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style=”background-color:rgb(245,245,245);color:rgb(0,0,0);”> 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 9300

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1)

style=”background-color:rgb(245,245,245);color:rgb(0,0,0);”> 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 9300

. The number of fish tripled in the first year. 

(a) Assuming that the size of the fish population satisfies the logistic equation

dP/dt = kP(1−P/K),

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)=……………….?……………………..

(b) How long will it take for the population to increase to 4650 (half of the carrying capacity)? 

It will take ………………?……………… years.

2) Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation

dP/dt=cln(K/P)P

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

(a) Solve this differential equation for c=0.15, K=5000, and initial population P(0)=200.

P(t)=…………………?…………………..

(b) Compute the limiting value of the size of the population. 

limt→∞P(t)= …………………?………………………

 
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