Response......Page 70
2.5 Competition......Page 76
2.6 Ecosystems Modelling......Page 80
2.7 Interacting Metapopulations......Page 84
2.7.1 Competition......Page 85
2.7.2 Predation......Page 87
2.7.3 Predator-mediated Coexistence of Competitors......Page 88
2.7.4 Effects of Habitat Destruction......Page 89
2.8 Conclusions......Page 91
3.1 Introduction......Page 93
3.2 The Simple Epidemic and SIS Diseases......Page 96
3.3 SIR Epidemics......Page 100
3.4 SIR Endemics......Page 106
3.4.1 No Disease-related Death......Page 107
3.4.2 Including Disease-related Death......Page 109
3.5 Eradication and Control......Page 110
3.6.1 The Equations......Page 113
3.6.2 Steady State......Page 115
3.7 Vector-borne Diseases......Page 117
3.8 Basic Model for Macroparasitic Diseases......Page 119
3.9 Evolutionary Aspects......Page 123
3.10 Conclusions......Page 125
4.1 Introduction......Page 127
4.2 Mendelian Genetics in Populations with Non-overlapping Gen- erations......Page 129
4.3 Selection Pressure......Page 133
4.4.2 Selection for a Recessive Allele......Page 137
4.4.4 The Additive Case......Page 139
4.5 Analytical Approach for Weak Selection......Page 140
4.6 The Balance Between Selection and Mutation......Page 141
4.7 Wright's Adaptive Topography......Page 143
4.8 Evolution of the Genetic System......Page 144
4.9 Game Theory......Page 146
4.10 Replicator Dynamics......Page 152
4.11 Conclusions......Page 155
5.1 Introduction......Page 157
5.2.1 General Derivation......Page 158
5.2.2 Some Particular Cases......Page 161
5.3 Directed Motion, or Taxis......Page 164
5.4 Steady State Equations and Transit Times......Page 166
5.4.2 Transit Times......Page 167
5.5 Biological Invasions: A Model for Muskrat Dispersal......Page 170
5.6 Travelling Wave Solutions of General Reaction-diffusion Equa- tions......Page 174
5.6.1 Node-saddle Orbits (the Monostable Equation)......Page 176
5.7 Travelling Wave Solutions of Systems of Reaction-diffusion Equations: Spatial Spread of Epidemics......Page 178
5.8 Conclusions......Page 182
6.1 Introduction......Page 185
6.2 Biochemical Kinetics......Page 186
6.3 Metabolic Pathways......Page 193
6.3.1 Activation and Inhibition......Page 194
6.3.2 Cooperative Phenomena......Page 196
6.4 Neural Modelling......Page 201
6.5 Immunology and AIDS......Page 207
6.6 Conclusions......Page 212
7.1 Introduction......Page 215
7.2 Turing Instability......Page 216
7.3 Turing Bifurcations......Page 221
7.4.1 Conditions for Turing Instability......Page 224
7.4.3 Do Activator-inhibitor Systems Explain Biological Pat- tern Formation?......Page 233
7.5 Bifurcations with Domain Size......Page 234
7.6 Incorporating Biological Movement......Page 239
7.8 Conclusions......Page 243
8.1 Introduction......Page 245
8.2 Phenomenological Models......Page 247
8.3 Nutrients: the Diffusion-limited Stage......Page 250
8.4 Moving Boundary Problems......Page 252
8.5 Growth Promoters and Inhibitors......Page 255
8.6 Vascularisation......Page 257
8.7 Metastasis......Page 258
8.8 Immune System Response......Page 259
8.9 Conclusions......Page 261
Further Reading......Page 263
A.1.1 Graphical Analysis......Page 267
A.1.2 Linearisation......Page 268
A.2.1 Saddle-node Bifurcations......Page 270
A.2.2 Transcritical Bifurcations......Page 271
A.2.3 Pitchfork Bifurcations......Page 272
A.2.4 Period-doubling or Flip Bifurcations......Page 273
A.3 Systems of Linear Equations: Jury Conditions......Page 276
A.4 Systems of Nonlinear Difference Equations......Page 277
A.4.2 Bifurcation for Systems......Page 278
B.1.1 Geometric Analysis......Page 281
B.1.3 Linearisation......Page 282
B.2.1 Geometric Analysis (Phase Plane)......Page 283
B.2.2 Linearisation......Page 284
B.2.3 Poincaé-Bendixson Theory......Page 286
B.3 Some Results and Techniques for mth Order Systems......Page 287
B.3.2 Lyapunov Functions......Page 288
B.4.1 Bifurcations with Eigenvalue Zero......Page 289
B.4.2 Hopf Bifurcations......Page 290
C.1 First-order Partial Differential Equations and Characteristics......Page 293
C.2.1 The Fundamental Solution......Page 294
C.2.2 Connection with Probabilities......Page 297
C.2.3 Other Coordinate Systems......Page 298
C.3 Some Spectral Theory for Laplace's Equation......Page 299
C.4 Separation of Variables in Partial Differential Equations......Page 301
C.5 Systems of Diffusion Equations with Linear Kinetics......Page 305
C.6 Separating the Spatial Variables from Each Other......Page 307
D.1 Perron-Frobenius Theory......Page 309
E. Hints for Exercises......Page 311
Index......Page 339
