Math 410(510) Assignment

[E-K] is the textbook by Edelstein-keshet,
[S] is the notes written by Shi (available at CourseInfo:course documents)
[PBA] is the segment of textbook by Polking, Boggess and Arnold
(available at CourseInfo:course documents)
[BC] is a book by Brauer and Castillo-Chavez "Mathematical Models
       in population biology and epidemilology"

8/30 Concept of DE, ODE, initial value problem
        Malthus exponential model, Logistic model
Reading: [E-K] page 115-120, 212-215,  [PBA] page 123-130
              [S] Notes 1  page 1-2
Web reading:
http://www.sosmath.com/diffeq/modeling/modeling.html
http://www.sosmath.com/diffeq/first/application/population/population.html

9/4  Solve simple equations
       Graphing the equations: direction fields and softwares
Reading: [S] Notes 2 [E-K] page 165-171
Web-reading:
http://www.sosmath.com/diffeq/first/separable/separable.html
http://www.sosmath.com/diffeq/slope/slope1.html
Web-software:
http://math.rice.edu/~dfield/ (click DFIELD 2001.2)
http://www.aw.com/ide/Media/MainMenu/Part1.html (Try 2 and 3)

Homework 1: (due 9/6)
[E-K] page 152: (1 or 2), page 257: 2(c), [PBA] page 134: 1,3,9,11,
Other problems:
(1) Solve P'=sin(t) (P+1), P(0)=1
(2) Slove P'=-y^2, y(0)=0.5
(3) the problem at http://www.sosmath.com/diffeq/first/separable/example2/example.html
(4) the problem at http://www.sosmath.com/diffeq/slope/example1/example.html
     (note: answer is available online for (3) and (4))
(5) Use computer to generate the direction field of dP/dt=P(10-P)-sin(t), submit a printout
      of the graph (hint: choose appropriate window)
 

9/6: Phase line, Euler method
Reading: [S] Notes 2,  [E-K] page 212-215
Web-reading:
http://www.sosmath.com/diffeq/first/phaseline/phaseline.html
http://www.sosmath.com/diffeq/first/numerical/numerical.html
Optional Web-reading:
(1) Original paper by Malthus (in Course Document folder)
(2) An article about Velhust's logistic equation (in Course Document folder)
Web-software:
http://mss.math.vanderbilt.edu/~pscrooke/MSS/euler.html (do Euler's method)
http://math.rice.edu/~dfield/ (click DFIELD 2001.2)
 

9/11: Data fitting, and Allee effect
[PBA] page 131-134, 138-142
Optional Web-reading:
(1) Original book by Gause (http://www.ggause.com/Contgau.htm), especially chapter IV
Web-software:
http://www.btinternet.com/~endeavour/stats.html (do linear regression)
http://www.phys.unsw.edu.au/3rdyearlab/graphing/graph.html (another linear regression)

Homework 2: (due 9/13)
(1) [E-K] page 152 (5d,e,f)
(2) [E-K] page 257 (3) Find the phase lines, equilibrium solutions, and sketch a few solutions on direction fields.
(3) [PBA] page 135 (8)
(5) [PBA] page 136 (13) estimate k and N using the method introduced in class
(6) [PBA] page 137 (14) estimate k and N using the method introduced in class
(7) Compute by hand using Euler's method: y'=t+y, y(0)=1, step size=0.1, n=5. Check your answer using the internet software.

9/13: Nondimensionalization and harvesting
Readings: [S] Notes 1 and 3

9/18: Stability, bifurcation and harvesting
Readings: handout in class (page 28-32 of [BC])
Web-Reading:
http://www.sosmath.com/diffeq/first/phaseline/phaseline.html (about stability of equilibrium pts)
http://www.sosmath.com/diffeq/first/bifurcation/bifurcation.html (about bifurcation and harvesting)
http://math.bu.edu/DYSYS/ode-bif/ode-bif.html (stability, bifurcation and harvesting)

Homework 3: (due 9/20)

(1) [PBA] page 135 (10a,b)
(2) [PBA] page 135 (15)
(3) [PBA] page 138 (18,19,20)
(4) [E-K] page 258 (6)
(5) [S] Notes 3 (exercises 1,2,3)
(6) Solve the following logistic equation with harvesting:
      P'=4P(1-P/4)-3, P(0)=2.

9/20: Outbreak of spruce budworm
Reading: [PBA] page 158-161
Webreading: Paper on spruce budworm at courseinfo(a huge pdf file)
http://www.entomology.umn.edu/cues/dx/sk/spruce2.htm (pictures of Budworm)
http://www.ento.vt.edu/~sharov/PopEcol/lec13/chorist.html (pictures of Budworm)
http://www.na.fs.fed.us/spfo/pubs/fidls/sbw/budworm.htm
(an article from US Department of Agriculture Forest Service)
Websoftware: http://mss.math.vanderbilt.edu/~pscrooke/MSS/plotseveral.html
                      (maybe useful for you in other situations, it draws graphs of functions)

9/25: Computer Lab and Test 1 (take home)

No homework this week, Test 1 due 9/28.

9/27: Outbreak of spruce budworm

10/2: Comments on test problems. Predator-prey models.

10/4: Mathematics of systems: phase portrait, equilibrium points, nullclines,
10/9: Mathematics of systems:  linearizations, types of equilibrium points, periodic solutions, separatrices.
Reading: [E-K] page 171-193
Web reading: (if you have not taken Math 302)
http://www.sosmath.com/diffeq/system/linear/basicdef/basicdef.html
http://www.sosmath.com/diffeq/system/linear/graphical/graphical.html
http://www.sosmath.com/diffeq/system/linear/eigenvalue/eigenvalue.html
http://www.sosmath.com/diffeq/system/linear/qualin/qualin.html
Web software:
ttp://math.rice.edu/~dfield/ (click PPLANE 2001.2)

Homework 4: (due 10/11)

(1) [E-K] page 201 (5a,b,e) (hand drawing, also draw the arrows in the regions bounded by nullclines)
(2) [E-K] page 201 (6a,b,e)
(3) [E-K] page 201 (7a,b,e)
(4) [E-K] page 201 (9)

10/11: Analysis of predator-prey and competition models.

10/18: Chemotaxis model (1)

Homework 5: (due 10/25)

(2) [E-K] page 201 (6a,b,e)
(3) [E-K] page 201 (7a,b,e)
(4) [E-K] page 201 (9)