Question

The Pacific halibut fishery has been modeled by the differential equation dy = kyl 1-y dt M = ky (1 台) 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 = 7 * 107 kg, and k = 0.76 per year. (a) If y(0) = 2 x 107 kg, find the biomass a year later. (Round your answer to two decimal places.) 3.65 X * 107 kg (b) How long will it take for the biomass to reach 4 x 107 kg? (Round your answer to two decimal places.) 1.29 X yr Need Help?

Answer 1

The question involves using the logistic growth model to calculate the future biomass of the Pacific halibut fishery. For part (a), the biomass after one year is found through the logistic equation given the initial biomass, growth rate, and carrying capacity. For part (b), it requires solving for the time it takes the biomass to reach a specified amount.

Step-by-step explanation:

The student's question involves applying the logistic growth model to calculate the future biomass of the Pacific halibut fishery, given a differential equation, the initial biomass, and the carrying capacity. The equation provided is dy/dt = ky(1 - y/M), where y(t) is the biomass at time t, k is the growth rate, and M is the carrying capacity.

Part (a)

To find the biomass a year later, one would use the logistic growth equation to update the biomass given the initial conditions y(0) = 2 x 107 kg, k = 0.76 per year, and M = 7 x 107 kg. Using integration techniques or numerical methods, we could find y(1), the biomass one year later, and round the answer to two decimal places.

Part (b)

To determine how long it will take for the biomass to reach 4 x 107 kg, we would solve the logistic equation for time t when y(t) equals the desired biomass. This may require using numerical solvers or iterative methods as it's likely the solution cannot be found by simple algebraic manipulation.