Explorations of Mathematical Models in Biology with Maple

Explore and analyze the solutions of mathematical models from diverse disciplines

As biology increasingly depends on data, algorithms, and models, it has become necessary to use a computing language, such as the user-friendly MapleTM, to focus more on building and analyzing models as opposed to configuring tedious calculations. Explorations of Mathematical Models in Biology with Maple provides an introduction to model creation using Maple, followed by the translation, analysis, interpretation, and observation of the models.

With an integrated and interdisciplinary approach that embeds mathematical modeling into biological applications, the book illustrates numerous applications of mathematical techniques within biology, ecology, and environmental sciences. Featuring a quantitative, computational, and mathematical approach, the book includes:

• Examples of real-world applications, such as population dynamics, genetics, drug administration, interacting species, and the spread of contagious diseases, to showcase the relevancy and wide applicability of abstract mathematical techniques
• Discussion of various mathematical concepts, such as Markov chains, matrix algebra, eigenvalues, eigenvectors, first-order linear difference equations, and nonlinear first-order difference equations
• Coverage of difference equations to model a wide range of real-life discrete time situations in diverse areas as well as discussions on matrices to model linear problems
• Solutions to selected exercises and additional Maple codes

MAZEN SHAHIN, PhD, is Professor in the Department of Mathematical Sciences at Delaware State University. He has extensive background and experience in designing interdisciplinary instructional materials that integrate mathematics and other disciplines, such as biology, ecology, and finance. Dr. Shahin’s research interests include boundary value problems, dynamical systems, impulsive differential equations, and mathematics education. Dr. Shahin is the author of Explorations of Mathematical Models in Biology with MATLAB®, also published by Wiley.

Preface vii

Acknowledgments xi

1 Overview of Discrete Dynamical Modeling and Maple™ 1

1.1 Introduction to Modeling and Difference Equations 1

1.2 The Modeling Process 8

1.3 Getting Started with Maple 12

2 Modeling with First-Order Difference Equations 37

2.1 Modeling with First-Order Linear Homogeneous Difference Equations with Constant Coefficients 37

2.2 Modeling with Nonhomogeneous First-Order Linear Difference Equations 53

2.3 Modeling with Nonlinear Difference Equations: Logistic Growth Models 72

2.4 Logistic Equations and Chaos 89

3 Modeling with Matrices 100

3.1 Systems of Linear Equations Having Unique Solutions 100

3.2 The Gauss–Jordan Elimination Method with Models 114

3.3 Introduction to Matrices 136

3.4 Determinants and Systems of Linear Equations 164

3.5 Eigenvalues and Eigenvectors 176

3.6 Eigenvalues and Stability of Linear Models 199

4 Modeling with Systems of Linear Difference Equations 209

4.1 Modeling with Markov Chains 209

4.2 Age-Structured Population Models 230

4.3 Modeling with Second-Order Linear Difference Equations 240

5 Modeling with Nonlinear Systems of Difference Equations 258

5.1 Modeling of Interacting Species 258

5.2 The SIR Model of Infectious Diseases 272

5.3 Modeling with Second-Order Nonlinear Difference Equations 280

Bibliography 286

Index 288

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