Math 49001
Partial Differential Equations and Mathematical
Biology
Spring 2009
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
Darwin Day : (Feb 12, 2009)
Official Website
W&M Dept
Biology Darwin day activities
Science
Magazine Feb 6, 2009 issue
National
Geographic Feb 2009 issue
Darwin: American
Museum of Natural History
PBS special
about Darwin
Complete Work of Charles
Darwin Online
Lecture Notes
Chapter 1: Derivation of
reactiondiffusion
equations (18 pages)
Chapter 2: Diffusion equation on
a bounded
domain (22 pages)
Chapter 3: Diffusion with point
source
Chapter 4: Nonlinear scalar reactiondiffusion
equations
Chapter 5: Reaction diffusion systems
Lecture slides
1/22 Introduction Lecture 1
1/27 Review of calculus, 1D conservation law, advection (1.2) Lecture 2 {discussion problems for 1/29: [L] 1.2
(1,5,8,10)}
1/29 Advection equation, characteristic, (1.2)
2/3 diffusion equation, random walk, mixing problem, heat
transfer (1.3,1.4) Lecture 3
2/5 derivation of
reactiondiffusion
models (1.4,1.7) {discussion problems for 2/10:[L] 1.4 (3,7,9), 1.7 (4)}
2/10 Nondimensionalization, More discussion of
diffusion models, boundary value
problem
2/12 (Special) Darwin day: Fisher equation
2/17 Fourier series solution of diffusion equation, Application
of Fourier series solutions, 2D and 3D
2/19 Fundamental solution of diffusion equation
2/24 Traveling wave solution
2/26 Presentation 1 (Kate Levy: epidemic models)
3/3 Presentation 1 (Matt Peppe:
selforganization in cellular and
developmental biology; Christiaan Kroesen: bacteria Colonies)
3/5 Presentation 1 (Will
JordanCooley: Oyster population model)
3/17 Numerical scheme for diffusion equation (explicit finite
difference)
3/19 Numerical scheme for diffusion equation (implicit finite
difference) Maple PDE solver
3/24 Fisher equation: stability, bifurcation
3/26 Chemotaxis powerpoint
slides
3/31 Agestructure model
4/2 Turing instability and Turing bifurcation
(slides: mathematics,
animal
pattern)
4/7 Hopf bifurcation (slides)
4/9 CIMA reaction model
analysis
4/14 Numerical bifurcation
4/16 No class (working on numerical code, paper)
4/21 No class (working on numerical
code, paper)
4/23 No class (working
on numerical code, paper)
4/28 Presentation 2 Christiaan
Kroesen: Numeircal simulation of RD model, Matt Peppe:Oscillatory
reactions conducted in a grid of cells
4/30 Presentation 2 Will
JordanCooley: Oyster Larval Behavior in the Vertical Water Column,
Kate Levy: Agestructured epidemic models
Homework assignment
([S] is Shi's lecture notes, [L] is Logan's book)
Homework 1 (due 2/5, Thursday): [L] 1.2 (3,7), 1.3(4,5), [S]
Chapter 1
(3,4)
Homework 2 (due 2/12, Thursday): [L] 1.4(4,10) [S] Chapter 1
(8,9,10,16)
Homework 3 (due 2/24, Tuesday): [L] 4.1(1,3) [S] Chapter 2(1,4,6,7,9)
Howework 4 (due 3/26, Thursday): [L] 2.8(1a,2) 4.8(2,3) [S]
Chapter 4 (2) You do not need to write the actual Maple code, but
write the "psedocode" which reflects the Robin boundary conditions.
Homework 5 (due 4/10 Thursday): [L]
5.1(1), 5.3(3,4)
Projects/Presentations
Project 1:
Classical papers:
 Diffusion equation from Brownian motion (original
paper by Robert Brown in 1827, paper (and English
translation) or this
version by Albert Einstein in 1905, a lecture
notes from MIT open course, see Lecture 1 there)
 Fisher, RA 1937 The wave of advance of advantageous genes. Annals
of Eugenics, 7:355369 (paper)
 Skellam, JG, 1951 Random dispersal in theoretical populations,
Biometrika (paper)
 Turing, AM, 1953 The Chemical Basis of Morphogenesis,
Philosophical Transactions of the Royal Society of London. Series B,
Biological Sciences (paper)
 Kierstead L and Slobodkin LB, 1953, The size of water masses
containing plankton blooms. J. Mar. Res. 37,
pp. 1–7. (paper)
 Anderson RM and May RM, 1979, Population biology of
infectious diseases: Part I, Nature
280, 361  367 (paper);
Part II, Nature 280, 455461 (paper)
More recent interests:
 Fourier transform in diffusion equation and music (http://www.mathreference.com/laxffour,intro.html)
 Derive exact solution of a
population model with densitydependent migration and Allee effect,
predatorprey model ( Petrovskii, Sergei; Li, BaiLian, An
exactly solvable model of
population dynamics with densitydependent migrations and the Allee
effect. Math. Biosci. 186 (2003), no. 1,
7991. paper; Petrovskii, Sergei;
Malchow, Horst; Li, BaiLian, An exact solution of
a diffusive predatorprey system. Proc. R. Soc. Lond. Ser.
A Math. Phys. Eng. Sci. 461
(2005), no. 2056, 10291053. paper)
 CIMA reaction (Lengyel, István; Epstein, Irving R., Diffusioninduced instability in chemically reacting
systems: steadystate multiplicity, oscillation, and chaos. Chaos
1 (1991), no. 1, 6976. (paper);
Lengyel, I. & Epstein, I. R. (1991) Science 251, 650652. (SWEM
library or ask Prof Shi for a copy), I Lengyel, IR Epstein 
Proceedings of the National Academy of Sciences, 1992, paper)
(research project, a paper
by Prof Shi)
 Bacteria Colonies (BenJacob,
E., et.al., 1994, Nature 368, 46; AL Lin, et.al.
Biophysical Journal, 2004)
 Regular and irregular patterns in semiarid vegetation (paper
by Klausmeier,
Science,
Vol. 284, 18261828, 1999)
 Diversity of vegetation patterns and desertification (paper
by von
Hardenberg,
et.al. Phys. Rev. Let, 198101, 2001)
 Factors that make an infectious disease outbreak controllable.
Fraser C; Riley S; Anderson RM; Ferguson NM., PNAS, 101:61466151
(2004). (paper)
 A recent survey paper relevent to material in this course: Baker,
R. E.; Gaffney, E. A.; Maini, P. K. Partial differential equations for
selforganization in cellular and developmental biology.
Nonlinearity 21 (2008), no. 11, R251R290. (paper)
 Your own suggestion
Project 2:
 Derive and solve BlackScholes PDE
in finance.
 Tumour modeling ([B] Chapter 8)
 Chemotaxis ([B] 5.3 and 7.6 and others)
 Derive and analyze FitzHughNagumo
equation ([B] 6.4 and others) .
 Traveling wave in epidemic models ([B]
Chapter 3 and 5.7 and others)
 Traveling wave in periodic environment
 Reactiondiffusion in heterogeneous
environment (different diffusion rate corresponding to quality of
habitat)
 Invasion and the evolution of speed in
Australian cane toads (Nature Feb
16, 2006 article)
 A reactionadvectiondiffusion equation from chaotic chemical
mixing (paper by Neufeld, et al,
Chaos, Vol
12, 426438, 2002, paper
by Menon, et al, Phys. Rev. E. Vol 71, 066201, 2005, and preprint by
Prof. Shi)
 Bifurcation and periodc solution in reactiondiffusion systems of
predatorprey interaction
 Autocatalytic
chemical reactions (paper
by
Rovinsky et al Phys Rev A, Vol 46, 63156322, 1992; paper by Horvath, et.al, 1997)
 Integrodifferential equation with logistic growth from
population growth
 Nonlocal logistic equation (paper
by Schnerb, Phys. Rev. E Vol 69, 061917 (2004), paper by Fuentes, et al Phys. Rev.
Lett. 91, 158104 (2003))
 Synchronization in reactiondiffusion models of neural
conduction.
 Wave type solutions for Fisher equation in higher dimension.
 Numerical methods of reactiondiffusion equations and systems
 Turing Pattern: (a) Fish
pattern A
reaction–diffusion wave on the skin of the marine angelfish Pomacanthus
Shigeru Kondo & Rihito Asai, Nature 376,
765768
(31 August 1995)
 Turing Pattern: (b) Bifurcation of
patterns in chemotaxis model: Maini,
P.K., Myerscough, M.R., Winters, K.H. and Murray, J.D., 1991.
Bifurcating spatially heterogeneous solutions in a chemotaxis model for
biological pattern generation. Bull Math Biol 53,
pp. 701–719.
 More on chemotaxis models: T.
Hillen and K.J. Painter. (2009). A user's guide to PDE models for
chemotaxis. Journal of Mathematical Biology. 58, 183217.
 M. J. Tindall,
S. L. Porter, P. K. Maini, G. Gaglia and
J. P. Armitage, Overview of Mathematical Approaches Used to
Model Bacterial Chemotaxis I: The Single Cell; II: Bulletin of Mathematical Biology, 70, 2008.
paper
1, paper
2
 Numerical simulation for RD systems: Garvie, Marcus R. Finitedifference schemes for reactiondiffusion
equations modeling predatorprey interactions in MATLAB. Bull.
Math. Biol. 69 (2007), no. 3, 931956. (Matlab code
can be found from Dr. Garvie's website) A long survey article with a
lot of simulation: Medvinsky,
Alexander B; Petrovskii, Sergei; Tikhonova, Irene; Malchow, Horst; Li,
BaiLian, Spatiotemporal complexity of plankton and
fish dynamics. SIAM Rev. 44 (2002), no. 3, 311370
 Twinkling eyes paper: (1) Lingfa Yang and Irving R.
Epstein, Oscillatory
Turing Patterns in ReactionDiffusion Systems with Two Coupled Layers.
Phys. Rev. Lett. 90, 178303 (2003); (2) Lingfa Yang, Milos Dolnik,
Anatol M. Zhabotinsky, and Irving R. Epstein, Spatial
Resonances and Superposition Patterns in a ReactionDiffusion Model
with Interacting Turing Modes. Phys. Rev. Lett. 88, 208303 (2002) Lingfa Yang's website
 Another RD Matlab package: Reaction
Diffusion Toolbox
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
3d
graphing: Demonstrate Maple commands
for 3d graphing
Characterisic
for advection equation
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 2d:
spatial patterns of eigenfunctions of Laplacian on a square
Diffusion
with a point source:
simulation of the fundamental solutions in 1d and 2d
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(1u^m), where D>0, k>0,
and m>0.
Selfsimilar solution of diffusion
equation:
calculate the selfsimilar solutions of linear and nonlinear diffusion
equations
Gypsymoth problem: calculate the invasion
speed of gypsy moth in east United States
Difference equations for gene evolution prog1 prog2
Use
PDE solver in Maple
Numerical simulations for reactiondiffusion 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 advectionreactiondiffusion
equation
Matlab programs simulating RD equations and systems:
Programs
by Marcus Garvie (Florida State University)
Programs
by Julijana Gjorgjieva (Harvey
Mudd College)
simple program by J. Shi
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
GiererMeinhardtXmorphia
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
http://hopf.chem.brandeis.edu/yanglingfa/pattern/index.html
Animal Pictures by Tony Northrup
Reference Books in Mathematical Biology
 Mathematical Biology, Vol. 1: An Introduction. By James
Dickson
Murray, SpringerVerlag, New York, (2002).
 Mathematical Biology, Vol. 2: Spatial Models and
Biomedical
Applications.
By James Dickson Murray, SpringerVerlag, New York, (2002).
 Mathematical Models in Biology. By Leah
EdelsteinKeshet,
McGrawHill,
Boston, (1988). SIAM, (2005).
 Elements of Mathematical Ecology. By Mark Kot,
Cambridge
University
Press, (2001).
 Diffusion and Ecological Problems: Modern Perspectives.
By
Akira
Okubo, Simon A. Levin, SpringerVerlag, New York, (2001).
 Quantitative Analysis of Movement: Measuring and Modeling
Population
Redistribution in Animals and Plants. By Peter Turchin, Sinauer
Associates,
Inc, (1998).

Modeling Differential
Equations in Biology By Clifford
Henry Taubes, Prentice
Hall (2000).
 Growth and Diffusion Phenomena: Mathematical Frameworks
and
Applications.
By Robert Banks, SpringerVerlag, New York, (1993).
 Spatial Ecology via ReactionDiffusion 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,
SpringerVerlag,
London, (2003).
 Complex Population Dynamics :
A Theoretical/Empirical Synthesis,by Peter Turchin, Princeton
University Press, (2003).

Mathematics in Population
Biology, by
Horst R. Thieme, Princeton University Press, (2003).
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): 8087, 1988.