Math 336: Differential equations in biology

Fall 03

Instructor: Patrick De Leenheer DIMACS/CoRE Building room 413 email: leenheer@math.rutgers.edu (by far the preferred way for communication) phone: 732-445-4580 appointments: just send me email and mention best times for you to meet.
Textbook: Mathematical models in biology by L. Edelstein-Keshet, McGraw-Hill, 1988.
Where and when: Scott Hall 214, College Avenue Campus, Tuesday/Friday 9:50-11:10.

7-2-03:

Welcome to the webpage of this course! If you would like to learn more about mathematical models in biology then this could be the right place for you. The common property of the models is that they are described by differential equations. That explains why you should be familiar with these types of equations: prerequisites for the course are Calc 4 and Linear Algebra. If you are not quite sure about your expertise in this matter, there are some notes and problems + answers available on the following page. You should feel comfortable with this material (ask me for help if that is not the case!).The mentioned webpage has been set-up by professor Eduardo Sontag who taught the course in the fall of 2002. In broad terms I expect to follow his format, so if you wish you can check out his overheads to already get some idea of what we will discuss in class.The order of the topics could change, and certain subjects could be replaced by others, but the 'style' should be similar.
Important note regarding the textbook: It is out of print! However, the author gave permission to copy the material we will cover in class and I will provide it to you.
Some info about me: Although my background is in control theory, I have recently done some work on predator-prey models, virus models (from the within host-viewpoint; in particular HIV models) and chemostat models ('chemostat' is a fancy word for a biological reactor in which some organisms are competing for nutrient). I will be working at DIMACS on mathematical models in biology and I am supposed to organize a seminar. I'll keep you informed about our seminar guests during the semester. This way, you'll get the opportunity to attend talks of experts in mathematical biology from around the country as well as abroad. Also, DIMACS offers workshops on specific topics which could be of interest to our class.

8-18-03:

A copy of the textbook will be on reserve at the Alexander library.
If you like to check out some of my work, have a look at my old webpage.
The course grading will be calculated from bi-weekly (approximately) quizzes and homework, 2 tests, 1 final and class attendance/participation. Tentative weights of each of the above: 20%, 20%, 30%, 20% and 10%.
Some review papers on: epidemiology, HIV dynamics

8-31-03:

Transparancies of the course.

9-7-03:

The first quiz is on Friday 9-12. Study everything covered in class before 9-12 AND the notes modeling.pdf from the table below. Quiz problems will bear a strong resemblance with HW problems.
Important note: There will not be any make-ups for quizzes or exams!
I will bring the textbook to class. Feel free to browse through it during class time.

9-17-03:

Check out the program of the BIOMAPS/DIMACS seminar series at this webpage (Wednesdays, free lunch at 12:30pm; seminar: 1pm-2pm)

10-02-03:
The second quiz is on Friday 10-10. The material covered will be on linear ODE's and stability, compartmental models, nullclines. NOT included are: drug infusion, epidemics.
The format will be the same as that of the first quiz.
10-05-03:
For the slides on chemical reactions, taught by professor Sontag, go here.
10-08-03:
Date change for test1: October 21 (not October 28 as announced before)
10-17-03:
Practice problems to write down differential equations of chemical reactions. There will be a problem about this on the test of 10-21.
There will NOT be any problems on either quasi-steady state approximation or cell differentiation mechanisms on this test.
In summary: on the test is everything covered in class so far, except for the two topics just mentioned.
11-25-03:
The material for test 2 begins on p. 42 (Quasi-Steady State Approximation) and covers everything discussed in class up to p. 90. This includes
the transport equation and chemotaxis, but NOT diffusion.

Date	sections	topics	homework/suggested problems	notes
9/2	4.1-4.5	ODE's, growth	7 and even # in notes/11,12,13	modeling, A
9/5	4.1-4.5	# parameters		more on non-dimensionalization1,2,3
9/9	4.6-4.7	steady-states & linearization
9/12	4.8-4.10	linear ODE, stability
9/16	4.11	drug infusion; compartments	25a,b, 29/ 25c,d,e,f,28	B
9/19	5.2-5.4	phase plane		solving/plotting ODE's using MAPLE, java applet for phase plane/2 dim ODE's
9/23	5.7-5.8	linear phase planes
9/26	5.5,5.10	nullclines, chemostat phase plane	6,11/5,7	C
9/30	6.6	epidemics	30/29 Maple project -> extra credit	C
10/3	7.1-7.2	chemical kinetics
10/7	7.1-7.2, ctd 7.3-7.4	fast/slow time, Michelis-Menten sigmoidal responses	reading assignment: 7.3,7.4 (report -> extra credit)
10/10	7.5-7.6 7.7	bifurcations and switches conditions for switching
10/14	7.8	activator-inhibitor qualitatively	8,16,20/no suggested problems	D
10/17		review
10/21	test1	chap 4-chap 7, open book
10/24	8.3-8.4	limit cycles, Poincaré-Bendixon
10/28	8.3,8.4 ctd	cubic nullclines, relaxation
10/31	8.1-8.2 9.1	excitable sys; Hodgkin-Huxley multivariable calculus review	slides p 68 -> extra credit 4 and problems below/no suggested problems	E
11/4	9.2-9.3	densities, conservation equation
11/7	9.4 (+notes)	transport eqn, travelling waves
11/11	9.4 (+notes)	gradients, attraction and repulsion	slides p 86, 1a,b/no suggested problems	F
11/14	9.4-9.9+notes	diffusion equation
11/18	(ctd')	random walks, diffussion times
11/21	(ctd')	transit times; separation of variables
11/25	10.1-10.4	density-dep. diff., PDE systems
11/28	Thanksgiving
12/2	test 2
12/5	10.5-10.7	reaction-diffusion: waves
12/9	11.4-11.8	morphogenesis, pattern formation
12/10	Last day of classes
12/12	Reading Day
12/15	Final Exam	8-11am Scott Hall 214

A: Homework for chapter 4 is due 9/19.
B: Homework is due 9/26.
C: Homework is due 10/17.
D: Homework is due 10/31.
E: Homework is due 11/14.
F: Homework is due 11/26.

Problems for 10/31:
Consider the vector field F(x,y)= (f(x,y),g(x,y)) defined on the unit square [0,1]x[0,1].
For which of the following cases is F(x,y) not pointing outward on the boundary of the unit square
(so that the unit square is a trapping region and thus the Poincare-Bendixson theorem may be applied):
1. f(x,y)=2y-x, g(x,y)=x+3y
2. f(x,y)=y-xy, g(x,y)=x-xy
3. f(x,y)=1-x^2-y^2, g(x,y)=-x^2y

Math 336: Differential equations in biology

Fall 03

Instructor: Patrick De Leenheer DIMACS/CoRE Building room 413 email: leenheer@math.rutgers.edu (by far the preferred way for communication) phone: 732-445-4580 appointments: just send me email and mention best times for you to meet.

Textbook: Mathematical models in biology by L. Edelstein-Keshet, McGraw-Hill, 1988.

Where and when: Scott Hall 214, College Avenue Campus, Tuesday/Friday 9:50-11:10.

7-2-03:

8-18-03:

Practice problems on the diffusion equation + answers: problems, answer

Answers to homework problems on the transport equations (p86 in slides):answers