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
4/11 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
Projects/Presentations
Project 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)
- Fourier transform in diffusion equation and music (http://www.mathreference.com/la-xf-four,intro.html)
- Fisher, RA 1937 The wave of advance of advantageous genes. Annals
of Eugenics, 7:355-369 (paper)
- Derive animal aggreration
model (paper of Turchin: Population
consequences of aggregative movement. Journal of Animal Ecology 58,
(1989), 75-100.)
- 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).)
- Skellam, JG, 1951 Random dispersal in theoretical populations,
Biometrika (paper)
- Eigenfunctions of Laplacian
for balls in 2-d and 3-d
- 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
- 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)
- 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)
- 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:
- Derive and solve Black-Scholes PDE
in finance. (P. Dunlap)
- Tumour modeling ([B] Chapter 8) (M. Zuk)
- Chemotaxis ([B] 5.3 and 7.6 and others)
- Derive and analyze FitzHugh-Nagumo
equation ([B] 6.4 and others) . (P. Lucey)
- Traveling wave in epidemic models ([B]
Chapter 3 and 5.7 and others)
- Traveling wave in periodic environment
- Reaction-diffusion 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) (D.
Bigelow)
- Regular and irregular patterns in semiarid vegetation (paper by Klausmeier,
Science,
Vol. 284, 1826--1828, 1999) (D.
LaMontagne)
- Diversity of vegetation patterns and desertification (paper by von
Hardenberg,
et.al. Phys. Rev. Let, 198101, 2001) (T.
Little)
- 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)
- Bifurcation and periodc solution in reaction-diffusion systems of
predator-prey interaction
- Autocatalytic
chemical reactions (paper
by
Rovinsky et al Phys Rev A, Vol 46, 6315--6322, 1992; paper by Horvath, et.al, 1997) (R. Carpenter)
- Integro-differential 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 reaction-diffusion models of neural
conduction. (L. Osborne)
- Wave type solutions for Fisher equation in higher dimension. (F. Hirata)
- 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
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, 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). 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, 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).
-
Modeling Differential
Equations in Biology By Clifford
Henry Taubes, Prentice
Hall (2000).
- 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).
- 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): 80-87, 1988.
