Mathematical modeling is the most effective bridge connecting mathematics and many disciplines such as physics, biology, computer science, engineering, and social sciences. A mathematical model, which is a mathematical description of a real system, can potentially help to explain a system, to uncover the underlying mechanisms via hypotheses and data fitting, to examine the effects of different components, and to make predictions.
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Mathematical Modeling I – preliminary is designed for undergraduate students. Two other followup books, Mathematical Modeling II – advanced and Mathematical Modeling III – case studies in biology, will be published. II and III will be designed for both graduate students and undergraduate students. All the three books are independent and useful for study and application of mathematical modeling in any discipline.
- Introduction
- Discrete-time models
- Motivation
- An example – bacterial reproduction
- Solution and equilibrium of a discrete model
- Cobwebbing
- General theory and analytical methods
- Optimization of discrete models
- Continuous-time models
- Motivation and derivation of continuous models
- Differential equation models
- Separation of variables
- Linear equations
- Optimization of continuous models
- Sensitivity analysis
- Systems of difference equations (discrete)
- Analytical methods
- Some examples
- Systems of differential equations (continuous)
- Some motivation examples
- Nondimensionalization
- Analytical methods
- Higher dimensional systems
- Bifurcation analysis
- Saddle-node bifurcation
- Transcritical bifurcation
- Pitchfork bifurcation
- Generic saddle-node bifurcation
- Saddle-node, transcritical, and pitchfork bifurcations in two-dimensional systems
- Introduction of Hopf bifurcations
- Normal form of Hopf bifurcation
- Generic Hopf bifurcation
- Matlab programming
- Data fitting