=========================preview======================
(MATH365)2009_s_midterm.pdf
Back to MATH365 Login to download
======================================================
Midterm Exam
MATH 365: Mathematical Biology
J. R. Chasnov
13 April 2007
Answer ALL questions Full mark: 40; each question carries 10 marks. Time allowed C 60 minutes

Directions CThisisa closedbook examination.Youmaywriteonthe frontandbackofthe exampapers.
Student Name:
Student Number:


Total


Consider the following di.erential equation obtained from a genetics model: dp = sp .h+(1. 3h)p. (1. 2h)p 2..
dn
(a)(4pts) Findall .xedpointsof p.

(b)(2 pts)If0 <= s,h,p <=1, which .xedpoints arevalid solutions?
(c)(4pts) Determinethe stabilityofthe .xedpoints which arevalid solutions.
C1 C Consider a .sh tank with .sh dying with constant rate d. No .sh areborn. Assume N0 initial .sh.

(a)
(5 pts) Derive the di.erential equation model for pN (t), de.ned as the probability that the .sh tank population is of size N at time t.

(b)
(5 pts) Solve for pN0 (t)andpN0.1(t). You may use the following formula for the solution of dy/dt+ay =


g(t), with y(0) = y0: t .at as

y(t)= ey(0)+ eg(s)ds .
0
C2 C




Construct an age-structured model for the HKUST studentbody. First, assume threeyear classes,years 1-3,whereallstudentsmustleavetheUniversityafter3years. Furthermore, assumethatthe probabilityof remaining in the University fromyear class i. 1 toyear class i is givenby si, and that eachyear, exactly N students enter the .rstyear class. Assume no transfer students.
(a)(5pts)Inthe steady-state, determinethenumberof studentsinyear classes 1-3.
Now,modelthepost-2012 transition froma threetoa fouryear University. Assumein academicyear 2012 only, N students are simultaneously admitted to each ofyear class0 andyear class 1.
(b) (5 pts) Determine the number of students in the four year classes in 2012 and subsequent years. At whichyear does the steady-state distribution .rst attain?
C3 C Consider a generalized Lotka-Volterra predator-prey model with two predators and one prey. Let U(t)be the number of prey and V1(t), V2(t)be the numbers of predators at time t. Use parameters , 1, 2, e1, e2, 1 and 2 to generalize the Lotka-Volterra equations. You may assume all these parameters are di.erent and no special equalities hold.




(a)(4pts)Writedownthe generalized Lotka-Volterra equations.

(b)(2pts) Determineallthe .xedpointsofthe generalized equations.
(c) (4 pts) Determine the condition on the parameters for whichpredator two (with number V2)is expected to go extinct.
C4 C