Chapter 5: Applications of Integration
Section 5.6: Differential Equations
A species undergoes logistic growth, governed by the formula developed in Example 5.6.7. Observation yields the following three data points.
[Time in yearsPopulation Size113003187042070]
Determine the carrying capacity c, the initial population y0, and the rate constant k, if it is known that k>0.
If yt=c y0y0+c−y0e−c k t, then the three constants c,y0, and k can be determined from the three equations
These equations are
Figure 5.6.8(a) Logistic curve from data
c y0y0+c−y0e−c k
c y0y0+c−y0e−3 c k
c y0y0+c−y0e−4 c k
with (numeric) solution c≐2451,k≐0.000214,y0≐983. Hence, the desired logistic curve is
In Figure 5.6.8(a) this solution is graphed in black; the asymptote c≐2451 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.
Write yt=… from Example 5.6.7.
Be sure to use the exponential "e".
Context Panel: Assign Function
yt=c y0y0+c−y0ⅇ−c k t→assign as functiony
Write the equations y1=…, etc.
Press the Enter key.
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 yt at the parameter values.
Context Panel: Evaluate and Display Inline
ytx=a|f(x) = 2.408022604⁢106982.5017593+1468.407444⁢ⅇ−0.5236436096⁢t
The graph in Figure 5.6.8(a) can be obtained with the Plot Builder, invoked from the Context Panel.
<< Previous Example Section 5.6
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)