Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Math 563: Advanced Mathematical Methods in
Document related concepts
Transcript
Math 563: Advanced Mathematical Methods in Biology Instructor: Dr. Trachette Jackson Tentative Syllabus∗ This course is an exploration in applications of mathematics to various biological, ecological, physiological, and medical problems. What follows is a tentative list of topics to be covered including Population Dispersal, Geographic Spread of Epidemics, Waves of Pursuit in Predator-Prey Systems, Angiogenesis, Transport of Macromolecules in Tumors and Pattern Formation in Developmental Biology. Part I: Partial Differential Equations in Biological Settings Week 1 (Jan 6): Course Logistics/Motivation/Review • Motivation: Why is it worthwhile to model biological systems? • Review: Functions of Several Variables, Partial Differential Equations Weeks 2-3 (Jan 11-20th): Where do PDE models in Biology Come From? • Derivation of the Conservation Equation • Convection, Diffusion, Chemotaxis in terms of Flux • Random Walks • Properties of and Solution Techniques for PDEs Weeks 4-6 (Jan 25-Feb 10th): Applications • Population Dispersal • Macrophages and Bacteria • Facilitated Diffusion of Oxygen • Spatio-temporal Models of Tumor Growth • Transport of Macromolecules in Tissue • Angiogenesis • Slime Mold Aggregation Week 7 (February 15-17th): In Class Presentations Week 8 (Feb 22-24): Review, Final Project Outline Deadline, Exam Week 9 (March 1-3): SPRING BREAK Part II Traveling Waves and Models for Development and Pattern Formation in Biological Systems Weeks 10-11 (March 8-17th): Traveling Waves and Traveling Pulses • Background: The Traveling Wave Form • Fisher’s Equation: The Spread of Genes in a Population • Asymptotic Solution and Stability of Wavefront Solutions • The Spread and Control of Insect Populations • Waves of Pursuit and Invasion in Predator-Prey Systems • Spatial Spread of Epidemics • Tumor Encapsulation Weeks 12-14 (March 22-31): Spatial Pattern Formation with Reaction-Diffusion and Mechanical Mechanisms • Role of Pattern in Developmental Biology • The Turing Mechanism • Linear Stability Analysis • Dispersion Relations • Pattern Generation with Single Species Models • Mammalian Coat Patterns • Butterfly Wings • Mechanical Models for Generating Pattern Weeks 14-15 (April 5-14): Student Presentations/Review Week 16 (April 19): Exam Delay and Age-Structured Models (If Time Permits) • Linear Stability Analysis • Delay Models in Physiology: Dynamic Diseases • The Cell Cycle • Age of HIV Infection
