An Introduction to Mathematical Biology by Linda J.s. AllenAn Introduction to Mathematical Biology by Linda J.s. Allen

An Introduction to Mathematical Biology

byLinda J.s. Allen

Paperback | July 19, 2006

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KEY BENEFIT: This reference 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. Contains applications of mathematical theory to biological examples in each chapter. Focuses on deterministic mathematical models with an emphasis on predicting the qualitative solution behavior over time. Discusses classical mathematical models from population , including the Leslie matrix model, the Nicholson-Bailey model, and the Lotka-Volterra predator-prey model. Also discusses more recent models, such as a model for the Human Immunodeficiency Virus - HIV and a model for flour beetles. KEY MARKET: Readers seeking a solid background in the mathematics behind modeling in biology and exposure to a wide variety of mathematical models in biology.
Title:An Introduction to Mathematical BiologyFormat:PaperbackDimensions:368 pages, 9.9 × 7.9 × 0.9 inPublished:July 19, 2006Publisher:Pearson EducationLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0130352160

ISBN - 13:9780130352163

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Education and Development of Infants, Toddlers, and Preschoolers presents points of view, philosophies, and real-life program vignettes relating to teaching, caring for, working with, and parenting children from birth through preschool. Concepts, ideas, research, and applications assist readers in acquiring the knowledge and skills for working with young children. The intended audience includes child-care providers and students in early childhood education, nursing, social work, psychology, and home economics. Parents and prospective parents will discover implications for parenting as they learn about the developmental process. Throughout the text, current research supports new trends and classic studies clarify traditional practice. Research is used to heighten interest, illustrate concepts, clarify content, and provide a basis for decision making. Quotations from studies relate research to a particular topic. When care-givers act on the basis of evidence, they increase the chances that they will contribute positively to children's development and education. Child development and early childhood topics are part of everyone's life, whether or not he or she is a parent. Many contemporary magazines contain articles about the care and education of young children. These articles sometimes emphasize the bizarre and the sensational. The public is more aware than ever of child development issues. I show how contemporary issues, practices, and ideas relate to and, in many instances, grow out of current knowledge about young children. Another theme is the public attitude toward the very young. Public policies resulting from specific state and federal legislation affect how people rear and educate children. Laws relating to child abuse, compulsory school attendance, adoption, and education of the handicapped are just a few examples of public policy in action. These and other public policy programs are discussed in light of their effects on people and institutions and their resulting influence on practices relating to educating and developing young children. People need information and knowledge about young children in order to conduct quality programs, create good environments, and be effective care-givers -and service providers. The text does not advocate "one best way" of caring for or working with young children. Instead, different philosophies, models, and points of view are described to help readers understand the many opinions about what is "best." Real-life vignettes assist the reader in understanding how theories and ideas are translated into practice. Implications for Care-givers sections throughout the text provide reasonable and practical suggestions that enable those who work with the very young to conduct developmentally appropriate practices. Indeed, a hallmark of the text is the application of theory to everyday interactions with young children. Considerable attention is given to how people such as parents, teachers, siblings, and peers interact with developing children. The influences of environmental settings—home, school, peer group—are also emphasized. Special Features Reality-Based Discussions. The text is based on the realities of care-giving at all levels from birth through preschool. Specific techniques and activities are identified that care-givers can use with young children and their parents. The program vignettes in particular add a realistic focus to the text. Comprehensive and Extensive Coverage. This text integrates both development and curriculum. As a result, readers understand developmental concepts and theory as well as the applied curriculum and care-giving implications of theory. Comprehensive coverage of the major topics relating to the education and development of young children are included: involving parents, meeting young children's special needs, and guiding behavior. Current and Up-to-Date Information. The latest thinking of professionals in the field for the 1980s and '90s is presented. Material on infant care-giving provides theory and care-giver activities for this fast-growing area of early childhood education. Chapter-Opening Questions. Study questions help readers develop a learning set as a prelude to their reading. They alert readers to chapter content and help focus purpose and intent. In addition, they provide a basis for review and reflection. Program Vignettes. Vignettes of real-life programs spark the reader's interest and illuminate textual material. They add a realistic base to theory and assist readers in seeing what life is like for children and care-givers in varied settings. End-of-Chapter Activities. Each chapter ends with thought-provoking and challenging activities designed to help readers enrich and extend their knowledge and understanding about what is involved in being a good care-giver. End-of-Chapter Readings. The suggested readings provide direction for readers who want to pursue a topic in greater detail. The readings are interesting and informative and will contribute to readers' professional growth. The annotations also provide insights and additional useful information. Readability. The text is presented in an orderly, logical, and interesting manner. Visual materials such as charts and tables summarize and highlight important information and concepts.

Table of Contents

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® 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 Poincaré-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