The given logistic function is:

$P(t) = \frac{210}{1 + e^{-0.54t}}$

This equation describes the population $P(t)$ of fish in a pond as a function of time $t$ in years.

Key Features of the Logistic Function:

1. Carrying Capacity (Maximum Population): The maximum population the environment can sustain, given by the numerator (210 in this case). As $t$ approaches infinity, $P(t)$ approaches 210.

2. Initial Population: The initial population $P(0)$ can be found by substituting $t = 0$ into the equation: $P(0) = \frac{210}{1 + e^{0}} = \frac{210}{2} = 105$ So, the initial population is 105 fish.

3. Growth Rate: The term $e^{-0.54t}$ indicates the rate of change. The coefficient $0.54$ represents the growth rate, and the negative sign indicates that as $t$ increases, $e^{-0.54t}$ decreases, which gradually slows down the growth.

Analysis:

• As $t$ increases: The exponential term $e^{-0.54t}$ decreases, making the denominator approach 1, and $P(t)$ approaches the carrying capacity of 210 fish.
• At $t = 0$: The population is half of the carrying capacity due to the initial exponential term being 1.

Summary of the Population Function:

• Initial Population: 105 fish
• Maximum Population (Carrying Capacity): 210 fish
• Growth Rate: $0.54$ per year

Would you like further details or have any questions?

Here are 5 related questions:

1. How does the logistic model differ from an exponential growth model?
2. What happens to the population if the growth rate $0.54$ is increased or decreased?
3. How can we calculate the time $t$ when the population reaches half of the carrying capacity?
4. How does the population change as $t$ approaches infinity?
5. Can the logistic model be used to predict population decline?

Tip: The logistic function is widely used in ecology to model populations that grow rapidly at first but slow down as they approach a maximum limit, reflecting limited resources.