Chapter 9.4, Problem 19E

Let’s modify the logistic differential equation of Example 1 as follows:

$\frac{dP}{dt}=0.08P\left(1-\frac{P}{1000}\right)-15$

(a) Suppose $P\left(t\right)$ represents a fish population at time t, where t is measured in weeks. Explain the meaning of the final term in the equation $\left(-15\right)$.

(b) Draw a direction field for this differential equation.

(c) What are the equilibrium solutions?

(d) Use the direction field to sketch several solution curves. Describe what happens to the fish population for various initial populations.

(e) Solve this differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial populations 200 and 300. Graph the solutions and compare with your sketches in part (d).