On January 1, 1879, records show that 500 of a fish called Atlantic striped bass were introduced into the San Francisco Bay. In 1899, the first year fishing for bass was allowed, 100,000 of these bass were caught, representing 10% of the population at the start of 1899. Owing to reproduction, at any time the bass population is growing at a rate proportional to the population at that moment. Assume that when fishing starts in 1899, the rate at which bass are caught is proportional to the square of the population with constant of proportionality . What happens to the bass population in the long run?
A)It approaches 0.
B)It grows without bound.
C)It approaches .
D)It approaches .

Question

On January 1, 1879, records show that 500 of a fish called Atlantic striped bass were introduced into the San Francisco Bay. In 1899, the first year fishing for bass was allowed, 100,000 of these bass were caught, representing 10% of the population at the start of 1899. Owing to reproduction, at any time the bass population is growing at a rate proportional to the population at that moment. Assume that when fishing starts in 1899, the rate at which bass are caught is proportional to the square of the population with constant of proportionality  . What happens to the bass population in the long run?
A)It approaches 0. 
B)It grows without bound. 
C)It approaches . 
D)It approaches .

Quizplus · Accepted Answer

This is a logistic differential equation, and its solution can be found using separation of variables and integration. The solution is a sigmoid function, which approaches a maximum carrying capacity as time goes to infinity. In this case, the carrying capacity is given by the constant of proportionality, which is 11eb69ff_fde8_ee65_af94_e94aeb89fc6e_TB8190_11 . Therefore, the correct answer is D.

On January 1, 1879, Records Show That 500 of a Fish