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Question 7.2.6: Solving a Logistic Growth Problem Given a maximum sustainabl...

Solving a Logistic Growth Problem

Given a maximum sustainable population of M = 1000 (this could be measured in millions or tons, etc.) and growth rate k = 0.007, find an expression for the population at any time t, given an initial population of y(0) = 350 and assuming logistic growth.

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From the solution (2.6) of the logistic equation, we have k M=7 and

y=\frac{1000 A e^{7 t}}{1+A e^{7 t}} .

From the initial condition, we have

350=y(0)=\frac{1000 A}{1+A} .

Solving for A, we obtain A = 35\65 , which gives us the solution of the IVP

y=\frac{35,000 e^{7 t}}{65+35 e^{7 t}} \text {. }

This solution is plotted in Figure 7.10.

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