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Slope Fields

Part 4: Slope Fields for Limited Population Growth

Now we study the slope field for the limited population growth model,

dP/dt = c P (M - P).

Recall that our model for fruit fly growth had c = 0.000098 (with time measured in days) and M = 1000.

  1. Redefine the right-hand-side function, and draw the slope field for this model.

  2. What geometric feature of the slope field reflects the fact that the slope depends only on P, not on t?

  3. Describe geometric features of the slope field that reflect symbolic (algebraic) features of the slope function.

  4. In the Limited Population Growth module, we constructed an approximate solution of the differential equation, starting with an initial population of 111 flies. Overlay this approximate solution on the slope field. Does it look as though it fits?

  5. Add to your picture the "automatic" solution through (0,111) generated by your helper application. Describe any deviations you see from the approximate solution.

  6. What does the picture suggest would happen if there were more than 1000 flies in this environment at time 0? If you're not sure, generate a solution starting from, say, 1500 flies. How reasonable is the model in this case?
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Last modified: October 7, 1997