Math 336: Differential equations in biology

Fall 03

  1.  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.

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

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

7-2-03:

8-18-03:

8-31-03:
9-7-03:
9-17-03:
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

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

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