MATH 563 - Advanced Mathematical Methods for the Biological Sciences
   
Instructor: Forger

3 credits

Prerequisites: Math 217 and Math 450 or 454.

Background and Goals:  
Natural systems behave in a way that reflects an underlying spatial pattern.  This course focuses on discovering the way in which spatial variation influences the motion, dispersion, and persistence of species.  The concepts underlying spatially dependent processes and the partial differential equations which model them will be discussed in a general manner with specific applications taken from molecular, cellular, and population biology.  This course is centered on modeling in three major areas i) Models of Motion: Diffusion, Convection, Chemotaxis and Haptotaxis; ii) Biological Pattern Formation; and iii) Delay-differential Equations and Age-structured Models.

Course Content: 
This course will introduce and explore partial differential equation modeling in biological settings.  Students should have some experience with solution techniques for partial differential equations as well as an interest in biomedical applications.  There will also be a brief introduction to delay differential equations and age-structured models; however no previous background in these areas is required. Mathematical topics covered include derivation of relevant PDEs from first principles; reduction of PDEs to ODEs under steady state, quasi-steady state, and traveling wave assumptions; solution techniques for PDEs and concepts of spatial stability and instability.  These concepts will be introduced within the setting of classical and current problems in biology and the biomedical sciences such as cell motion, transport of biological substances, and biological pattern formation.  Above all, this course aims to enhance the interdisciplinary training of advanced undergraduate and graduate students from mathematics and other disciplines by introducing fundamental properties of partial differential equations in the context of interesting biological phenomena. Grades will be based on the completion of a research project and weekly (or biweekly) homework assignments, computer lab assignments, and in class presentations.