An Introduction to mathematical biology
著者
書誌事項
An Introduction to mathematical biology
Pearson Prentice Hall, c2007
大学図書館所蔵 全11件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
Includes bibliographical references and index
内容説明・目次
内容説明
For advanced undergraduate and beginning graduate courses on Modeling offered in departments of Mathematics. This text introduces a variety of mathematical models for biological systems, and presents the mathematical theory and techniques useful in analyzing those models. Material is organized according to the mathematical theory rather than the biological application. Undergraduate courses in calculus, linear algebra, and differential equations are assumed.
目次
Preface xi
1 LINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES 1
1.1 Introduction 1
1.2 Basic Definitions and Notation 2
1.3 First-Order Equations 6
1.4 Second-Order and Higher-Order Equations 8
1.5 First-Order Linear Systems 14
1.6 An Example: Leslie's Age-Structured Model 18
1.7 Properties of the Leslie Matrix 20
1.8 Exercises for Chapter 1 28
1.9 References for Chapter 1 33
1.10 Appendix for Chapter 1 34
1.10.1 Maple Program:Turtle Model 34
1.10.2 MATLAB (R) Program:Turtle Model 34
2 NONLINEAR DIFFERENCE EQUATIONS, THEORY, AND EXAMPLES 36
2.1 Introduction 36
2.2 Basic Definitions and Notation 37
2.3 Local Stability in First-Order Equations 40
2.4 Cobwebbing Method for First-Order Equations 45
2.5 Global Stability in First-Order Equations 46
2.6 The Approximate Logistic Equation 52
2.7 Bifurcation Theory 55
2.7.1 Types of Bifurcations 56
2.7.2 Liapunov Exponents 60
2.8 Stability in First-Order Systems 62
2.9 Jury Conditions 67
2.10 An Example: Epidemic Model 69
2.11 Delay Difference Equations 73
2.12 Exercises for Chapter 2 76
2.13 References for Chapter 2 82
2.14 Appendix for Chapter 2 84
2.14.1 Proof of Theorem 2.1 84
2.14.2 A Definition of Chaos 86
2.14.3 Jury Conditions (Schur-Cohn Criteria) 86
2.14.4 Liapunov Exponents for Systems of Difference Equations 87
2.14.5 MATLAB Program: SIR Epidemic Model 88
3 BIOLOGICAL APPLICATIONS OF DIFFERENCE EQUATIONS 89
3.1 Introduction 89
3.2 Population Models 90
3.3 Nicholson-Bailey Model 92
3.4 Other Host-Parasitoid Models 96
3.5 Host-Parasite Model 98
3.6 Predator-Prey Model 99
3.7 Population Genetics Models 103
3.8 Nonlinear Structured Models 110
3.8.1 Density-Dependent Leslie Matrix Models 110
3.8.2 Structured Model for Flour Beetle Populations 116
3.8.3 Structured Model for the Northern Spotted Owl 118
3.8.4 Two-Sex Model 121
3.9 Measles Model with Vaccination 123
3.10 Exercises for Chapter 3 127
3.11 References for Chapter 3 134
3.12 Appendix for Chapter 3 138
3.12.1 Maple Program: Nicholson-Bailey Model 138
3.12.2 Whooping Crane Data 138
3.12.3 Waterfowl Data 139
4 LINEAR DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES 141
4.1 Introduction 141
4.2 Basic Definitions and Notation 142
4.3 First-Order Linear Differential Equations 144
4.4 Higher-Order Linear Differential Equations 145
4.4.1 Constant Coefficients 146
4.5 Routh-Hurwitz Criteria 150
4.6 Converting Higher-Order Equations to First-OrderSystems 152
4.7 First-Order Linear Systems 154
4.7.1 Constant Coefficients 155
4.8 Phase-Plane Analysis 157
4.9 Gershgorin's Theorem 162
4.10 An Example: Pharmacokinetics Model 163
4.11 Discrete and Continuous Time Delays 165
4.12 Exercises for Chapter 4 169
4.13 References for Chapter 4 172
4.14 Appendix for Chapter 4 173
4.14.1 Exponential of a Matrix 173
4.14.2 Maple Program: Pharmacokinetics Model 175
5 NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS: THEORY AND EXAMPLES 176
5.1 Introduction 176
5.2 Basic Definitions and Notation 177
5.3 Local Stability in First-Order Equations 180
5.3.1 Application to Population Growth Models 181
5.4 Phase Line Diagrams 184
5.5 Local Stability in First-Order Systems 186
5.6 Phase Plane Analysis 191
5.7 Periodic Solutions 194
5.7.1 Poincare-Bendixson Theorem 194
5.7.2 Bendixson's and Dulac's Criteria 197
5.8 Bifurcations 199
5.8.1 First-Order Equations 200
5.8.2 Hopf Bifurcation Theorem 201
5.9 Delay Logistic Equation 204
5.10 Stability Using Qualitative Matrix Stability 211
5.11 Global Stability and Liapunov Functions 216
5.12 Persistence and Extinction Theory 221
5.13 Exercises for Chapter 5 224
5.14 References for Chapter 5 232
5.15 Appendix for Chapter 5 234
5.15.1 Subcritical and Supercritical Hopf Bifurcations 234
5.15.2 Strong Delay Kernel 235
6 BIOLOGICAL APPLICATIONS OF DIFFERENTIAL EQUATIONS 237
6.1 Introduction 237
6.2 Harvesting a Single Population 238
6.3 Predator-Prey Models 240
6.4 Competition Models 248
6.4.1 Two Species 248
6.4.2 Three Species 250
6.5 Spruce Budworm Model 254
6.6 Metapopulation and Patch Models 260
6.7 Chemostat Model 263
6.7.1 Michaelis-Menten Kinetics 263
6.7.2 Bacterial Growth in a Chemostat 266
6.8 Epidemic Models 271
6.8.1 SI, SIS, and SIR Epidemic Models 271
6.8.2 Cellular Dynamics of HIV 276
6.9 Excitable Systems 279
6.9.1 Van der Pol Equation 279
6.9.2 Hodgkin-Huxley and FitzHugh-Nagumo Models 280
6.10 Exercises for Chapter 6 283
6.11 References for Chapter 6 292
6.12 Appendix for Chapter 6 296
6.12.1 Lynx and Fox Data 296
6.12.2 Extinction in Metapopulation Models 296
7 PARTIAL DIFFERENTIAL EQUATIONS: THEORY, EXAMPLES, AND APPLICATIONS 299
7.1 Introduction 299
7.2 Continuous Age-Structured Model 300
7.2.1 Method of Characteristics 302
7.2.2 Analysis of the Continuous Age-Structured Model 306
7.3 Reaction-Diffusion Equations 309
7.4 Equilibrium and Traveling Wave Solutions 316
7.5 Critical Patch Size 319
7.6 Spread of Genes and Traveling Waves 321
7.7 Pattern Formation 325
7.8 Integrodifference Equations 330
7.9 Exercises for Chapter 7 331
7.10 References for Chapter 7 336
Index 339
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