Problem 26 - Applied Calculus

Question

In \(1923,\) koalas were introduced on Kangaroo Island off the coast of Australia. In \(1996,\) the population was \(5000 .\) By \(2005,\) the population had grown to \(27,000,\) prompting a debate on how to control their growth and avoid koalas dying of starvation." Assuming exponential growth, find the (continuous) rate of growth of the koala population between 1996 and \(2005 .\) Find a formula for the population as a function of the number of years since \(1996,\) and estimate the population in the year \(2020 .\)

Step-by-Step Solution

Answer

The continuous growth rate is approximately 0.187, and the estimated population in 2020 is about 444,650.

Step 1: Understand the exponential growth model

The population growth of the koalas can be described by the exponential model \( P(t) = P_0 \, e^{rt} \). Here, \( P_0 \) is the initial population in 1996, \( r \) is the continuous growth rate, and \( t \) is the time in years since 1996.

Step 2: Set up the known values

In 1996, the population \( P(0) = 5000 \). In 2005, the population \( P(9) = 27000 \). We will use these values to determine the growth rate \( r \).

Step 3: Solve for the growth rate the equation using known values

We use the equation \( 27000 = 5000 \, e^{9r} \) to solve for \( r \). Divide both sides by 5000 to isolate \( e^{9r} \): \( \frac{27000}{5000} = e^{9r} \). Calculate: \( 5.4 = e^{9r} \).

Step 4: Use natural logarithms to find the growth rate

Take the natural logarithm of both sides: \( \ln(5.4) = 9r \). Calculate \( \ln(5.4) \approx 1.686 \). Solve for \( r \): \( r = \frac{1.686}{9} \approx 0.187 \).

Step 5: Write the formula for population function

Now that we have \( r \), the formula for the population is \( P(t) = 5000 \, e^{0.187t} \).

Step 6: Estimate the population in 2020

Calculate the number of years since 1996 to 2020: \( t = 2020 - 1996 = 24 \). Substitute \( t = 24 \) into the population formula: \( P(24) = 5000 \, e^{0.187 \times 24} \). Calculate \( e^{0.187 \times 24} \approx e^{4.488} \approx 88.93 \). \( P(24) \approx 5000 \times 88.93 \approx 444650 \).

Key Concepts

Population Modeling

Population modeling is much like creating a simplified representation of a complex real-world scenario. In our case, it's all about understanding how the population of koalas changes over time. Using population models, we can predict future changes based on current data and assumptions.
For the koalas on Kangaroo Island, an exponential growth model was used. Exponential growth occurs when the growth rate of a population is proportional to its current size. This is often seen in ecosystems when resources are unlimited, leading to a rapid increase in population size.
By setting a mathematical formula, like the exponential growth model, scientists and conservationists can predict how populations might evolve, allowing for better decision-making regarding wildlife management and conservation efforts. The model requires an initial population size and a known growth rate, which is something we can deduce when we have at least two points of known population data, as in the exercise with the koalas from 1996 and 2005.

Continuous Growth Rate

The continuous growth rate is a key parameter in exponential growth models. It's represented by the variable \( r \) in the exponential growth formula \( P(t) = P_0 e^{rt} \). This rate gives us an understanding of how fast the population is increasing over time.
Unlike a simple percentage growth rate, which might be calculated yearly, the continuous growth rate assumes that growth is happening at every instant. This is more realistic for natural populations, as organisms reproduce continuously, not just once per year.
To calculate the continuous growth rate, we need to find \( r \) from known population values at two different points in time. In the koala example, the equation \( 27000 = 5000 \, e^{9r} \) was used, and then solved by taking logarithms, resulting in \( r \approx 0.187 \). This rate tells us that the koala population grew by approximately \( 18.7\% \) each year, continuously.

Exponential Growth Formula

The exponential growth formula is a powerful tool in ecology and other fields for predicting future population sizes. It is expressed as \( P(t) = P_0 \, e^{rt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population size, \( r \) is the continuous growth rate, and \( t \) is the time elapsed since the starting point.
This formula assumes that resources are unlimited, which is why it can often predict rapid population increases over short periods. While this might not be sustainable in the long term due to environmental limits, it can be very accurate in the short to medium term for populations in well-resourced environments.
The beauty of the exponential growth model is its simplicity and wide applicability. In our koala example, once the growth rate \( r \) was known, predicting the population at any future time involved merely plugging into the equation. For 2020, we calculated \( P(24) \) using the initial population and the growth rate, yielding an estimated population of 444,650 koalas, highlighting the effectiveness of the model in predicting future conditions.