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  3. part a) biologists stocked a lake with 400 fish an

PART A) Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population...

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Advance Math

PART A)

Biologists stocked a lake with 400 fish and estimated thecarrying capacity (the maximal population for the fish of thatspecies in that lake) to be 5300. The number of fish doubled in thefirst year.

Assuming that the size of the fish population satisfies thelogistic equation

dPdt=kP(1?PK),

determine the constant k, and then solve the equation to find anexpression for the size of the population after t years.
k=_______________  
P(t)=______________

How long will it take for the population to increase to 2650 (halfof the carrying capacity)?
It will take ________________ years.

PART B)

Another model for a growth function for a limited population isgiven by the Gompertz function, which is a solution of thedifferential equation

dPdt=cln(KP)P

where c is a constant and K is the carrying capacity.

Solve this differential equation for c=0.2, K=3000 and initialpopulation P0=200
P(t)=__________ .

Compute the limiting value of the size of the population.
limt??P(t)=_________

At what value of PP does PP grow fastest? ___________
P=__________

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