If P0 > c (which implies that
−1 < a < 0),
then the logistics function
P(t) =
c
1 + ae−bt
decreases as t increases. Biologists often use this type of logistic function to model populations that decrease over time. See the following figure. Apply this information to the exercise.
A biologist finds that the fish population in a small lake can be closely modeled by the logistic function
P(t) =
3000
1 + (−0.6667)e−0.05t
where t is the time, in years, since the lake was first stocked with fish.
(a) What was the fish population when the lake was first stocked with fish?
(b) According to the logistic model, what will the fish population approach in the long-term future?
To find the fish population when the lake was first stocked with fish, we need to evaluate the logistic function at t = 0.
(a) Substituting t = 0 into the logistic function:
P(0) = 3000 / (1 + (-0.6667)e^(-0.05*0))
P(0) = 3000 / (1 + 0.6667)
P(0) = 3000 / 1.6667
P(0) ≈ 1800
Therefore, the fish population when the lake was first stocked with fish was approximately 1800.
(b) According to the logistic model, in the long-term future, the fish population will approach the carrying capacity, which is represented by the constant c in the logistic function.
In this case, the carrying capacity is represented by c = 3000.
Therefore, in the long-term future, the fish population will approach 3000.
To find the fish population when the lake was first stocked with fish, we need to evaluate the logistic function at t = 0.
(a) Substitute t = 0 into the logistic function:
P(0) = 3000 / (1 + (-0.6667)e^(-0.05*0))
Since e^0 = 1, this simplifies to:
P(0) = 3000 / (1 + (-0.6667) * 1)
P(0) = 3000 / (1 - 0.6667)
P(0) = 3000 / 0.3333
P(0) ≈ 9000
Therefore, the fish population when the lake was first stocked with fish was approximately 9000.
(b) To determine the long-term fish population according to the logistic model, we need to consider the behavior of the function as t increases towards infinity.
The logistic function approaches its carrying capacity (c) as t approaches infinity. In this case, the carrying capacity is given as P0, which is 3000.
So, as t increases towards infinity, P(t) will approach 3000.
Hence, according to the logistic model, the fish population will approach 3000 in the long-term future.