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:

9/4  Solve simple equations
       Graphing the equations: direction fields and softwares
Reading: [S] Notes 2 [E-K] page 165-171
Web-software: (click DFIELD 2001.2) (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
(4) the problem at
     (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
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: (do Euler's method) (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 (, especially chapter IV
Web-software: (do linear regression) (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: (about stability of equilibrium pts) (about bifurcation and harvesting) (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) (pictures of Budworm) (pictures of Budworm)
(an article from US Department of Agriculture Forest Service)
                      (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)
Web software:
ttp:// (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)