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Chapter 5: Problem 13

Gray wolves recolonized in the Upper Peninsula of Michigan beginning in 1990 . Their population has been documented as shown in the accompanying table. \({ }^{49}\) a. Explain why one would expect an exponential model to be appropriate for these data. b. Find an exponential model for the data given. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Wolves } \\ \hline 1990 & 6 \\ \hline 1991 & 17 \\ \hline 1992 & 21 \\ \hline 1993 & 30 \\ \hline 1994 & 57 \\ \hline 1995 & 80 \\ \hline 1996 & 116 \\ \hline 1997 & 112 \\ \hline 1998 & 140 \\ \hline 1999 & 174 \\ \hline 2000 & 216 \\ \hline \end{array} $$ c. Graph the data and the exponential model. Would it be better to use a piecewise-defined function? d. Find an exponential model for 1990 through 1996 and another for 1997 through \(2000 .\) e. Write a formula for the number of wolves as a piecewise-defined function using the two exponential models. Is the combined model a better fit?

Short Answer

Expert verified
Exponential growth models are appropriate; piecewise models fit better over separate periods due to changes in growth trends.

Step by step solution

01

Justifying Exponential Model

Given the data from 1990 to 2000, we observe the wolf population grew significantly. An exponential model may be appropriate if the population increases by a fixed percentage rather than a fixed amount. Comparing the growth across years, each subsequent year sees a larger increase, suggesting exponential growth.
02

Constructing Exponential Model (1990-2000)

An exponential model has the form \( P(t) = P_0 \, e^{kt} \), where \( P_0 \) is the initial population and \( k \) is the growth rate. First, use the initial condition from 1990: \( P_0 = 6 \). Then, choose another data point, say 2000: \( P(10) = 216 \). Solve for \( k \) using the equation \( 216 = 6 \, e^{10k} \): \\[ e^{10k} = \frac{216}{6} = 36 \] \Taking the natural log of both sides, \\[ 10k = \ln(36) \] \\[ k = \frac{\ln(36)}{10} \approx 0.183 \] \The model is \( P(t) = 6 \, e^{0.183t} \).
03

Graphing Data and Model

Plot the data points on a graph from 1990 to 2000. Superimpose the exponential model \( P(t) = 6 \, e^{0.183t} \). Observing the fit, note that if the data deviates noticeably in particular periods (e.g., 1997 shows a dip), a piecewise-defined function may better represent the growth across these years.
04

Constructing Two Exponential Models for Different Periods

For 1990 to 1996, use initial data (1990, 6) and another year (1996, 116). Solving \( 116 = 6 \, e^{6k_1} \) gives: \\[ e^{6k_1} = \frac{116}{6} \Rightarrow k_1 \approx \frac{\ln(\frac{116}{6})}{6} \approx 0.519 \] \Model (1990-1996): \( P_1(t) = 6 \, e^{0.519t} \). For 1997 to 2000, find \( P_0 \) at 1997 (initial population is 112) and use 2000 (216): \\[ 216 = 112 \, e^{3k_2} \] \\[ k_2 \approx \frac{\ln(\frac{216}{112})}{3} \approx 0.248 \] \Model (1997-2000): \( P_2(t) = 112 \, e^{0.248(t-7)} \).
05

Writing Piecewise-Defined Function

Combine the two exponential models into a piecewise function: \\[ P(t) = \begin{cases} 6 e^{0.519t}, & \text{for } 0 \leq t \leq 6 \ 112 e^{0.248(t-7)}, & \text{for } 7 \leq t \leq 10 \end{cases} \] \This piecewise function reflects different growth rates in different periods.
06

Evaluating the Fit of the Piecewise Model

The combined model accounts for changes in growth rate over time, potentially offering a closer fit to specific data trends. If deviations exist in the single model's predictions, the piecewise model likely reduces this error, especially if anomalies (like slower growth) are evident in sections of the data.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Piecewise Function
In mathematical modeling, a piecewise function is a type of function that is defined by different expressions over different intervals of its domain. This is particularly useful when the trend you are modeling has varying behaviors over time.

Imagine recording wolf population growth that behaves differently across decades. If the first period shows rapid growth, while the next period's growth slows down, a single formula might not give an accurate representation. Instead, a piecewise function is perfect here, as it allows two different formulas to be used, capturing each phase accurately.

To set it up, you write separate formulas for each interval and define them such that each formula applies only within its respective time frame. This yields a more nuanced and authentic model of the changing dynamics you're observing.

This approach recognizes the nuances in growth patterns and ensures that each segment is modeled in the way that most accurately represents its specific characteristics, providing a more precise representation of reality.
Wolf Population Modeling
Modeling the wolf population in the Upper Peninsula of Michigan helps us understand how wolf numbers have changed over time. This can reflect broader ecological dynamics and inform conservation strategies.

From 1990 to 2000, wolf population data suggest an increase, characteristic of rebound mammal species after declining to very low numbers. Note that the population numbers aren't just increasing steadily by the same amount every year, suggesting simple linear growth isn't a suitable choice.

Through population modeling, researchers and wildlife managers can forecast future population changes and assess the effectiveness of conservation efforts. Wolf population modeling demonstrates common population dynamics like population bottlenecks, recovery phases, and eventual stabilization.
Exponential Models
The basis of exponential models is the premise that growth doesn't happen at a constant rate, but rather the rate of growth itself increases over time. In the context of wolf populations, this means that as there are more wolves, the population increases even more rapidly.

Why Exponential?
Exponential growth is evident when a quantity grows by a consistent factor over equal intervals of time. For the wolves, years show increasingly larger jumps in population numbers. This tells us the population doesn't just grow by adding a set number of wolves each year, but rather multiplies by a consistent growth factor.

Creating Exponential Models
To create an exponential model, we use the formula \( P(t) = P_0 \times e^{kt} \), where:
  • \( P(t) \) is the population at time \( t \)
  • \( P_0 \) is the initial population at the start of the time period
  • \( e \) is Euler's number (approximately 2.71828)
  • \( k \) is the constant growth rate
For example, finding the growth rate \( k \) involves taking the natural logarithm of the growth factor over a specific time span鈥攁 fundamental step in crafting an accurate model.

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Most popular questions from this chapter

For traffic moving along a highway, we use \(q\) to denote the mean flow rate. That is the average number of vehicles per hour passing a certain point. We let \(q_{m}\) denote the maximum flow rate, \(k\) the mean traffic density (that is, the average number of vehicles per mile), and \(k_{m}\) the density at which flow rate is a maximum (that is, the value of \(k\) when \(q=q_{m}\) ). a. An important measurement of traffic on a highway is the relative density \(R\), which is defined as $$ R=\frac{k}{k_{m}}. $$ i. What does a value of \(R<1\) indicate about traffic on a highway? ii. What does a value of \(R>1\) indicate about traffic on a highway? b. Let \(u\) denote the mean speed of vehicles on the road and \(u_{f}\) the free speed -that is, the speed when there is no traffic congestion at all. One study \({ }^{43}\) proposes the following relation between density and speed: $$ u=u_{f} e^{-0.5 R^{2}}. $$ Use function composition to find a formula that directly relates mean speed to mean traffic density. c. Make a graph of mean speed versus mean traffic density, assuming that \(k_{m}\) is 122 cars per mile and \(u_{f}\) is 75 miles per hour. (Include values of mean traffic density up to 250 vehicles per mile.) Paying particular attention to concavity, explain the significance of the point \(k=122\) on the graph. d. Traffic is considered to be seriously congested if the mean speed drops to 35 miles per hour. Use the graph from part c to determine what density will result in serious congestion.

Many science fiction movies feature animals such as ants, spiders, or apes growing to monstrous sizes and threatening defenseless Earthlings. (Of course, they are in the end defeated by the hero and heroine.) Biologists use power functions as a rough guide to relate body weight and cross-sectional area of limbs to length or height. Generally, weight is thought to be proportional to the cube of length, whereas cross-sectional area of limbs is proportional to the square of length. Suppose an ant, having been exposed to "radiation," is enlarged to 500 times its normal length. (Such an event can occur only in Hollywood fantasy. Radiation is utterly incapable of causing such a reaction.) a. By how much will its weight be increased? b. By how much will the cross-sectional area of its legs be increased? c. Pressure on a limb is weight divided by crosssectional area. By how much has the pressure on a leg of the giant ant increased? What do you think is likely to happen to the unfortunate ant? \({ }^{15}\)

If we view a star now, and then view it again 6 months later, our position will have changed by the diameter of the Earth's orbit around the sun. (See Figure 5.68.) For stars within about 100 light-years of Earth, the change in viewing location is sufficient to make the star appear to be in a different location in the sky. Half of the angle from one location to the next is known as the parallax angle. Even for nearby stars, the parallax angle is very \(\operatorname{small}^{36}\) and is normally measured in seconds of arc. The distance to a star can be determined from the parallax angle. The table below gives parallax angle \(p\) measured in seconds of arc and the distance \(d\) from the sun measured in light-years. $$ \begin{array}{|l|c|c|} \hline \text { Star } & \text { Parallax angle } & \text { Distance } \\ \hline \text { Markab } & 0.030 & 109 \\ \hline \text { Al Na'ir } & 0.051 & 64 \\ \hline \text { Alderamin } & 0.063 & 52 \\ \hline \text { Altair } & 0.198 & 16.5 \\ \hline \text { Vega } & 0.123 & 26.5 \\ \hline \text { Rasalhague } & 0.056 & 58 \\ \hline \end{array} $$ a. Make a plot of \(\ln d\) against \(\ln p\), and determine whether it is reasonable to model the data with a power function. b. Make a power function model of the data for \(d\) in terms of \(p\). c. If one star has a parallax angle twice that of a second, how do their distances compare? d. The star Mergez has a parallax angle of \(0.052\) second of arc. Use functional notation to express how far away Mergez is, and then calculate that value. e. The star Sabik is 69 light-years from the sun. What is its parallax angle?

The power \(P\) required for level flight by an airplane is a function of the speed \(u\) of flight. Consideration of drag on the plane yields the model $$ P=\frac{u^{3}}{a}+\frac{b}{u}. $$ Here \(a\) and \(b\) are constants that depend on the characteristics of the airplane. This model may also be applied to the flight of a bird such as the budgerigar (a type of parakeet), where we take \(a=7800\) and \(b=600\). Here the flight speed \(u\) is measured in kilometers per hour, and the power \(P\) is the rate of oxygen consumption in cubic centimeters per gram per hour. \({ }^{68}\) a. Make a graph of \(P\) as a function of \(u\) for the budgerigar. Include flight speeds between 25 and 45 kilometers per hour. b. Calculate \(P(39)\) and explain what your answer means in practical terms. c. At what flight speed is the required power minimized?

The following table shows the number \(H\) of cases of perinatal HIV infections in the U.S. as reported by the Centers for Disease Control and Prevention. Here \(t\) denotes years since 1985 . $$ \begin{array}{|c|c|} \hline t & H \\ \hline 0 & 210 \\ \hline 1 & 380 \\ \hline 2 & 500 \\ \hline 6 & 780 \\ \hline 8 & 770 \\ \hline 9 & 680 \\ \hline 11 & 490 \\ \hline 12 & 300 \\ \hline \end{array} $$ a. Make a plot of the data. b. Use regression to find a quadratic model for \(H\) as a function of \(t\). c. Add the plot of the quadratic model to the data plot in part a. d. When does the model show a maximum number of cases of perinatal HIV infection?
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