## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

**Approved:**2006-09-01 (revised 2013-06-01, A. Mogilner)

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Search by ISBN on Amazon: 978-0898715545

**Prerequisites:**

**Suggested Schedule:**

Lecture(s) |
Sections |
Comments/Topics |

1 |
Linear Difference Equations & Systems: Methods |
Exponential solutions; Eigenvalues and qualitative behavior. |

1.5 |
Linear Difference Equations & Systems: Applications |
Population dynamics – plant propagation and insect generations; Red blood cell dynamics; Control of respiratory volume. |

2 |
Nonlinear Difference Equations: Methods |
Fixed points and their stability; Bifurcations, stable oscillations and period doubling; Graphical methods – cobwebbing. |

1 |
Nonlinear Difference Equations: Applications |
Logistic growth; Density-dependent population dynamics. |

1 |
Systems of Nonlinear Difference Equations: Methods |
Matrix representations; Fixed points and stability criteria. |

1 |
Systems of Nonlinear Difference Equations: Applications |
Host-parasitoid systems; CO |

2 |
ODEs and Systems (Linear Equations and Systems): Methods |
Fixed points of ODEs and their stability (graphical methods); Matrix methods for linear systems; Eigenvalues and qualitative behavior; Phase plane and analysis. |

1.5 |
ODEs and Systems (Linear Equations and Systems): Applications |
Growth in a chemostat; Compartmental models in physiology. |

2 |
ODEs and Systems (Nonlinear Equations and Systems): Methods |
Fixed points and analysis of their stability; Nullclines and phase plane methods; Dimensional analysis. |

3 |
ODEs and Systems (Nonlinear Equations and Systems): Applications |
1. Enzyme kinetics - Michaelis-Menten; cooperativity; threshold phenomena; chemotherapy models. 2. Population dynamics - Predator prey systems; interspecies competition; mutualism; effects of fishing or hunting. 3. Epidemiology - SIR models, effects of vaccination |

2-3 |
ODEs and Systems (Limit Cycles, Oscillations, and Excitable Systems): Methods |
Poincare-Bendixson Theorem - existence of stable cycles; Cubic nullclines; Hopf bifurcation. |

2 |
ODEs and Systems (Limit Cycles, Oscillations, and Excitable Systems): Applications |
Transmission of action potentials in neurons - Hodgkin-Huxley equations; Fitzhugh-Nagumo analysis; Oscillations in population biology; Oscillatory chemical and biochemical systems; Circadian rhythms |

2 |
PDEs (Transport Processes): Methods |
The diffusion equation; Laminar hydrodynamics. |

3 |
PDEs (Transport Processes): Applications |
Diffusion in disease models, Diffusive transport in physiology; Hemodynamics. |

**Additional Notes:**

**Learning Goals:**

**Assessment:**