James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 11.2 Series 707
laboratory Project
CAS
Logistic Sequences
A sequence that arises in ecology as a model for population growth is defined by the logistic
difference equation
p n11 − kp ns1 2 p nd
where p n measures the size of the population of the nth generation of a single species. To keep
the numbers manageable, p n is a fraction of the maximal size of the population, so 0 < p n < 1.
Notice that the form of this equation is similar to the logistic differential equation in Section 9.4.
The discrete model—with sequences instead of continuous functions—is preferable for modeling
insect populations, where mating and death occur in a periodic fashion.
An ecologist is interested in predicting the size of the population as time goes on, and asks
these questions: Will it stabilize at a limiting value? Will it change in a cyclical fashion? Or will
it exhibit random behavior?
Write a program to compute the first n terms of this sequence starting with an initial population
p 0, where 0 , p 0 , 1. Use this program to do the following.
1. Calculate 20 or 30 terms of the sequence for p 0 − 1 2 and for two values of k such that
1 , k , 3. Graph each sequence. Do the sequences appear to converge? Repeat for a different
value of p 0 between 0 and 1. Does the limit depend on the choice of p 0? Does it depend on
the choice of k?
2. Calculate terms of the sequence for a value of k between 3 and 3.4 and plot them. What do
you notice about the behavior of the terms?
3. Experiment with values of k between 3.4 and 3.5. What happens to the terms?
4. For values of k between 3.6 and 4, compute and plot at least 100 terms and comment on the
behavior of the sequence. What happens if you change p 0 by 0.001? This type of behavior is
called chaotic and is exhibited by insect populations under certain conditions.
The current record for computing a
decimal approximation for was
obtained by Shigeru Kondo and Alexander
Yee in 2011 and contains more
than 10 trillion decimal places.
What do we mean when we express a number as an infinite decimal? For instance, what
does it mean to write
− 3.14159 26535 89793 23846 26433 83279 50288 . . .
The convention behind our decimal notation is that any number can be written as an infinite
sum. Here it means that
− 3 1 1 10 1 4
10 2 1 1
10 3 1 5
10 4 1 9
10 5 1 2
10 6 1 6
10 7 1 5
10 8 1 ∙ ∙ ∙
where the three dots s∙ ∙ ∙d indicate that the sum continues forever, and the more terms we
add, the closer we get to the actual value of .
`
In general, if we try to add the terms of an infinite sequence ha n j n−1 we get an expression
of the form
1
a 1 1 a 2 1 a 3 1 ∙ ∙ ∙ 1 a n 1 ∙ ∙ ∙
which is called an infinite series (or just a series) and is denoted, for short, by the symbol
ò a n or o a n
n−1
Does it make sense to talk about the sum of infinitely many terms?
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