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Lectures: Old Engineering Hall 316 MWF 10:00-10:50

Instructor: Bard Ermentrout

Office Hours: TBA
Office: Thackeray 502
Phone: 624-8324
EMAIL: bard@pitt.edu
WWW: http://www.pitt.edu/~phase

• Written homework 40%
• Project 20%
• Two Exams 40%

The book is Mathematical Models in Biology by Leah Edelstein-Keshet, but I will be supplementing the material with lots of other things.

• Discrete time systems
  • Chapter 1.1-1.2, cell division, annual plants, fireflies, and economic forecasts
  • Systems of discrete time equations 1.3-1.4
  • Analysis of the above models and other models
  • Lecture 1
  • Lecture 2
  • HOMEWORK DUE 9/10
  • Some java scripts for cobwebbing:
    • Cobweb java 1
    • Cobweb java 2
    • Linear cobweb script
  • XPP file for cobwebbing the logistic map
  • Computer software
    • So much of modeling cannot be done analytically, so we usually must resort to the computer. I expect that there are many useful packages available for MatLab, Mathematica, etc. I will provide code and examples from my own package, XPP-AUT which is free, small, and stable. You can download it for Windows , Macs , or Linux The main website gives instructions for Macs.
    • No mater what you use, you should get used to using the computer since we will do a lot of numerical examples!
    • Try the XPP for maps tutorial
• Lecture 3 and HOMEWORK for Sep 17
• Host parasite model in XPP
• Lecture 4 and HOMEWORK for Sep 24
• Cicada model - play with k, the period, and nu, the predator death
• Differential equations models Chapters 4-6 in the book
• Lectures on mechanics and HOMEWORK for Oct 8
  • Page 1 from lect notes....
  • XPP file for double pendulum
  • XPP animation file for double pendulum
• Linear planar problems with XPP
• Nonlinear planar problems with XPP
• Predator-prey model with allee effect for prey
  • Run this problem in XPP or an equivalent program
  • The parameter d sets the death rate of the predator (y). Draw the nullclines for d=1. Integrate with initial conditions x(0)=1.125, y(0)=0.05.
  • Repeat for d=0.55
  • Repeat for d=0.45; what is different between the two
  • Try d=0.25.
• Lecture notes and HOMEWORK for Oct 22
• MIDTERM EXAM Oct 22
  • Know chapter 1-4 in the book
  • Solutions to linear discrete systems
  • Cobwebbing
  • Creating models
  • Finding equilibria and stability of discrete models
  • Making a model dimensionless
  • This will be an easy exam!
• Homework for Oct 29
  • page 258 problem 8
  • Do cases (b,d) in figure 6.6 that I did not do in class without looking at figure 6.7 :)
  • page 261, problem 17. PLEASE NOTE the equations are wrong and the correct equations are on page xxxvii in the errata of the book!! For part b, choose, r=1,k1=k2=1,alpha=0.25,beta=1 to do the phaseplane
  • page 261 prob 23.
• HOMEWORK FOR NOV 5TH
• Lectures from a week or so agao
• Lectures from this week
• HW due next week - just 2 problems!
• HOMEWORK DUE FRIDAY AFTER THANKSGIVING BREAK
  • Chapt 7:17,18,19,22;
  • Chapt 8: 6a-f,19 a,b.
  • Show that g(x,t)=1/(2 sqrt(Pi R t)) * exp(-x^2/(4 R t)) solves u_t = R u_xx . Set R=1 and plot this as a function of x on [-10,10] for t=0.25, t=1, t=10. For each values of t>0, the plot has an inflection point with respect to x, that is, g_xx(x,t)=0. Find the position of the inflection point where g_xx=0 as a function of time, t.
  • Let u_t(x,t)= a u(x,t) + R u_xx(x,t) on 0 < x < Pi.

    Suppose u_x(0,t)=u_x(Pi,t)=0.

    Show that u(x,t) = c_n(t) cos(n x) where n is an integer solves this equation and write down the differential equation for c_n(t). For what values of (a,n,R) do solutions c_n(t) decay to zero as t increases?

• Gierer-Meinhardt Pattern formation model How to use it is in the file. Try it to see pattern formation!
• FINAL EXAM!!
• Probabilistic models
• Choosing parameters

Additional Info

• Differential equations using the computer

  • linear 2D systems
  • nonlinear 2D systems

• Project

  A central part of the course will be a project in which you choose some sort of physical, social, or biological system and create a mathematical model of it to explain some aspect of the system. For example, in the past, some students have looked at chemical oscillation models, some have modeled simple mechanical toys, some have modeled vampires. By the middle of the term, you should be thinking about a project.

• Homework

  There will homework of several varieties. Much of it will be taken from the book but there will be a substantial amount of homework which you can do on the computer that will allow you to simulate different systems. I will describe these later on in the term, but they will likely involve your using the computer and your brains to solve some applied problems that arise in applications.

• Software

  XPPaut info I have not specified any particular software package to use for the course but I will introduce you to a piece of software that I wrote that allows you to solve and animate ODEs, PDEs, and all the kinds of equations you will encounter in the course. It is available for Mac, Windows, and Unix platforms and is free. Some of the problem may be easier with the use of software; I will introduce its use to you during the course of the semester. The software is available in some of the labs, but I can show you how to put it on your laptops.