Course Topics
Learning Objectives
After successfully completing the course, the student will be able to
- Give examples of well-known mathematical models of biological systems
- Explain how mathematical approaches have informed biological understanding
- Formulate an ordinary differential equation model for a given biological system
- Use mathematical approaches to analyze ordinary differential equation models, including finding analytic solutions, qualitative analyses and linear stability analyses
- Numerically simulate a given ordinary differential equation model, using both prepackaged web applications and MATLAB
Course Topics
- Introduction to differential equations, including compartmental models and example biological contexts
- Initial value problems
- One dimensional models, including examples: exponential and logistic growth, Allee effect model, SI/SIS models, drug elimination model
- Separable equations
- First order linear equations and integrating factors
- Examples
- *Exact equations (online module)
- Autonomous models
- Slope field, numerical exploration using web applet
- Numerical integration
- Qualitative theory, equilibria and linear stability
- Bifurcations: saddle-node and transcritical
- Systems of ODEs, largely focused on 2D models. Examples to include predator-prey, competition, SIR model, HIV model
- Phase planes and nullclines, numerical exploration using PPLANE
- Recap: matrices and vectors, differentiation of sine and cosine
- Eigenvalues and eigenvectors, including online application
- Linear 2D system
- Stability; Linearization of 2D system
- 2nd order systems and reduction to 1st order
- Periodic solutions and limit cycles
- Periodically forced systems
- Introduction to MATLAB
- Examples