Math 490-01
Partial Differential Equations and Mathematical Biology
Spring 2004

Instructor: Professor Junping Shi

Syllabus   Course Schedule

Mathematical Biology Links

Mathematical Biology Journals in College of William and Mary library and network

Lecture Notes

Chapter 1: Derivation of reaction-diffusion equations (14 pages)
Chapter 2: Diffusion equation on a bounded domain (22 pages)
Chapter 3: Diffusion with point source (8 pages)
Chapter 4: Nonlinear scalar reaction-diffusion equations (38 pages)
Chapter 5: Reaction diffusion systems (9 pages)

Lecture slides

1/21 (Introduction)
        Handouts: Preface of [Murray], Chapter 1 of [Stewart]
1/23 (Review of multi-variable calculus)
4/14 Turing patterns in animal coats (powerpointpdf)
4/16 Chemotactic systems (powerpoint)
4/19 Spatial epidemic models (pdf)
4/21 Overview of other mathematical biology (pdf) Biomath graduate programs

Homework assignment
(problems from lecture notes)

Homework 1 (due 2/6, Friday): Chapter 1 (1,2,3,4,5,6ac,8,9) answer(pdf, problem 4-9)  Maple program for problem 1-3
Homework 2 (due 2/16, Monday): Chapter 2 (1,2,4,5,6,7,8,9,11)
Homework 3 (due 2/25, Wednesday): Chapter 2 (12,13), Chapter 3(1,2)
Homwwork 4 (due 3/26, Friday): Chapter 4(1,4,7,9,10a,12,13)
Homwwork 5 (due 4/12, Monday): Chapter 5(2,3) (use LaTeX to type the answer of 2)

Midterm Tests (take-home)

Test 1 (due 3/5 Friday)
Test 2 (cancelled, the percentage of Project 2 is now 30%)

Projects Material

Project 1: Paper on diffusion mechanisms

March 1:
Chris and Andrew: project 3(Probability) and Brian and Divid Weissenberger: project 4 (Constant flux)
March 3:
Zayd and Young: project 1(Robin boundary problem) and Michael and David Pluim: project 6 (agrregation and random walk with attraction)
March 5:
Vicki and Eileen: project 7 (CO2 in leaves) and Lena and Jacky: project 5 (nonlinear diffusion and animal dispersal)

Project 2: Analysis and Applications of Reaction-Diffusion Equations

April 23 (Friday) Bifurcation diagrams of equilibrium solutions of nonlinear-diffusion models
Young He Lee, Lena Sherbakov and Jacky Taber

April 26 (Monday) Animal aggregation and nonlinear diffusion models
Michael Deal, Brian Van Hise and David Pluim

April 28 (Wednesday) Numerical computations of reaction-diffusion systems
Zayd Khoury, Chris Leonetti and Andrew Todd

April 30 (Friday) Two-patch diffusive fishery harvesting model
Victoria Dyer, Eileen Tschetter and David Weissenberger

LaTeX information

MikTeX (TeX system for Windows)    WinEdt (TeX Editor for Windows)

LaTeX in W&M Math network   TeX Users Group (TUG) (information for all level of TeX users)

Inventor of TeX: Donald E. Knuth

A LaTeX sample file

Maple Programs

3-d graphing: Demonstrate Maple commands for 3-d graphing
First look of diffusion: simulation of a special solution of diffusion equation
Homework 1: answer of Homework 1 (prob 1-3), and solve differential equations
Fourier series of a solution of diffusion equation: Demonstrate the smothering effect of diffusion
Differential equations: Demonstrate how to solve initial value problem, boundary value problem of ODE, and PDE
Chemical problem: show how to solve the chemical mixing problem
Diffusive Malthus model:  show the effect of different growth rate on the fate of population which lives in a bounded region
Patterns of eigenfunction in 2-d: spatial patterns of eigenfunctions of Laplacian on a square
Diffusion with a point source: simulation of the fundamental solutions in 1-d and 2-d
Diffusion with a continuous source: simulation of solution of diffusion equation on a half line with fixed value at x=0
Fuel spill problem: solve the fuel spill problem in Section 3.3
Muskrat dispersal: use data fitting function to match the muskrat population growth
Homework 3
Numerical simulations for reaction-diffusion equations in an interval:
     Diffusion equation with Dirichlet boundary condition
     Diffusive logistic equation with Dirichlet boundary condition
     An unstable iteration (Diffusive logistic equation with Dirichlet boundary condition)
     Diffusive cubic(Allee effect) equation with Dirichlet boundary condition
     Diffusive logistic equation with Neumann boundary condition
     Diffusive logistic equation with Neumann boundary condition (show a "traveling wave")
     Diffusive Allen-Cahn equation with Neumann boundary condition (show super-slow motion)
Algebraic perturbation equation
Use perturbation method to solve diffusive logistic equation
Bifurcation diagram of equilibrium solutions (Using time-mapping)
     Diffusive logistic equation
     Diffusive cubic(strong Allee effect) equation
     Diffusive cubic(weak Allee effect) equation
Numerical simulations for reaction-diffusion systems in an interval:
     Gray-Scott model (Neumann boundary condition)
     Predator-prey model (Neumann boundary condition)

Biological Pattern Gallery

Brownian motion and random walk simulations:
Random walk simulation   Random walk advective-diffusion animation   A Video clip of random walk     Brownian motion of gas molecules

Pattern formation

Alan Turing Home Page

Gierer-MeinhardtXmorphia Fur coat pattern formation of exotic vertebrates   Gray Scott Model of Reaction Diffusion

Patterns and Spatiotemporal Chaos - Java Simulations    Nonlinear Kinetics Group in University of Leeds

Modelling Pigmentation Patterns

Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis
K. J. Painter, P. K. Maini, and H. G. Othmer

Videos of cellular slime mold aggregations

Animal Pictures by Tony Northrup

Reference Books

  • Mathematical Biology, Vol. 1: An Introduction. By James Dickson Murray, Springer-Verlag,  New York, (2002).
  • Mathematical Biology, Vol. 2: Spatial Models and Biomedical Applications. By James Dickson Murray, Springer-Verlag, New York, (2002).
  • Mathematical Models in Biology. By Leah Edelstein-Keshet, McGraw-Hill, Boston, (1988).
  • Elements of Mathematical Ecology. By Mark Kot, Cambridge University Press, (2001).
  • Diffusion and Ecological Problems: Modern Perspectives. By Akira Okubo, Simon A. Levin, Springer-Verlag, New York, (2001).
  • Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. By Peter Turchin, Sinauer Associates, Inc, (1998).
  • Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. By Robert Banks, Springer-Verlag, New York, (1993).
  • Spatial Ecology via Reaction-Diffusion Equations. By Stephen Cantrell, Christopher Cosner, Wiley, John & Sons, Inc., (2003).
  • Life's Other Secret: The New Mathematics of the Living World. By Ian Stewart, Wiley, John & Sons, Inc., (1999).
  • Essential Mathematical Biology, By Nicholas F. Britton, Springer-Verlag, London, (2003).
  • General Articles in Mathematical biology

    Modeling of Biological Systems, A Workshop at the National Science Foundation in 1996

    Why Is Mathematical Biology So Hard?  Michael C. Reed, Notices of American Mathematical Society, March, 2004.

    Uses and Abuses of Mathematics in Biology  Robert M. May, Science,  February 6, 2004.
    A webpage about Brahe, Kepler and Newton's story

    Getting Started in Mathematical Biology  Frank Hoppensteadt, Notices of American Mathematical Society,  Sept. 1995.

    Some Advice to Young Mathematical Biologists  Kenneth Lange, (from internet), date unknown.

    How the leopard gets its spots?  James Murray, Scientific American, 258(3): 80-87, 1988.

    Other related courses on web:

    VIGRE Minicourse The Mathematics Behind Biological Invasions Fred Adler, Department of Mathematics, Salt Lake City, Utah, USA

    Environmental Fluid Mechanics I: Mass Transfer and Diffusion  Environmental Fluid Mechanics II: Stratified Flow and Buoyant Mixing
    Institute for Hydromechanics under the Department of Civil Engineering, Univerisity of Karlsruhe, Germany

    Mathematics of Reaction-Diffusion Systems
    Prof. Renato Feres, Department of Mathematics, Washington University, St. Louis, Missouri, USA

    1. Dynamical Systems in Chemical Engineering and Biology
    2. Analytical Methods in Chemical Engineering
    3. Patterns and Interfaces in Dissipative Dynamics
    Prof. Len Pisman, Department of Chemical Engineering Technion, Israel Institute of Technology, Haifa, Israel

    Random Walks and Diffusion
    Prof. Martin Bazant, Department of Mathematics, MIT, Cambridge, MA, USA

    MIT, Cambridge, MA, USA

    Pattern Formation
    Dr. Joceline Lega, Department of Mathematics, University of Arizona, AZ, USA

    A. Metapopulation dynamics and habitat fragmentation

    D. Two-patch model

    Levin 1974, The American naturalist

    F. Biological Invasion

    Sanchirico, James N.; Wilen, James E.
    Dynamics of spatial exploitation: a metapopulation approach. (English. English summary)
    Natur. Resource Modeling 14 (2001), no. 3, 391--418.

    Roughgarden, Jonathan; Iwasa, Yoh
    Dynamics of a metapopulation with space-limited subpopulations.
    Theoret. Population Biol. 29 (1986), no. 2, 235--261.

    Holt, Robert D.
    Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution.
    Theoret. Population Biol. 28 (1985), no. 2, 181--208.

    Freedman, H. I.; Waltman, Paul Mathematical models of population interactions with dispersal.
    I. Stability of two habitats with and without a predator. SIAM J. Appl. Math. 32 (1977), no. 3, 631--648.