**Instructor: Professor Junping
Shi**

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**

**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)**

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

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**

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).

** 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.

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

INTRODUCTION
TO TRANSPORT IN FLUID SYSTEMS

MIT, Cambridge, MA, USA

Pattern
Formation

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

**A.** **Metapopulation dynamics and habitat fragmentation**

http://www.esd.ornl.gov/programs/SERDP/EcoModels/metapop.html

**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.