points$)$

Assuming $P0,$ suppose that a population develops according to the logistic equation

$\frac{dP}{dt}=0\mathrm{.}05P-0\mathrm{.}0005P^2$

where $t$ is measured in weeks. Answer the following questions.

The carrying capacity is the limit $\underset{t}{lim}P(t)$ of the population size after a very long time. What is the carrying capacity?

Carrying Capacity $=$

When $P$ is very small its growth is approximately exponential $P(t)$~~$A{e}^{kt}$ for some constants A and $k$ Here $k$ represents the "exponential growth rate". In this problem what is the value of $k?$

$k=$

For what values of $P$ is the population increasing?

Answer $($in $$ notation$)$: $$

$4.$ For what values of $P$ is the population decreasing?

Answer $($in $$ notation$)$: $$