Question

a population grows according to the logistic growth model, with growth parameter ( r = 0.9 ). starting with an initial population given by ( p_0 = 0.3 ), answer parts (a) through (c)
(a) find ( p_1 )
( p_1 = 0.189 ) (round to three decimal places as needed.)
(b) find ( p_2 )
( p_2 = 0.1380 ) (round to four decimal places as needed.)
(c) determine what percent of the habitats carrying capacity is taken up by the third generation.
the percent of the carrying capacity taken up by the third generation is ( square % ).
(round to three decimal places as needed.)

Explanation:

Step1: Recall the logistic growth formula

The logistic growth formula is \(p_{n + 1}=rp_{n}(1 - p_{n})\). Given \(r = 0.9\) and \(p_{0}=0.3\), we first found \(p_{1}=0.9\times0.3\times(1 - 0.3)=0.189\) and \(p_{2}=0.9\times0.189\times(1 - 0.189)=0.1380\). Now for \(p_{3}\), we use the formula \(p_{3}=rp_{2}(1 - p_{2})\).

Step2: Calculate \(p_{3}\)

Substitute \(r = 0.9\) and \(p_{2}=0.1380\) into the formula:

$$
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$$

Answer:

\(10.7\)