Math 490-01
Partial Differential Equations and Mathematical Biology
Spring 2009

Instructor: Professor Junping Shi

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

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

Complete Work of Charles Darwin Online

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

1/22 Introduction Lecture 1
1/27 Review of calculus, 1-D conservation law, advection (1.2)  Lecture 2 {discussion problems for 1/29: [L] 1.2 (1,5,8,10)}

2/3
diffusion equation, random walk, mixing problem, heat transfer (1.3,1.4) Lecture 3
2/5   derivation of reaction-diffusion 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:
self-organization in cellular and developmental biology; Christiaan Kroesen: bacteria Colonies)
3/5 Presentation 1 (Will Jordan-Cooley: 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
Age-structure 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 R-D model, Matt Peppe:Oscillatory reactions conducted in a grid of cells
4/30 Presentation 2 Will Jordan-Cooley: Oyster Larval Behavior in the Vertical Water Column, Kate Levy: Age-structured 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 "psedo-code" 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:

1. 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)
2. Fisher, RA 1937 The wave of advance of advantageous genes. Annals of Eugenics, 7:355-369 (paper)
3. Skellam, JG, 1951 Random dispersal in theoretical populations, Biometrika (paper)
4. Turing, AM, 1953 The Chemical Basis of Morphogenesis, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences (paper)
5. Kierstead L and Slobodkin LB, 1953, The size of water masses containing plankton blooms. J. Mar. Res. 37, pp. 1–7. (paper)
6. Anderson RM and  May RM, 1979, Population biology of infectious diseases: Part I, Nature 280, 361 - 367 (paper);  Part II,  Nature 280, 455--461 (paper)
More recent interests:
1. Fourier transform in diffusion equation and music (http://www.mathreference.com/la-xf-four,intro.html)
2. Derive exact solution of a population model with density-dependent migration and Allee effect, predator-prey model ( 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; Petrovskii, Sergei; Malchow, Horst; Li, Bai-Lian, An exact solution of a diffusive predator-prey system. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), no. 2056, 1029--1053. paper)
3. CIMA reaction (Lengyel, István; Epstein, Irving R., Diffusion-induced instability in chemically reacting systems: steady-state multiplicity, oscillation, and chaos. Chaos 1 (1991), no. 1, 69--76. (paper); Lengyel, I. & Epstein, I. R. (1991) Science 251, 650-652. (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)
4. Bacteria Colonies (Ben-Jacob, E., et.al., 1994, Nature 368, 46; AL Lin, et.al. Biophysical Journal, 2004)
5. Regular and irregular patterns in semiarid vegetation (paper by Klausmeier, Science, Vol. 284, 1826--1828, 1999)
6. Diversity of vegetation patterns and desertification (paper by von Hardenberg, et.al. Phys. Rev. Let, 198101, 2001)
7. Factors that make an infectious disease outbreak controllable. Fraser C; Riley S; Anderson RM; Ferguson NM., PNAS, 101:6146-6151 (2004). (paper)
8. A recent survey paper relevent to material in this course: Baker, R. E.; Gaffney, E. A.; Maini, P. K. Partial differential equations for self-organization in cellular and developmental biology. Nonlinearity 21 (2008), no. 11, R251--R290. (paper)

Project 2:

1. Derive and solve Black-Scholes PDE in finance.
2. Tumour modeling ([B] Chapter 8)
3. Chemotaxis ([B] 5.3 and 7.6 and others)
4. Derive and analyze FitzHugh-Nagumo equation ([B] 6.4 and others) .
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
9. 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)
10. Bifurcation and periodc solution in reaction-diffusion systems of predator-prey interaction
11. Autocatalytic chemical reactions (paper by Rovinsky  et al Phys Rev A, Vol 46, 6315--6322, 1992; paper by Horvath, et.al, 1997)
12. Integro-differential equation with logistic growth from population growth
13. Nonlocal logistic equation (paper by Schnerb,  Phys. Rev. E Vol 69, 061917 (2004), paper by Fuentes, et al Phys. Rev. Lett. 91, 158104 (2003))
14. Synchronization in reaction-diffusion models of neural conduction.
15. Wave type solutions for Fisher equation in higher dimension.
16. Numerical methods of reaction-diffusion equations and systems
17. Turing Pattern: (a) Fish pattern A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus Shigeru Kondo & Rihito Asai, Nature 376, 765-768 (31 August 1995)
18. 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.
19. 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, 183--217.
20. 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
21. Numerical simulation for R-D systems: Garvie, Marcus R. Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB. Bull. Math. Biol. 69 (2007), no. 3, 931--956. (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, Bai-Lian, Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44 (2002), no. 3, 311--370
22. Twinkling eyes paper:  (1) Lingfa Yang and Irving R. Epstein, Oscillatory Turing Patterns in Reaction-Diffusion 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 Reaction-Diffusion Model with Interacting Turing Modes. Phys. Rev. Lett. 88, 208303 (2002) Lingfa Yang's website
23. Another R-D Matlab package: Reaction Diffusion Toolbox

LaTeX information

A LaTeX sample file

Maple and Matlab Programs

3-d graphing: Demonstrate Maple commands for 3-d graphing
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

Use PDE solver in Maple
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

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

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

Pattern formation

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.