91Ó°ÊÓ

(Requires technology to find a best-fit function.) In \(1911,\) reindeer were introduced to St. Paul Island, one of the Pribilof Islands, off the coast of Alaska in the Bering Sea. There was plenty of food and no hunting or reindeer predators. The size of the reindeer herd grew rapidly for a number of years, as given in the accompanying table. $$ \begin{array}{cr|rr} \hline & \text { Population } & & \text { Population } \\ \text { Year } & \text { Size } & \text { Year } & \text { Size } \\ \hline 1911 & 17 & 1925 & 246 \\ 1912 & 20 & 1926 & 254 \\ 1913 & 42 & 1927 & 254 \\ 1914 & 76 & 1928 & 314 \\ 1915 & 93 & 1929 & 339 \\ 1916 & 110 & 1930 & 415 \\ 1917 & 136 & 1931 & 466 \\ 1918 & 153 & 1932 & 525 \\ 1919 & 170 & 1933 & 670 \\ 1920 & 203 & 1934 & 831 \\ 1921 & 280 & 1935 & 1186 \\ 1922 & 229 & 1936 & 1415 \\ 1923 & 161 & 1937 & 1737 \\ 1924 & 212 & 1938 & 2034 \\ \hline \end{array} $$ a. Use the reindeer data file (in Excel or graph link form) to plot the data. b. Find a best-fit exponential function. c. How does the predicted population from part (b) differ from the observed ones? d. Does your answer in part (c) give you any insights into why the model does not fit the observed data perfectly? e. Estimate the doubling time of this population.

Step by step solution

01

Input the Data

First, open your Excel file or graphing software and input the years in one column and the corresponding reindeer population sizes in the adjacent column.

02

Plot the Data

Create a scatter plot by selecting the two columns of data and using the scatter plot function in your software. This will help visualize how the population changes over time.

03

Fit an Exponential Function

Use the trendline or best-fit function tool available in your software. Choose 'exponential function' as the type of trendline. The software will provide an equation of the form: \[ P(t) = P_0 \times e^{kt} \] where \(P(t)\) represents population size at time \(t\), \(P_0\) is the initial population size, and \(k\) is the growth rate.

04

Interpret the Best-Fit Function

Once the best-fit exponential function is generated, note down the values of \(P_0\) and \(k\) given by the software. Compare these values with the observed population sizes.

05

Compare Predicted and Observed Data

Calculate the predicted population sizes using the best-fit function for each year and compare them to the observed values in the table. This can be done by substituting the year values into the exponential equation derived.

06

Analyze Differences

Examine the differences between the predicted and observed population sizes. Consider possible reasons for any discrepancies, such as environmental factors, limitations of the exponential model, or other influences not accounted for in the model.

07

Estimate Doubling Time

The doubling time can be calculated using the growth rate \(k\). The formula for doubling time \(T_d\) in terms of \(k\) is: \[ T_d = \frac{\ln(2)}{k} \]. Substitute the value of \(k\) obtained from your best-fit function to find the doubling time.

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Key Concepts

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

Data Plotting

Data plotting is the first step in analyzing reindeer population growth. You'll start by inputting your data into a spreadsheet or graphing software like Excel. In one column, list the years starting from 1911 to 1938. In the adjacent column, input the corresponding population sizes for each year. Next, create a scatter plot by selecting these columns and using the scatter plot function. This visualization helps to see how the population changes over time. Notice the points and how they trend upwards indicating growth. This sets the foundation for further analysis.

Best-Fit Function

To analyze how the population grows, we need to find a best-fit function. In most graphing software, there's a feature for trendlines. Select 'exponential function' for the trendline type. The software will generate an equation of the form:
\[ P(t) = P_0 \times e^{kt} \]
Here, \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population size, and \(k\) is the growth rate. This equation best fits the scatter plot data by minimizing the distance between the predicted and actual values. This best-fit function is crucial for making predictions and understanding the growth pattern.

Exponential Function

An exponential function describes how a quantity grows rapidly over time. In our context, the function:
\[ P(t) = P_0 \times e^{kt} \]
The term \(P_0\) represents the population size at year zero (1911 in our dataset). The factor \(e\) is Euler's number (approximately 2.718), a constant in exponential growth. The growth rate \(k\) shows how fast the population increases. This type of function is characterized by a rate proportional to its current value, leading to exponential increases over time. When modeling biological populations, exponential functions are powerful tools because they account for rapid, compounded growth.

Doubling Time

Doubling time is a crucial concept to understand how quickly a population size doubles. For exponential growth, doubling time \(T_d\) can be calculated using:
\[ T_d = \frac{\text{ln}(2)}{k} \]
Here, \(\text{ln}(2)\) is the natural logarithm of 2, approximately 0.693, and \(k\) is the growth rate from our best-fit function. Doubling time provides a simple measure to grasp how fast the population grows. For instance, if the doubling time is 5 years, the reindeer population will double every 5 years if the conditions remain constant. Understanding doubling time is pivotal for predicting future population sizes.

Model Analysis

Analyzing the model involves comparing the predicted population sizes to the observed data. After calculating the predicted populations using the best-fit exponential function, you may notice some discrepancies with the actual values. These differences can be due to factors not accounted for by the model, like environmental changes or food supply variations. Exponential models assume continuous growth without limits, but real-world scenarios often include constraints. Despite these differences, the model provides valuable insights into the general growth pattern and helps identify when the exponential model may not fully capture the dynamics.

Growth Rate Calculation

The growth rate is a key parameter in our best-fit exponential function. It shows how quickly the population grows. By analyzing the best-fit equation:
\[ P(t) = P_0 \times e^{kt} \]
the value of \(k\) indicates the growth per unit time (usually per year). To find this, graphing software usually calculates it automatically based on the given data. Alternatively, you can manually adjust \(k\) to minimize the differences between predicted and observed populations. A higher \(k\) value means faster growth. Understanding the growth rate helps in making predictions and planning for future population management.