Preface ....................................................... vii
Acknowledgments ................................................ xi
1 Overview of Discrete Dynamical Modeling and MapleТМ .......... 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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