## Sustainable Forest Management

sustainable management of resources is a very desirable feature of land and sea management. It is desirable for resources to be harvested as fast as nature replenishes them. Suppose a forest contains 1200 tree, and 100 saplings are planted every year.If 10 of the forest is cut down every year, what will be the number of trees in the forest be in the long term?Suppose initially there are 1200 trees in the forest. Let the number of trees at the start of each spring be

\[y_n\]

then the number in year \[n+1\]

be \[y_{n+1}\]

, then \[y_{n+1}=100+0.9y_n\]

- 100 saplings are planted ND 10% of trees are harvested so 90% are left.\[y_0=1200\]

\[y_1=100+0.9 y_0=100+0.9 \times 1200=1180\]

\[y_2=100+0.9 y_1=100+0.9 \times 1180=1162\]

\[y_3=100+0.9 y_2=100+0.9 \times 1162=1145.8\]

and so on. If the number of trees is to reach a limit

\[l\]

then \[y_{n+1} \rightarrow y_n \rightarrow l\]

as \[n \rightarrow \infty\]

.\[l=100+0.9l \rightarrow l-0.9l=100 \rightarrow 0.1l=100 \rightarrow l=\frac{100}{0.1}=1000\]

.In the long term the forest will contain 1000 trees and 100 trees will be harvested every year.