Problem 21 - College Algebra

Question

Use this scenario: The population \(P\) of an endangered species habitat for wolves is modeled by the function \(P(x)=\frac{558}{1+54.8 e^{-0.462 x}},\) where \(x\) is given in years. Graph the population model to show the population over a span of 10 years.

Step-by-Step Solution

Answer

Graph the plotted points from years 0 to 10 to sketch the logistic growth curve.

Step 1: Understand the function

The function given is a logistic model of population growth, which is a common mathematical model for population that eventually levels off at a carrying capacity. Here, the function is \( P(x) = \frac{558}{1 + 54.8 e^{-0.462 x}} \), where \( P(x) \) represents the population at year \( x \).

Step 2: Determine the input values

We want to graph the population over a span of 10 years, meaning we need input values \( x \) from 0 to 10 years. So we will calculate \( P(x) \) for \( x = 0, 1, 2, \, \ldots, 10 \).

Step 3: Calculate population values

Calculate \( P(x) \) for each year, \( x \): 1. \( x = 0: P(0) = \frac{558}{1 + 54.8 e^{0}} = \frac{558}{1 + 54.8} \) 2. \( x = 1: P(1) = \frac{558}{1 + 54.8 e^{-0.462\times1}} \) 3. Continue this calculation up to \( x = 10 \) using a calculator for precision.

Step 4: Plot the values

On graph paper or using software, plot the calculated population values \( P(x) \) on the vertical axis against the year values \( x \) on the horizontal axis. Ensure consistent scaling to accurately represent the curve.

Step 5: Draw the curve

Connect the plotted points smoothly to represent the logistic growth curve. This should initially rise quickly and then level off, showing the population reaching its carrying capacity over time.

Key Concepts

Population Modeling

Population modeling is a mathematical approach used to represent how populations grow and change over time. The logistic growth model is a common choice for this purpose because it considers the growth of the population to slow down as it approaches a certain limit, known as the carrying capacity.
The logistic model takes the form:

• \( P(x) = \frac{K}{1 + Ae^{-rx}} \)

Here:

• \( P(x) \) denotes the population at time \( x \).
• \( K \) is the carrying capacity.
• \( A \) is a constant determined by initial conditions.
• \( r \) is the growth rate.

The logistic model is particularly effective because it mirrors real-world biological limits. As resources become less available—like food or space—population growth naturally slows.

Endangered Species

Endangered species face a significant risk of extinction in the near future. Recognizing their plight, it's crucial to monitor population numbers to aid in conservation efforts. Population modeling becomes vital, as it offers insights into growth patterns, allowing scientists to predict future trends and recommend action steps.
For example, using models like the logistic growth function helps understand how a species may recover if conditions improve. It gives a clearer picture of population changes and potential reproduction rates under different scenarios. This understanding can influence policy, helping protect these species from human activities, habitat destruction, and other threats.

Graphing Functions

Graphing functions is a critical skill in mathematics that visually represents data for easier analysis. In the case of the logistic growth model, graphing helps us see how the population of an endangered species might expand over time.
When sketching such a graph each point represents the population at a specific time. By...

• Placing years (\( x \)) on the horizontal axis.
• Plotting population size (\( P(x) \)) on the vertical axis.

The result is a S-shaped curve, known as the sigmoid curve. Initially, this curve starts steep, showing rapid growth. Eventually, it begins to flatten as the carrying capacity is approached. This visualization not only showcases trends over time but also aids in assessing future population resilience.

Carrying Capacity

Carrying capacity is the maximum population size an environment can support without ecological degradation. This concept recognizes that resources are limited and that populations cannot grow indefinitely.
When applied in the context of logistic growth models, carrying capacity (often denoted as \( K \)) appears as the upper limit that the graph approaches but rarely exceeds. This is because as population increases, the availability of resources such as food, water, and space decreases. By observing this plateau, scientists can gauge the long-term sustainability of a species in its habitat.

• Carrying capacity helps determine the balance between species growth and environmental health.
• It is pivotal for setting conservation goals and managing wildlife populations effectively.

Understanding this concept is crucial for ecological studies, as it assists in making informed decisions about conservation and resource allocation.