Question

The Pacific halibut fishery has been modeled by the differential equation dy dt = ky 1 − y M where y(t) is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be M = 9 × 107 kg, and k = 0.74 per year. (a) If y(0) = 2 × 107 kg, find the biomass a year later. (Round your answer to two decimal places.)

Answer 1

3.37 x 10^7 Kg

Explanation:

Starting with the general solution equation for differential equation involving exponential population growth

`y =(c)/(1 + Ae^(-kt) )`

c = 9 x 10^7Kg

k = 0.74 per year

`y =(9 * 10^(7) )/(1 + Ae^(-0.74t) )`

`A=(c- y(0) )/(y(0) ) = (9*10^(7) - 2*10^(7) )/(2*10^(7)) \\\\A=3.5`

`y =(9 * 10^(7) )/(1 + Ae^(-0.74t) ) \\\\y =(9 * 10^(7) )/(1 + 3.5e^(-0.74t) ) \\\\A year later , t= 1\\y =(9 * 10^(7) )/(1 + 3.5e^(-0.74*1) ) = 33709144.04`

3.37 x 10^7 Kg