To estimate the deer population after one year, we can use the logistic growth model, which accounts for population limits due to resource constraints. The model is described by:

$P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}}$

Where:

• $P(t)$ is the population at time $t$,
• $K$ is the carrying capacity (300 deer),
• $P_0$ is the initial population (50 deer),
• $r$ is the intrinsic growth rate (80% = 0.8 per year),
• $t$ is the time in years.

Step 1: Set up the equation

Given: $P_0 = 50, \quad K = 300, \quad r = 0.8, \quad t = 1$

The population after 1 year is: $P(1) = \frac{300}{1 + \left(\frac{300 - 50}{50}\right)e^{-0.8 \cdot 1}}$

Step 2: Simplify

First, calculate the fraction inside: $\frac{K - P_0}{P_0} = \frac{300 - 50}{50} = 5$

Substitute back into the equation: $P(1) = \frac{300}{1 + 5e^{-0.8}}$

Step 3: Compute $e^{-0.8}$

Using an approximation: $e^{-0.8} \approx 0.449$

Substitute: $P(1) = \frac{300}{1 + 5(0.449)} = \frac{300}{1 + 2.245} = \frac{300}{3.245}$

Step 4: Final calculation

$P(1) \approx 92.4$

Conclusion

After one year, the estimated population is approximately 92 deer.

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Do you want further clarifications or details about this calculation? Here are 5 related questions to deepen your understanding:

1. How does the logistic model differ from exponential growth in terms of population predictions?
2. What would the population be after 2 years under the same conditions?
3. How is the carrying capacity $K$ determined in real-life scenarios?
4. What happens if $r$ is higher or lower? How does it affect the growth rate?
5. How do environmental factors influence the assumptions of logistic growth?

Tip: Always double-check whether resource constraints like carrying capacity are realistic when applying population growth models!