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