Slope Fields

Part 2: Slope Fields for Natural Growth

We now apply the slope field concept to the natural growth equation,

where k is a positive constant.

Suppose, for sake of illustration, that P(t) is the number of fruit flies in a laboratory colony at time t, measured in days from some starting time at which the size of the colony is known. (This is similar to the scenario in the Limited Population Growth module, but we are not assuming any limit on the size of the population.) Suppose further that observation of the colony shows that k = 0.1. That is, the population is growing "continuously" at a proportional rate of ten percent per day.

1. Redefine the right-hand-side function to match this scenario, and construct a slope field for the differential equation.
2. What geometric feature of the slope field reflects the fact that the slope depends only on P, not on t?
3. Add solution curves to your plot for three different starting populations.
4. Write down formulas for the three solutions you just generated. Replace the solution curves generated "automatically" by your helper application with curves drawn from your formulas. Is the picture the same?

Last modified: October 7, 1997