   A population of fish in a pond is described by the logistic function P(t)=2101+e−0.54tPt=2101+e−0.54t, where P(t)Pt is the number of fish in tt years.

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.

Explore how the logistic function P(t) = 210 / (1 + e^(-0.54t)) describes the population of fish in a pond over time. Learn about the carrying capacity, growth rate, and initial population with detailed explanations and examples.

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