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(MATH365)2007final.pdf
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Final Exam
MATH 365: Mathematical Biology
J. R. Chasnov
22 May2007
Answer ALL questions Full mark: 60; each question carries 10 marks. Time allowed C 3 hours
Directions CThisisa closedbook examination.Youmaywriteonthe frontandbackofthe exampapers.
Student Name:
Student Number:
Question No. (mark) Marks
Total
Considertwo countries: one rich, onepoor. Excluding immigration, thepoor country hasper capitapopu-lation growth rate g and the rich country hasper capitapopulation decay rate d. Model immigration from thepoor countryto the rich countryby aper capita immigration rate i. Assume that the poor countrys population continues to grow despite immigration.
(a)(4 pts) Determine di.erential equations for thepopulation sizes N and M of thepoor and rich countries.
(b)(4 pts) With initialpopulation sizes N0 and M0, solve for N(t)andM(t). It mayhelp to recall that the solution to By +ay = g(t)is givenby
t
.at as
y(t)= ey(0)+ eg(s)ds .
0
(c)(2 pts) Determine the asymptotic long-timepopulation ratio M/N.
C1 C
Consider an age-structuredpopulation model with females and males modeled separately. Let ui,n and vi,nbe the number of females and males in age class i at census n, respectively. Furthermore, de.ne si and ti tobe the fraction of females and males surviving from age class i.1to i, and mi the number of o.spring from a female in age class i. Assume that a fraction of births are females and1 . are males. For simplicity, assume only three age classes.
(a)(5 pts) Determine the evolution equation for {ui,n+1}i=1,2,3 in terms of thepopulation structure at census
n.
(b) (5 pts) Determine the evolution equation for {vi,n+1}i=1,2,3 in termsofthepopulation structureat census
n.
C2 C Consider a population of replicating molecules. There is a unique molecule M that reproduces with rate R, while all other molecules, denoted by S, reproduce at a slower rate r. Suppose molecule M can only reproduce an exact copy of itself with probability q, and that all molecules die with rate d. Denote the number of molecules of type M and S by their names.
(a) (4 pts) Determine di.erential equations for M and S.
(b)(2pts)Supposethatthe deathrate adjustsinthelongrunsuchthatthetotalpopulationsize M +S is constant. Determine d in terms of M and S that is required tokeep thepopulation size constant.
(c) (4 pts) With the value of d determined above, .nd the minimum value of q for M to survive foreverby deriving the condition on q such that M =0 is an unstable equilibrium.
C3 C Considerapopulation growth model with constant death rate d, a birth rate b= b(N)that depends on the population size N, and a carrying capacity K.
(a)
(5 pts) Find a simple model for b(N)of the form c1/(c2 +c3N)such that (i) the birth rate is , with >d, when thepopulation size is small; (ii) the birth rate equals the death rate when thepopulation size equals the carrying capacity, and; (iii) the birth rate goes to zero a