Chapter 6.1, Problem 52E

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function

$N^{\prime }(t)=r+A\sin \frac{2\pi \text{t}}{P}$

where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t = 0.

1. a. Suppose P = 10, A = 20, and r = 0. If the initial population is N(0) = 10, does the population ever become extinct? Explain.
2. b. Suppose P = 10, A = 20, and r = 0. If the initial population is N(0) = 100, does the population ever become extinct? Explain.
3. c. Suppose P = 10, A = 50, and r = 5. If the initial population is N(0) = 10, does the population ever become extinct? Explain.
4. d. Suppose P = 10, A = 50, and r = −5. Find the initial population N(0) needed to ensure that the population never becomes extinct.