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Syllabus

Homework assignments:
Note that late HW is not accepted!

due 1/30

HW 2, due 3/4: Do problems #3.9.4, #3.9.16. do both problems at the end of the notes on games and replicator dynamics(last version Feb 16, 3pm). Do the problem at the end of the notes on the genetic nework.

HW 3, due 4/13: The first problem is here (correction made on 4/6). Do both problems at the end of the notes on the diffusion equation. Do problem #4.5.5 from our text (for part (d), use the online Vector Field Analyzer).

Weekly HW (not collected):

1st day of class Wed 1/7

Wk 2 (ends Fr 1/16): Redo the least squares analysis in the simpler case that y=ax, and apply this to Gause's data set on Paramecium aurelia (see p11). Compare the slope you obtain (i.e. the value of a) to the value k given in the text. Read the notes below on cobwebbing as preparation for next week's class.

Wk 3 (ends Fr 1/23): Read the sections on the Beverton-Holt model and the Ricker model (p. 26-29, caution: a square is missing in the denominator of f'(x) on p.27). #2.4.6(slightly challenging), #2.4.10, #2.4.11 (Mon 1/19 MLK Day).

Wk 4 (ends Fr 1/30): no HW (HW I due on Fr 1/30) .

Wk 5 (ends Fr 2/6): #3.9.5, #3.9.18, read section 3.3.4 on non-dimensionalization (Ex I on Fr 2/6; review on Wed 2/4).

Wk 6 (ends Fr 2/13): Complete the case study on p. 3 of the notes on games and replicator dynamics(last version Feb 16, 3pm).

Wk 7 (ends Fr 2/20): Review section 3.4.1 in the text and complete the missing steps in the notes on the genetic nework. Check out this game.

Wk 8 (ends Fr 2/27): Explore the online Vector Field Analyzer (written by Matthias Kawski from Arizona State University) to study the genetic network, and the chemostat model discussed in class. Do the HW problems mentioned by Professor Ellermeyer in class. His lecture notes are here. (2 guest lectures about the chemostat on Mon 2/23 and Wed 2/25 by professor Sean Ellermeyer from Kennesaw State University)

Wk 9 (ends Fr 3/6): Read section 4.2. Start 3rd monthly HW assignment. (HW II due on Wed 3/4) .

Wk 10 (ends Fr 3/13): Spring Break.

Wk 11 (ends Fr 3/20): Let c>0. Consider the problem u_t+cu_x=0 with $u(x,0)=u_0(x)$. This is a transport problem where the species (with density u(x,t) at position x in R and time t>0) is transported with velocity c, perhaps by being suspended in a flow that moves with that velocity. Solve this problem using the method of characteristics. Repeat this if instead the equation is u_t+cu_x=-d(x) where d(x) is the death rate at position x. (Ex II on Fr 3/20; review on Wed 3/18).

Wk 12 (ends Fr 3/27): Do the problems in notes1 and notes2. Do problem #4.5.3.

Wk 13 (ends Fr 4/3): Do problem # 4.5.4. Read the online material regarding the cell cycle (see Announcements section below).

Wk 14 (ends Fr 4/10): no HW (Two guest lectures on Wed 4/8 and Fr 4/10 by professor Sergei Pilyugin from UF's Mathematics Department. Please prepare his lectures by reading the online material regarding the cell cycle which is mentioned in the Announcements section below.).

Wk 15 (ends Fr 4/17): (1 guest lectures on Mon 4/13 by professor Ben Bolker from UF's Department of Zoology. Student class presentations on Wed 4/15: Jessica+Erica and Nick+Alex and Fr 4/17:Cassidy+Karina and Lindsay+Mandi) (HW III due on Mon 4/13) .

Wk 16 (ends Fr 4/24): (Ex III on Wed 4/22; review on Mon 4/20).

Running list of papers for class presentations (for pairs of grad students only):

Experiments and discrete time population model of the flour beetle, known as the LPA model (Lindsay and Mandi on Fr 4/17).
The repeated prisoner's dilemma game, and the evolution of cooperation is here (Cassidy and Karina on Fr 4/17).
Prove that an ESS of the underlying game is an asymptotically stable fixed point of the replicator equation (see me for more details in case you want to do this project).
Construct the phase portrait for the rock-scissors-paper when the various payoffs are not equal (see me for more details in case you want to do this project).
Present the model in section 10.2.1 of our text. It explains the chemotactic paradox, and offers a model for its resolution. This project requires a bit of background: 1. Background on chemotaxis is presented in project 14 of chapter 9 (p.249). 2. Background on modeling of chemical reactions using the law of mass action is explained in section 3.3.1 (Alex and Nick on Wed 4/15)
Present the delayed chemostat model mentioned by professor Ellermeyer during his guest lecture. The model can be found here.
Present the model from this paper. It concerns tumor formation caused by acidic gradients and this process is modeled as a traveling wave solution of a system of 3 reaction diffusion equations (one for normal cells, one for tumor cells and one for the acid). (Jessica and Erica on Wed 4/15)

Announcements:

For notes on matrices and systems, click here.
For notes on the least squares method, click here.
Gause's short, online book The Struggle for Existence is here. This is ref [63] from our text and it contains the data set featured in chapter 2 to illustrate the least squares method.
For notes on cobwebbing, click here.
For errata of our textbook, go here.
For notes on the Jury conditions, click here.
For notes on the juvenile-adult population model, click here .
For notes on games and the replicator equation, click here (last version Feb 16, 3pm).
John Maynard Smith and Price introduced the hawk-dove game in this paper.
For notes on the genetic network, click here.
For notes on the chemostat (by S. Ellermeyer), click here.
Go here for Matthias Kawski's online Vector Field Analyzer, a useful tool to plot vector fields and solution curves of planar systems.
For notes on the fundamental solution of the diffusion equation, click here.
For notes on the diffusion equation in a compact interval, click here.
For notes on the cell cycle (by Sergei S. Pilyugin), click here.
For notes on stochastic models (by Ben Bolker), click here.
If you have time, try to attend some of the lectures of the session Modeling in Biomathematics on Tuesday March 3rd during the SIAM Student Conference.
If you get bored during Spring Break, you should attend some of the lectures of the Special Session in Biomathematics on Tuesday/Wednesday March 10/11 at the Ulam Centennial Conference. Titles and abstracts of the talks are listed here. Details of the schedule will be posted on the conference website soon.
Solutions to exam 1 are here.
One of our guest lecturers is professor Pilyugin who will talk about the rescaling method. Understanding this lecture requires some basic knowledge of the cell cycle. A good introduction is here. For a very simple cartoon movie go here, and a slightly more elaborate animation including audio is here.
I'd like to try something out. Mathematica offers what they call their Player, so students can interactively work on Mathematica files (called notebooks) made by other people -like their teacher- without actually having to purchase a Mathematica license. I've never tried this, and would like to see how well it works. You will first need to download Mathematica Player here, and then you should be able to work on a very short notebook I wrote relating to the critical domain size topic discussed in class. That notebook is here.
Since this Mathematica Player seems to work, here's another one related to the example of pattern formation discussed in class: click here. The 1952 paper by Alan Turing that heavily influenced much of the area of pattern formation is here.
If you're interested in participating in a math bio summer program at FSU, see here for details, then contact me.