# 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.

## MAT 124: Mathematical Biology

Approved: 2006-09-01 (revised 2013-06-01, A. Mogilner)
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Mathematical Models in Biology by Leah Edelstein-Keshet; SIAM 2005; \$57.00.
Search by ISBN on Amazon: 978-0898715545
Prerequisites:
(MAT 022A or MAT 027A or MAT 067 or BIS 027A); (MAT 022B or MAT 027B or BIS 027B).
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; CO2 and ventilation volume; Population genetics. 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.