Chapter 3.9, Problem 96E

Logistic growth Scientists often use the logistic growth function $P(t)=\frac{P_{0}K}{P_0+(K-P_0)e^{-r_{0}t}}$to model population growth, where P₀ is the initial population at time t = 0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model.

96. Population crash The logistic model can be used for situations in which the initial population P₀ is above the carrying capacity K. For example, consider a deer population of 1500 on an island where a fire has reduced the carrying capacity to 1000 deer.

1. a. Assuming a base growth rate of r₀ = 0.1 and an initial population of P(0) = 1500, write a logistic growth function for the deer population and graph it. Based on the graph, what happens to the deer population in the long run?
2. b. How fast (in deer per year) is the population declining immediately after the fire at t = 0?
3. c. How long does it take the deer population to decline to 1200 deer?