MATH 147
Quiz Key #10
4/21/2016
(1) A strain of bacteria reproduces asexually every 10 minutes. That is, every
10 minutes every bacterial cell splits into two cells.
(a) Find the recursion (difference equation) for this population.
(b) Find the solution of the recursion in part (a) if there are initially 3 bacterial
cells.
(c) How long will it take until there are 96 bacteria?
Solution: (a) If time is measured in 10-minute intervals, then the recursion is
given by the following: xt+1 = 2xt .
(b) The equation is given by P = 3 · 2t , where t is measured in 10-minute
intervals.
(c) We solve the equation:
96 = 3 · 2t .
32 = 2t .
25 = 2 t .
t = 5.
Since t is measured in 10-minute intervals, it will take 50 minutes for there to
be 96 bacteria.
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(2) Consider the difference equation:
x t +1 =
xt
.
0.5 + xt
(a) Find all equilibria of this difference equation.
(b) Use the stability criterion to characterize the stability of each equilibrium.
Solution: (a) We solve the equation:
x=
x
.
0.5 + x
0.5x + x2 = x.
x2 − 0.5x = 0.
x ( x − 0.5) = 0.
x = 0, 0.5.
Therefore, the equilibria are given by x ∗ = 0, 0.5.
(b) Now, letting f ( x ) = 0.5x+ x , we need to find f 0 ( x ) to test stability.
f 0 (x) =
0.5 + x − x
(0.5 + x )
2
=
0.5
(0.5 + x )2
.
0.5
= 2 and f 0 (0.5) = 0.5
= 0.5. Therefore, | f 0 (0)| < 1.
Therefore, f 0 (0) = 0.25
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Hence, 0 is an unstable equilibrium. However, | f 0 (0.5)| < 1 and f 0 (0.5) > 0,
so 0.5 is a locally stable equilibrium approached without oscillations.
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