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.
[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.
Solution
Mathematical Solution
If yt=c y0y0+c−y0e−c k t, then the three constants c,y0, and k can be determined from the three equations
y1=1300,y3=1870,y4=2070
These equations are
Figure 5.6.8(a) Logistic curve from data
c y0y0+c−y0e−c k
=1300
c y0y0+c−y0e−3 c k
=1870
c y0y0+c−y0e−4 c k
=2070
with (numeric) solution c≐2451,k≐0.000214,y0≐983. Hence, the desired logistic curve is
y≐2.408022604106982.5017593+1468.407444ⅇ−0.5236436096 t
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.
Maple Solution
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.
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
cy0y0+c−y0ⅇ−kc=1300,cy0y0+c−y0ⅇ−3kc=1870,cy0y0+c−y0ⅇ−4kc=2070
→solve
c=2450.909203,k=0.0002136527983,y0=982.5017593
Expression palette: Evaluation template Evaluate yt at the parameter values.
Context Panel: Evaluate and Display Inline
ytx=a|f(x) = 2.408022604106982.5017593+1468.407444ⅇ−0.5236436096t
The graph in Figure 5.6.8(a) can be obtained with the Plot Builder, invoked from the Context Panel.
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