Question

PART A)

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

Assuming that the size of the fish population satisfies the logistic equation

dPdt=kP(1−PK),

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

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

PART B)

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential 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 initial population 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=__________

Answer 1