Math 490-01
Partial Differential Equations and Mathematical Biology
Spring 2006

Instructor: Professor Junping Shi

Syllabus   Course Schedule

Mathematical Biology Links  Links by Eduardo Sontag  Links by Kuang Yang  Links by  Josef Hofbauer

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

Biomath graduate programs

Lecture Notes

Chapter 1: Derivation of reaction-diffusion equations (18 pages)
Chapter 2: Diffusion equation on a bounded domain (22 pages)
Chapter 3: Diffusion with point source
Chapter 4: Nonlinear scalar reaction-diffusion equations
Chapter 5: Reaction diffusion systems 

Lecture slides

1/19 Introduction Lecture 1
1/24 Review of calculus, review of basic population models Lecture 2
1/26 Nondimensionalization, derivation of reaction-diffusion models
Lecture 3
2/9   Brownian motion and diffusion (Paul Dunlap)
2/16 Fisher's 1937 paper (Ryan Carpenter)
2/21 Nonlinear diffusion equation (Daniel Grady)
2/23 Skellam's 1953 paper (Patrick Lucey)
2/28 An approximate solution for Fisher equation in high dimension (Fumie Hirata)
2/28 Exact traveling wave solution for diffusion equation with Allee effect  (Tina Little)
3/21 Evolution and dispersal of genes, Fisher equation Lecture 4 (powerpoint)
3/30 Stability in dynamical systems and Turing instability Lecture 5
4/4   Turing instability and bifurcation Lecture 6   Turing Patterns in animal coats
4/6   Chemotaxis and slime mold model
Lecture 7
A reaction-advection-diffusion equation from chaotic chemical mixing Lecture 8
4/13  Periodic solutions of systems Lecture 9

Homework assignment
([S] is Shi's lecture notes, [B] is Britton's book)

Homework 1 (due 1/31, Tuesday): [B] page 15 (1.6, 1.8), [S] Chapter 1 (1,2,3,4,10)
Homework 2 (due 2/8, Wednesday): [B] page 153 (5.5,5.7)  [S] Chapter 1 (8,9,13) Chapter 2 (2,4,5)
Homework 3 (due 2/15, Wednesday): [S] Chapter 2 (7(you can use result of 6), 12,14)
Howework 4 (due 2/22, Wednesday): [B] page 158 (5.9, 5.11)  and following problems:
        (1) Find an exact traveling wave of equation u_t=D u_{xx}+ku(1-u^m), where D>0, k>0,  and  m>0.
             (Hint: use Maple, and the form of the solution is u(x,t)=v(x-ct), v(z)=1/(1+exp(az))^{2/m}
        (2) (optional, I don't know the answer or if an answer exists)
Find an exact traveling wave of equation u_t=D u_{xx}+ku^m(1-u), where D>0, k>0,  and  m>1.
              (Hint: use Maple, and maybe also the same form as above, but I don't have solution for this one)
Homework 5 (due 3/29 Wednesday)
            (1) [S] Chapter 4 (5) You do not need to write the actual Maple code, but write the "psedo-code" which reflects the Robin boundary conditions.
            (2) [S] Chapter 4 (11) (Hint: use Maple to solve for u_2)

Test: test 1


Project 1:

  1. Diffusion equation from Brownian motion (original paper by Robert Brown in 1827, paper (and English translation) by Albert Einstein in 1905, a lecture notes from MIT open course, see Lecture 1 there)
  2. Fourier transform in diffusion equation and music (,intro.html)
  3. Fisher, RA 1937 The wave of advance of advantageous genes. Annals of Eugenics, 7:355-369 (paper)
  4. Derive animal aggreration model (paper of Turchin: Population consequences of aggregative movement. Journal of Animal Ecology 58, (1989), 75-100.)
  5. Solution of porous media equation (page 339-343, Elements of Mathematical Ecology. By Mark Kot, Cambridge University Press, (2001);  page 402-405, Mathematical Biology, Vol. 1: An Introduction. By James Dickson Murray, Springer-Verlag, New York, (2002).)
  6. Skellam, JG, 1951 Random dispersal in theoretical populations, Biometrika (paper)
  7. Eigenfunctions of Laplacian for balls in 2-d and 3-d
  8. Derive analytic solution of diffusive logistic equation with point source ( S. Puri, K. R. Elder and R. C. Desai, Approximate Asymptotic Solutions to the d-Dimensional Fisher Equation, Phys. Lett. A, 142, 357 (1989).) paper from Physics library
  9. Derive analytic solution of  traveling wave solution of  diffusive logistic equation (Fisher equation with density-dependent diffusion: special solutions, S Harris 2004 J. Phys. A 37 6267-6268 paper)
  10. Derive exact solution of a population model with density-dependent migration and Allee effect ( Petrovskii, Sergei; Li, Bai-Lian,  An exactly solvable model of population dynamics with density-dependent migrations and the Allee effect. Math. Biosci. 186 (2003), no. 1, 79--91. paper)
  11. Janet Raloff, Fishing for answers: deep trawls leave destruction in their wake---but for how long? Science News, 150 (1996) 268--271. (Section 16 of Taubes, Modeling Differential Equations in Biology, 2000, page 246-257)

Project 2:

  1. Derive and solve Black-Scholes PDE in finance. (P. Dunlap)
  2. Tumour modeling ([B] Chapter 8) (M. Zuk)
  3. Chemotaxis ([B] 5.3 and 7.6 and others)
  4. Derive and analyze FitzHugh-Nagumo equation ([B] 6.4 and others) . (P. Lucey)
  5. Traveling wave in epidemic models ([B] Chapter 3 and 5.7 and others)
  6. Traveling wave in periodic environment
  7. Reaction-diffusion in heterogeneous environment (different diffusion rate corresponding to quality of habitat)
  8. Invasion and the evolution of speed in Australian cane toads (Nature Feb 16, 2006 article) (D. Bigelow)
  9. Regular and irregular patterns in semiarid vegetation (paper by Klausmeier, Science, Vol. 284, 1826--1828, 1999) (D. LaMontagne)
  10. Diversity of vegetation patterns and desertification (paper by von Hardenberg, Phys. Rev. Let, 198101, 2001) (T. Little)
  11. A reaction-advection-diffusion equation from chaotic chemical mixing (paper by Neufeld, et al, Chaos, Vol 12, 426-438, 2002, paper by Menon, et al, Phys. Rev. E. Vol 71, 066201, 2005, and preprint by Prof. Shi)
  12. Bifurcation and periodc solution in reaction-diffusion systems of predator-prey interaction
  13. Autocatalytic chemical reactions (paper by Rovinsky  et al Phys Rev A, Vol 46, 6315--6322, 1992; paper by Horvath,, 1997) (R. Carpenter)
  14. Integro-differential equation with logistic growth from population growth
  15. Nonlocal logistic equation (paper by Schnerb,  Phys. Rev. E Vol 69, 061917 (2004), paper by Fuentes, et al Phys. Rev. Lett. 91, 158104 (2003))
  16. Synchronization in reaction-diffusion models of neural conduction. (L. Osborne)
  17. Wave type solutions for Fisher equation in higher dimension. (F. Hirata)
  18. Numerical methods of reaction-diffusion equations and systems

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

the pdf printout of the LaTeX sample file

Quick tutorial for LaTeX

Maple and Matlab Programs

3-d graphing: Demonstrate Maple commands for 3-d graphing
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
Boundary conditions and smoothering effect Demonstrate effect of different boundary conditions, and smoothering effect of diffusion
Chemical problem: show how to solve the chemical mixing problem
Robin boundary condition: calculation of Robin boundary eigenvalues, and critical patch size
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
Traveling wave of Fisher equation: show an exact traveling wave solution of Fisher equation
Traveling wave of generalized Fisher equation:
calculate an exact traveling wave solution of generalized Fisher equation
u_t=D u_{xx}+ku(1-u^m), where D>0, k>0,  and  m>0.

Self-similar solution of diffusion equation: calculate the self-similar solutions of linear and nonlinear diffusion equations
Gypsy-moth problem: calculate the invasion speed of gypsy moth in east United States

Difference equations for gene evolution prog1 prog2

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
Use perturbation method to solve diffusive logistic equation
Turing Bifurcation curves and unstable modes
Pulse solutions in advection-reaction-diffusion equation

Matlab programs simulating R-D equations and systems:
Programs by Marcus Garvie (Florida State University)
Programs by
Julijana Gjorgjieva (Harvey Mudd College)

Biological Pattern Gallery

Brownian motion and random walk simulations:
Random walk simulation random walk in one dimension   A Video clip of random walk     

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 in Mathematical Biology

General Articles in Mathematical biology

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

Mathematics, Biology, and Physics: Interactions and Interdependence  Michael C. Mackey and Moisés Santillán, Notices of American Mathematical Society, Sept, 2005.

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

Mathematical Challenges from Genomics and Molecular Biology Richard M. Karp, Notices of American Mathematical Society, May, 2002.

Mathematical Challenges in Spatial Ecology Claudia Neuhauser, Notices of American Mathematical Society, Dec. 2001.

Linking Mind to Brain: The Mathematics of Biological Intelligence  Stephen Grossberg, Notices of American Mathematical Society, Dec. 2000.

We Got Rhythm: Dynamical Systems of the Nervous System Nancy Kopell, Notices of American Mathematical Society, Jan. 2000.

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.