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Use a graphing calculator to graph the given examples of the various cases in Table 1 on page $379 .$ Logistic growth: $$ \begin{array}{l} N=\frac{1,000}{1+999 e^{-...

Question

Use a graphing calculator to graph the given examples of the various cases in Table 1 on page $379 .$ Logistic growth: $$ \begin{array}{l} N=\frac{1,000}{1+999 e^{-0.4 t}} \\ 0 \leq t \leq 40 \\ 0 \leq N \leq 1,000 \end{array} $$

Use a graphing calculator to graph the given examples of the various cases in Table 1 on page $379 .$ Logistic growth: $$ \begin{array}{l} N=\frac{1,000}{1+999 e^{-0.4 t}} \\ 0 \leq t \leq 40 \\ 0 \leq N \leq 1,000 \end{array} $$



Answers

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of differential equation
$ \frac
{dP}{dt} = c \ln (\frac {M}{P})P $
where $ c $ is a constant and $ M $ is the carrying capacity.
(a) Solve this differential equation.
(b) Compute lim $ _{t \to \infty} P(t). $
(c) Graph the Gompertz growth function for $ M = 1000, P_0 = 100, $ and $ c = 0.05, $
and compare it with the logistic function in Example 2. What are the similarities? What are differences?
(d) We know from Exercise 13 that the logistic function grows fastest when $ P = M/2. $ Use the Gompertz differential equation to show that the Gompertz function grows fastest $ P = M/e. $


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