$(1$ point$)$ Assuming P$0,$P$0,$ suppose that a population develops according to the logistic equation

dPdt$=0\mathrm{.}02$P$0\mathrm{.}0002$P$2$dPdt$=0\mathrm{.}02$P$0\mathrm{.}0002$P$2$

wherettis measured in weeks. Answer the following questions.

a$.$What is the carrying capacity?

Hint: The carrying capacity isKKin the logistic equationdPdt$=$rP$(1$PK$)$dPdt$=$rP$(1$PK$).$

Carrying Capacity:K$=$K$=$

b$.$ What is the value of rr$?$

Hint: When PP is small, the growth is approximately exponential since dPdt$=$rP$+$O$($P$2).$dPdt$=$rP$+$O$($P$2).$

Answer: r$=$r$=$ inverse weeks.

c$.$ For what values of PP is the population increasing?

Answer $($in interval notation$)$:

d$.$ For what values of PP is the population decreasing?

Answer $($in interval notation$)$: