Chapter 9.1, Problem 46E

Harvesting problems Consider the harvesting problem in Example 6.

If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

Example 6 A Harvesting Model

A simple model of o harvested resource (for example, timber or fish) assumes a competitor between the harvesting and the natural growth of the resource. This process may be described by the differential equation

where p(t) is the amount (or population) of the resource at time t ≥ 0, r > 0 is the natural growth rate of the resource, and H > 0 is the constant harvesting rate. An initial condition p(0) = p₀ is also specified to create an initial value problem. Notice that the rate of change p′(t) has a positive contribution from the natural growth rate one a negative contribution from the harvesting term.

1. a. For given constants p₀, r, and H. verify that the function

   $p\left(t\right)=\left(p_0-\frac{H}{r}\right)e^{rt}+\frac{H}{r}$

   is a solution of the initial value problem

2. b. Let p₀ = 1000 and r = 0.1. Graph the solutions for the harvesting rates H = 50, 90, 130, and 170. Describe and interpret the four curves.
3. c. What value of H gives a constant value of p, for all t ≥ 0?