生物数学引论 英文PDF电子书下载

1．Single Species Population Dynamics 1

1.1 Introduction 1

1.2 Linear and Nonlinear First Order Discrete Time Models 2

1.2.1 The Biology of Insect Population Dynamics 3

1.2.2 A Model for Insect Population Dynamics with Competition 4

1.3 Differential Equation Models 11

1.4 Evolutionary Aspects 16

1.5 Harvesting and Fisheries 17

1.6 Metapopulations 21

1.7 Delay Effects 24

1.8 Fibonacci's Rabbits 27

1.9 Leslie Matrices:Age-structured Populations in Discrete Time 30

1.10 Euler-Lotka Equations 34

1.10.1 Discrete Time 34

1.10.2 Continuous Time 38

1.11 The McKendrick Approach to Age Structure 41

1.12 Conclusions 44

2．Population Dynamics of Interacting Species 47

2.1 Introduction 48

2.2 Host-parasitoid Interactions 48

2.3 The Lotka-Volterra Prey-predator Equations 54

2.4 Modelling the Predator Functional Response 60

2.5 Competition 66

2.6 Ecosystems Modelling 70

2.7 Interacting Metapopulations 74

2.7.1 Competition 75

2.7.2 Predation 77

2.7.3 Predator-mediated Coexistence of Competitors 78

2.7.4 Effects of Habitat Destruction 79

2.8 Conclusions 81

3．Infectious Diseases 83

3.1 Introduction 83

3.2 The Simple Epidemic and SIS Diseases 86

3.3 SIR Epidemics 90

3.4 SIR Endemics 96

3.4.1 No Disease-related Death 97

3.4.2 Including Disease-related Death 99

3.5 Eradication and Control 100

3.6 Age-structured Populations 103

3.6.1 The Equations 103

3.6.2 Steady State 105

3.7 Vector-borne Diseases 107

3.8 Basic Model for Macroparasitic Diseases 109

3.9 Evolutionary Aspects 113

3.10 Conclusions 115

4．Population Genetics and Evolution 117

4.1 Introduction 117

4.2 Mendelian Genetics in Populations with Non-overlapping Gen-erations 119

4.3 Selection Pressure 123

4.4 Selection in Some Special Cases 127

4.4.1 Selection for a Dominant Allele 127

4.4.2 Selection for a Recessive Allele 127

4.4.3 Selection against Dominant and Recessive Alleles 129

4.4.4 The Additive Case 129

4.5 Analytical Approach for Weak Selection 130

4.6 The Balance Between Selection and Mutation 131

4.7 Wright's Adaptive Topography 133

4.8 Evolution of the Genetic System 134

4.9 Game Theory 136

4.10 Replicator Dynamics 142

4.11 Conclusions 145

5．Biological Motion 147

5.1 Introduction 147

5.2 Macroscopic Theory of Motion;A Continuum Approach 148

5.2.1 General Derivation 148

5.2.2 Some Particular Cases 151

5.3 Directed Motion,or Taxis 154

5.4 Steady State Equations and Transit Times 156

5.4.1 Steady State Equations in One Spatial Variable 156

5.4.2 Transit Times 157

5.4.3 Macrophages vs Bacteria 160

5.5 Biological Invasions:A Model for Muskrat Dispersal 160

5.6 Travelling Wave Solutions of General Reaction-diffusion Equa-tions 164

5.6.1 Node-saddle Orbits(the Monostable Equation) 166

5.6.2 Saddle-saddle Orbits(the Bistable Equation) 167

5.7 Travelling Wave Solutions of Systems of Reaction-diffusion Equations:Spatial Spread of Epidemics 168

5.8 Conclusions 172

6．Molecular and Cellular Biology 175

6.1 Introduction 175

6.2 Biochemical Kinetics 176

6.3 Metabolic Pathways 183

6.3.1 Activation and Inhibition 184

6.3.2 Cooperative Phenomena 186

6.4 Neural Modelling 191

6.5 Immunology and AIDS 197

6.6 Conclusions 202

7．Pattern Formation 205

7.1 Introduction 205

7.2 Turing Instability 206

7.3 Turing Bifurcations 211

7.4 Activator-inhibitor Systems 214

7.4.1 Conditions for Turing Instability 214

7.4.2 Short-range Activation,Long-range Inhibition 219

7.4.3 Do Activator-inhibitor Systems Explain Biological Pat-tern Formation? 223

7.5 Bifurcations with Domain Size 224

7.6 Incorporating Biological Movement 229

7.7 Mechanochemical Models 233

7.8 Conclusions 233

8．Tumour Modelling 235

8.1 Introduction 235

8.2 Phenomenological Models 237

8.3 Nutrients:the Diffusion-limited Stage 240

8.4 Moving Boundary Problems 242

8.5 Growth Promoters and Inhibitors 245

8.6 Vascularisation 247

8.7 Metastasis 248

8.8 Immune System Response 249

8.9 Conclusions 251

Further Reading 253

A．Some Techniques for Difference Equations 257

A.1 First-order Equations 257

A.1.1 Graphical Analysis 257

A.1.2 Linearisation 258

A.2 Bifurcations and Chaos for First-order Equations 260

A.2.1 Saddle-node Bifurcations 260

A.2.2 Transcritical Bifurcations 261

A.2.3 Pitchfork Bifurcations 262

A.2.4 Period-doubling or Flip Bifurcations 263

A.3 Systems of Linear Equations:Jury Conditions 266

A.4 Systems of Nonlinear Difference Equations 267

A.4.1 Linearisation of Systems 268

A.4.2 Bifurcation for Systems 268

B．Some Techniques for Ordinary Differential Equations 271

B.1 First-order Ordinary Differential Equations 271

B.1.1 Geometric Analysis 271

B.1.2 Integration 272

B.1.3 Linearisation 272

B.2 Second-order Ordinary Differential Equations 273

B.2.1 Geometric Analysis(Phase Plane) 273

B.2.2 Linearisation 274

B.2.3 Poincaré-Bendixson Theory 276

B.3 Some Results and Techniques for mth Order Systems 277

B.3.1 Linearisation 278

B.3.2 Lyapunov Functions 278

B.3.3 Some Miscellaneous Facts 279

B.4 Bifurcation Theory for Ordinary Differential Equations 279

B.4.1 Bifurcations with Eigenvalue Zero 279

B.4.2 Hopf Bifurcations 280

C．Some Techniques for Partial Differential Equations 283

C.1 First-order Partial Differential Equations and Characteristics 283

C.2 Some Results and Techniques for the Diffusion Equation 284

C.2.1 The Fundamental Solution 284

C.2.2 Connection with Probabilities 287

C.2.3 Other Coordinate Systems 288

C.3 Some Spectral Theory for Laplace's Equation 289

C.4 Separation of Variables in Partial Differential Equations 291

C.5 Systems of Diffusion Equations with Linear Kinetics 295

C.6 Separating the Spatial Variables from Each Other 297

D．Non-negative Matrices 299

D.1 Perron-Frobenius Theory 299

E．Hints for Exercises 301

Index 329

1.1 World human population growth over the last 2000 years 2

1.2 Cobweb maps for idealised contest and scramble competition 6

1.3 Cobweb maps for the Hassell equation 7

1.4 Parameter space for the Hassell equation 8

1.5 Cobwebbing variant 10

1.6 Logistic population growth 14

1.7 Yield-effort curves 19

1.8 The archetypal metapopulation model 22

1.9 The archetypal metapopulation growth rate 23

1.10 Instability in a delay equation 26

1.11 Fibonacci's rabbits 32

1.12 A Lexis diagram 41

1.13 Archetypal survivorship curves 43

2.1 The functions log R0 and 1-1/R0 51

2.2 Some numerical solutions of the Nicholson-Bailey host-parasitoid model 51

2.3 Negative binomial and Poisson distributions 53

2.4 Results of host-parasitoid model with negative binomial search func-tion 53

2.5 Winter moth host-parasitoid simulations and data 54

2.6 Results of Lotka-Volterra prey-predator model 57

2.7 Hare-lynx population data 58

2.8 Predator functional response curves 61

2.9 Rosenzweig-MacArthur phase planes 63

2.10 Lotka-Volterra competition phase planes 68

2.11 Parameter space for Lotka-Volterra competition equations 68

2.12 Numerical solution of a planktonic ecosystem model 73

2.13 Huffaker's prey-predator experiments 75

2.14 State transitions for the metapopulation model with two competitors 75

2.15 Numerical results for a metapopulation model with competition 76

2.16 State transitions for patches in a metapopulation model for predation 77

2.17 Numerical results for a metapopulation model of a predator-prey interaction 78

2.18 State transitions for a metapopulation model with two competing prey and one predator 79

2.19 Numerical results for a metapopulation model of predator-mediated coexistence 80

3.1 Progress of a disease within an individual 85

3.2 Phase planes for the SIR epidemic 91

3.3 The final size of an SIR epidemic 93

3.4 Bombay plague of 1906 94

3.5 Numerical solutions for an SIR endemic 98

3.6 A typical time series for measles 100

3.7 The basic reproductive ratio R0 as a function of the virulence c 114

4.1 Hardy-Weinberg genotype frequencies 121

4.2 Diagram of a typical life cycle 123

4.3 Numerical solutions for weak and strong selection against a domi-nant and a recessive allele 128

4.4 Central dogma of modern evolutionary biology 132

4.5 Wright's adaptive topography 135

4.6 Invasibility plots 140

4.7 The evolutionary trajectories for the rock-scissors-paper game 144

5.1 Flux in a general direction 149

5.2 Conservation in one dimension 152

5.3 Data for the invasion of muskrats in Europe 161

5.4 Reaction-diffusion invasion wave 162

5.5 Phase plane for the node-saddle orbit 166

5.6 The Black Death in Europe,1347-1350 169

5.7 Diagrammatic representation of a propagating epidemic 169

5.8 Proof of a travelling wave in the rabies model 171

6.1 Michaelis-Menten reaction dynamics 178

6.2 Michaelis-Menten and cooperative reaction velocity curves 186

6.3 FitzHugh-Nagumo phase plane 194

6.4 A numerical solution of the excitatory FitzHugh-Nagumo equations 195

6.5 A numerical solution of the oscillatory FitzHugh-Nagumo equations 196

7.1 The early stages of development of a vertebrate egg 206

7.2 Turing dispersion relations 212

7.3 Spatial eigenvalues for Turing instability 216

7.4 Marginal Turing instability in diffusion-coefficient space 219

7.5 Pure activation-inhibition and cross-activation-inhibition 220

7.6 Turing bifurcations with domain size 225

7.7 Numerical solutions and real tail patterns 227

7.8 Examples of biological patterns 229

7.9 Turing bifurcation with chemotaxis 231

8.1 Tumour growth models 238

8.2 The profile of the tumour growth inhibitor 246

A.1 Some examples of cobweb maps 259

A.2 Behaviour of solutions of a first-order linear difference equation 260

A.3 Saddle-node bifurcations 261

A.4 A transcritical bifurcation 262

A.5 A pitchfork bifurcation 263

A.6 Period-doubling bifurcations and the cascade to chaos 265

A.7 Naimark-Sacker bifurcation 268

B.1 Geometrical method for first-order ordinary differential equations 272

B.2 An example of a phase plane 274

B.3 Properties of the roots of a quadratic 276

B.4 A second example of a phase plane 277

B.5 Sub-and supercritical Hopf bifurcations 281

E.1 Cobweb maps for the sterile insect control equation 302

E.2 Leslie phase plane and time series 306

E.3 Phase planes for an ecosystem with photosynthesis 307

E.4 Leslie phase plane and time series 308

E.5 Population genetics of sickle cell anaemia 314

E.6 Evolutionary dynamics for the rock-scissors-paper game with reward 317

E.7 Turing instability in Gierer-Meinhardt parameter space 323

E.8 Mimura-Murray phase plane for ecological patchiness 324

E.9 Bifurcation diagram for tumour growth inhibitor 327

E.10 Bifurcation diagram for tumour immune response 328