To address this problem, we can model the population growth of the southern mountain caribou using the exponential growth model, which assumes that the per capita production rate, $r$, remains constant over time.

Exponential Growth Model:

The general equation for exponential growth is:

$P(t) = P_0 e^{rt}$

Where:

• $P(t)$ is the population at time $t$.
• $P_0$ is the initial population at time $t = 0$.
• $r$ is the per capita growth rate.
• $t$ is the time in years.

Given Data:

• In 1991, the population was $P_0 = 5071$.
• In 2024, the population was $P(2024) = 2626$.
• The critical population for survival is 572, which is the threshold we need to calculate when the population will reach this number.

We need to solve for $r$ first and then determine when the population will fall to 572.

Part (a): Determining the per capita production rate $r$

We can use the formula for exponential growth and plug in the known values:

$P(2024) = P_0 e^{r \cdot (2024 - 1991)}$

$2626 = 5071 e^{r \cdot 33}$

Now solve for $r$:

$\frac{2626}{5071} = e^{33r}$

$0.5177 = e^{33r}$

Taking the natural logarithm (ln) of both sides:

$\ln(0.5177) = 33r$

$-0.6594 = 33r$

$r = \frac{-0.6594}{33} \approx -0.01997 \quad \text{(per year)}$

Thus, the per capita production rate is approximately $r = -0.01997$ per year.

Part (b): Time when population reaches the critical threshold

To find when the population reaches the critical level of 572, we can use the same exponential growth equation. We need to find $t$ such that $P(t) = 572$.

$572 = 5071 e^{r \cdot t}$

Substitute $r \approx -0.01997$:

$572 = 5071 e^{-0.01997 \cdot t}$

Solve for $t$:

$\frac{572}{5071} = e^{-0.01997 \cdot t}$

$0.1128 = e^{-0.01997 \cdot t}$

Take the natural logarithm of both sides:

$\ln(0.1128) = -0.01997 \cdot t$

$-2.1787 = -0.01997 \cdot t$

Solve for $t$:

$t = \frac{-2.1787}{-0.01997} \approx 109.1 \quad \text{(years)}$

Since $t = 0$ corresponds to 1991, we add 109.1 years to 1991:

$1991 + 109.1 \approx 2100$

Thus, the caribou population will first fall to the critical level of 572 around the year 2100.

Final Answers:

(a) The per capita production rate $r$ is approximately $-0.01997$ per year.

(b) The population will first fall to the critical level of 572 after approximately 109 years, or in the year 2100.

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Would you like any further explanation or clarification on any step? Here are some additional questions to explore:

1. How would the model change if the per capita production rate was positive instead of negative?
2. What factors could affect the actual survival of the caribou in the real world, even with a model like this?
3. How can a population model be adapted to include the effects of inbreeding or environmental changes?
4. What are some methods for estimating the per capita production rate from real-world data?
5. If the caribou population were to experience a sudden increase in population size, how would the model need to be adjusted?

Tip: Always double-check the units of your time and population to ensure consistency when using growth models.