Chapter 7.5, Problem 46MS

Perplexing population predictions. Verhulst model describes a relation between the population density one year $\left(P_n\right)$ and the density the next year $\left(P_{n+1}\right).$

Here’s an example: $P_{n+1}=P_n+0.3(1-P_n).$ Thus, if $P_1=0.2,$ then $P_2=0.2+0.3(1-0.2)=0.2+0.3(0.8)=0.2+0.24=\mathrm{0.44.}$

In other words, from Year 1 to Year 2, the model predicts that the population density of this community went from 0.2 to 0.44. Using the formula and the prediction found for Year 2, what does the model predict for Year 3? Using the answer you just found, what does the model predict for Year 4?