The logistic population growth model in this case is given by:

$N(t) = \frac{37.5}{0.28 + 0.76t}$

where $N(t)$ is the population of foxes at time $t$ (in years). Let's go through each part step by step.

(a) How many foxes were introduced into the protected area?

To find the number of foxes introduced, we need to evaluate the function at $t = 0$, since at $t = 0$, this represents the initial population.

$N(0) = \frac{37.5}{0.28 + 0.76 \times 0} = \frac{37.5}{0.28}$

Now let's calculate that value.

$N(0) = \frac{37.5}{0.28} \approx 133.93$

Rounding to the nearest whole number, approximately 134 foxes were introduced into the protected area.

(b) Make a graph of $N(t)$ versus $t$

To graph $N(t) = \frac{37.5}{0.28 + 0.76t}$, you would plot the population of foxes $N(t)$ as a function of time $t$.

Explanation of population growth:

• At $t = 0$, the population starts at around 134 foxes.
• Over time, as $t$ increases, the denominator $0.28 + 0.76t$ increases, which makes $N(t)$ decrease.
• This pattern follows a logistic decline, meaning the population approaches zero as $t$ increases, but at a decreasing rate.

(c) When will the fox population reach 85 individuals?

To find when the population reaches 85 foxes, set $N(t) = 85$ and solve for $t$:

$85 = \frac{37.5}{0.28 + 0.76t}$

Multiply both sides by $(0.28 + 0.76t)$:

$85(0.28 + 0.76t) = 37.5$

Expanding this:

$23.8 + 64.6t = 37.5$

Now, solve for $t$:

$64.6t = 37.5 - 23.8 = 13.7$

$t = \frac{13.7}{64.6} \approx 0.212$

Thus, the fox population will reach 85 individuals after approximately 0.21 years, or about 2.5 months.

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Let me know if you'd like more details on any of these parts or further clarification!

Relative Questions:

1. What is the long-term behavior of the population as $t \to \infty$?
2. How would the population change if the growth rate constant $0.76$ was increased?
3. What effect does the initial population size have on the logistic model?
4. Can the population exceed the initial number of foxes introduced? Why or why not?
5. What happens to the population if the denominator was linear in $t$ rather than logistic?

Tip:

Logistic models are commonly used to describe population growth because they take into account the carrying capacity of the environment, leading to a more realistic representation of growth than exponential models.