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?
(b)3000
To find the fish population when the lake was first stocked with fish, we need to evaluate the logistic function P(t) at t = 0.
For part (a):
Substituting t = 0 into the equation, we get:
P(0) = 3000 / (1 + (-0.6667)e^(-0.05 * 0))
Since anything raised to the power of 0 equals 1, the equation simplifies to:
P(0) = 3000 / (1 + (-0.6667)(1))
Simplifying further:
P(0) = 3000 / (1 + 0.6667)
Doing the calculations:
P(0) = 3000 / 1.6667 ≈ 1799.88
Therefore, the fish population when the lake was first stocked with fish was approximately 1799.
For part (b):
To determine the long-term future population, we need to consider what happens as t approaches infinity. In the given equation, as t increases, the exponential term e^(-0.05t) approaches 0. As a result, the denominator of the logistic function approaches 1, making the population approach the value of c.
In this case, c = 3000. Therefore, according to the logistic model, the fish population will approach 3000 in the long-term future.