Chapter 5: Applications of Integration
Section 5.6: Differential Equations
Example 5.6.8
A species undergoes logistic growth, governed by the formula developed in Example 5.6.7. Observation yields the following three data points.
Determine the carrying capacity , the initial population , and the rate constant , if it is known that .
Solution
Mathematical Solution
If , then the three constants , and can be determined from the three equations
These equations are
Figure 5.6.8(a) Logistic curve from data
with (numeric) solution . Hence, the desired logistic curve is
In Figure 5.6.8(a) this solution is graphed in black; the asymptote is graphed in red. The astute reader will note that this problem required no calculus at all. It is strictly an algebraic problem of fitting a curve with three parameters to three pieces of data.
Maple Solution
Write from Example 5.6.7. Be sure to use the exponential "e".
Context Panel: Assign Function
Write the equations , etc. Press the Enter key.
Context Panel: Solve≻Solve Numerically from point (See Figure 5.6.8(b) for initial points.)
Figure 5.6.8(b) Initial points for numeric solution
Expression palette: Evaluation template Evaluate at the parameter values.
Context Panel: Evaluate and Display Inline
=
The graph in Figure 5.6.8(a) can be obtained with the Plot Builder, invoked from the Context Panel.
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