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:

Completion of courses MAT 22A or 67 and MAT 22B.

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.

Additional Notes:

This is 26 lectures of a total of 29. One will be devoted to the midterm exam, and two are for flexibility. If time runs out, consider shortening or eliminating lectures on limit cycles, oscillations, and excitable systems.

Learning Goals:

This course aims to help students to develop mathematical modeling skills, learn how to apply modeling to understanding biological systems, and, specifically, teaches them elements of dynamical systems, phase portrait analysis, scaling techniques, stability analysis, simulation of stochastic processes and diffusion-reaction-drift PDEs. The course discusses classical mathematical models of mathematical biology including logistic equation, Wolpert gene activation model, Ricker equation, Lotka-Volterra system, SIR model, Michaelis-Menten kinetics, Fitzhugh-Nagumo model, Turing system, Fisher equation for traveling waves, slime-mold aggregation, Bicoid distribution, dynamic instability models. Mastery of this course means ability to design mathematical models in biological applications, scale and non-dimensionalize equations, analyze equations qualitatively, apply diverse mathematical tools to solve nonlinear equations, communicate model assumptions and predictions, and to function on multi-disciplinary teams.

Assessment:

To assess the learning outcome for this course, there are 7-8 homeworks, 2 midterms, one final exam and one undergraduate project. For the project, the class is divided in groups of 4. Each group is given a recent paper that contains both experimental biological data and mathematical model. The students have to explain the paper to the class, write a short report on approaches used, offer critique of the paper and investigate analytically and numerically simplified model based on the paper.