According to United Press International, honeybee keepers lose 30% of their colo

Question

According to United Press International, honeybee keepers lose 30% of their colonies each winter due to an unknown illness. They also lose 5% of what remains because of other issues. Carlisle Honey reports that each spring, 4690 bees are born (no matter what happened over the winter). Suppose at the start of summer 2010, Carlisle Honey had 6000 bees.

a. Write a recursive equation describing the number of bees owned by Carlisle Honey at the start of year n (where n = 0 corresponds to 2010).

b. Find the number of bees at the start of summer 2020.

c. Over time, the number of bees will tend to approach what value?

d. Now choose a different number of bees to start the problem (your choice, but tell us what it is), and suppose the company still loses 30% each winter due to an unknown illness, loses 5% because of other reasons, and gets 4690 new bees each spring. Over time, the number of bees will approach what value?

e. Based upon your answers to (c) and (d), make a conjecture about the effect of the initial number of bees on the number of bees over time. Prove your conjecture.

Explanation / Answer

As we lose 30%+5% = 35%, we keep 65%. We also get 4690 new bees each year.

Thus, xn+1 = .65xn + 4690

b) We start with 6000. The chart below shows 10 iterations from 2010 to 2020. By 2020, we reach 13300.38

13300.38

c) If we do another 30 iterations, we see that it converges to 13400

d) I started with 20000. In 10 years, it reached 13488.85

If we do another 30 iterations, we see convergence to the same value, 13499.

e) No matter where we start, we reach 13400.

We can prove that, if there is a limit, it will be 13400, by showing that, for

x = .65x + 4690

.35x = 4690

x = 4690/.35

x = 13400

To prove that we do converge to this value, consider xn = 13400 + epsilon.

Then, xn+1 = .65(13400 + epsilon) + 4690

xn+1 = 13400 + .65 epsilon.

Then, xn+k = 13400 + (.65)^k epsilon.

Thus, it is clear that, as lim k -> inf (.65)^k = 0

lim k-> inf xn+k = 13400 + 0*epsilon = 13400

Thus, for any value of epsilon, or any initial value, the limiting value is 13400.

Year N 2010 6000 2011 8590 2012 10273.5 2013 11367.78 2014 12079.05 2015 12541.38 2016 12841.9 2017 13037.24 2018 13164.2 2019 13246.73 2020

13300.38

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