Explorations of Mathematical Models in Biology with MATLAB

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

Explorations of Mathematical Models in Biology with MATLAB is an ideal textbook for upper-undergraduate courses in mathematical models in biology, theoretical ecology, bioeconomics, forensic science, applied mathematics, and environmental science. The book is also an excellent reference for biologists, ecologists, mathematicians, biomathematicians, and environmental and resource economists.

PREFACE ix

ACKNOWLEDGMENTS xiii

1 OVERVIEW OF DISCRETE DYNAMICAL MODELING AND MATLAB® 1

1.1 Introduction to Modeling and Difference Equations 1

1.2 The Modeling Process 8

1.3 Getting Started with MATLAB 13

2 MODELING WITH FIRST-ORDER DIFFERENCE EQUATIONS 28

2.1 Modeling with First-Order Linear Homogenous Difference Equations with Constant Coefficients 28

2.2 Modeling with Nonhomogenous First-Order Linear Difference Equations 42

2.3 Modeling with Nonlinear Difference Equations: Logistic Growth Models 58

2.4 Logistic Equations and Chaos 74

3 MODELING WITH MATRICES 85

3.1 Systems of Linear Equations Having Unique Solutions 85

3.2 The Gauss-Jordan Elimination Method with Models 99

3.3 Introduction to Matrices 119

3.4 Determinants and Systems of Linear Equations 147

3.5 Eigenvalues and Eigenvectors 160

3.6 Eigenvalues and Stability of Linear Models 185

4 MODELING WITH SYSTEMS OF LINEAR DIFFERENCE EQUATIONS 195

4.1 Modeling with Markov Chains 195

4.2 Age-Structured Population Models 219

4.3 Modeling with Second-Order Linear Difference Equations 231

5 MODELING WITH NONLINEAR SYSTEMS OF DIFFERENCE EQUATIONS 249

5.1 Modeling of Interacting Species 249

5.2 The SIR Model of Infectious Disease 264

5.3 Modeling with Second-Order Nonlinear Difference Equations 270

REFERENCES 277

INDEX 279

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.