Problem 26 - Precalculus Mathematics for Calculus

Question

Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model $$n(t)=\frac{5600}{0.5+27.5 e^{-0.044 t}}$$ where \(t\) is measured in years. (a) Find the initial bird population. (b) Draw a graph of the function \(n(t)\) (c) What size does the population approach as time goes on?

Step-by-Step Solution

Answer

(a) 200; (b) S-shaped logistics curve; (c) 11200.

Step 1: Find the Initial Population

To find the initial bird population, we need to evaluate the function \( n(t) \) at \( t = 0 \). Substitute \( t = 0 \) into the logistic growth model: \[ n(0) = \frac{5600}{0.5 + 27.5 e^{-0.044 \times 0}} \] Since \( e^0 = 1 \), the expression simplifies to: \[ n(0) = \frac{5600}{0.5 + 27.5 \times 1} = \frac{5600}{28} \] Calculating \( \frac{5600}{28} \), we get \( n(0) = 200 \). So the initial bird population is 200.

Step 2: Draw the Graph of the Function

To draw the graph of the function \( n(t) = \frac{5600}{0.5 + 27.5 e^{-0.044t}} \), create a table of values by choosing different \( t \) values and computing \( n(t) \). Example points might include \( t = 0, 10, 20, 30, \ldots \). Plot these points on a graph, with \( t \) on the x-axis and \( n(t) \) on the y-axis. The graph should show an S-shaped curve characteristic of logistic growth, starting at an initial value of 200 and leveling off as \( t \) increases.

Step 3: Determine the Limit of Population Growth

To find out what size the population approaches as time goes on, calculate the limiting value of \( n(t) \) as \( t \to \infty \). When \( t \to \infty \), the term \( e^{-0.044t} \to 0 \). Substituting into the expression gives us: \[ n(\infty) = \frac{5600}{0.5 + 27.5 \cdot 0} = \frac{5600}{0.5} \] Calculating \( \frac{5600}{0.5} \), we find that \( n(\infty) = 11200 \). The population approaches a size of 11200.

Key Concepts

Initial Population

Understanding the initial population involves evaluating the bird population at the start, specifically when time \( t \) equals zero. To find this, substitute \( t = 0 \) into the logistic growth model:

• The equation given is \( n(t) = \frac{5600}{0.5 + 27.5 e^{-0.044 t}} \).
• When \( t = 0 \), the term \( e^{-0.044 \times 0} = 1 \) (because any number raised to the power of zero is 1).
• Thus, the equation simplifies to \( n(0) = \frac{5600}{0.5 + 27.5 \times 1} = \frac{5600}{28} \).
• Performing the division results in \( n(0) = 200 \).

The initial bird population is therefore 200. This value represents the population at the very beginning, giving us insight into the starting point from which the population will grow over time.

Graphing Functions

To visualize the bird population over time, we use graphing. The function \( n(t) = \frac{5600}{0.5 + 27.5 e^{-0.044t}} \) needs to be plotted over various time points:

• Choose sample points for \( t \) such as \( t = 0, 10, 20, 30, 40 \), and so on.
• Calculate \( n(t) \) for each of these \( t \) values to get corresponding \( n(t) \) values.
• Plot these \( t \) and \( n(t) \) pairs on a graph with \( t \) on the x-axis (years) and \( n(t) \) on the y-axis (populations).

The resulting graph will display an S-shaped curve. This curve is typical of logistic growth models:

• It starts off slowly, representing the early exponential growth phase.
• The growth becomes rapid as the population rises.
• Eventually, it levels off, showing the effect of limiting factors (such as habitat).

This visual representation helps in understanding how the population transitions from rapid growth to stabilization.

Population Limit

The population limit in a logistic growth model is the maximum population size that the environment can sustain. To find this limit, consider what happens as \( t \, \rightarrow \, \infty \):

• For large \( t \), the term \( e^{-0.044t} \rightarrow 0 \), simplifying the function to \( n(t) = \frac{5600}{0.5} \).
• So, \( n(t) \) approaches \( \frac{5600}{0.5} = 11200 \) as \( t \to \infty \).

This limiting value of 11200 represents the carrying capacity of the habitat. It indicates the largest population size that can be supported indefinitely, considering the limitations imposed by resources or space. Recognizing this limit is crucial in population studies, as it helps predict long-term population dynamics in an environment.