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Northern Yellowstone elk: The northern Yellowstone elk winter in the northern range of Yellowstone National Park. \({ }^{6}\) A moratorium on elk hunting was imposed in 1969 , and after that the growth of the elk population was approximately logistic for a time. The following table gives data on the growth. $$ \begin{array}{|c|c|} \hline \text { Year } & N \\ \hline 1968 & 3172 \\ \hline 1969 & 4305 \\ \hline 1970 & 5543 \\ \hline 1971 & 7281 \\ \hline 1972 & 8215 \\ \hline 1973 & 9981 \\ \hline 1974 & 10,529 \\ \hline \end{array} $$ a. Use regression to find a logistic model for this elk population. b. According to the model you made in part a, when would the elk population reach half of the carrying capacity? Note: At one time the gray wolf was a leading predator of the elk, but it was not a factor during this study period. The level at which the elk population stabilized suggests that food supply (and not just predators) can effectively regulate population size in this setting.

Step by step solution

01

Understand the Logistic Growth Model Formula

The logistic growth model is expressed with the formula: \[ N(t) = \frac{K}{1 + \frac{K-N_0}{N_0}e^{-rt}} \]where:- \(N(t)\) = Population at time \(t\),- \(K\) = Carrying capacity,- \(N_0\) = Initial population size,- \(r\) = Growth rate.

02

Model the Data with Regression

Use regression to fit the logistic model to the given data. Generally, this involves using transformations to linearize the data and then applying linear regression to find the parameters.Given the data, without computational tools or detailed calculations here, one would typically employ software like Excel, Python, or specialized statistical software for this task. This would yield values for \(K\), \(N_0\), and \(r\).

03

Identify Half of the Carrying Capacity

The problem asks for the time when the elk population reaches half of the carrying capacity. Once \(K\) is determined, half of the carrying capacity is \(\frac{K}{2}\).

04

Solve for Time Using the Logistic Growth Model

Rearrange the logistic growth formula to solve for \(t\) when \(N(t) = \frac{K}{2}\):\[ \frac{K}{2} = \frac{K}{1 + \frac{K-N_0}{N_0}e^{-rt}} \]Solve for \(t\) by isolating \(e^{-rt}\):\[ 1 + \frac{K-N_0}{N_0}e^{-rt} = 2 \]\[ e^{-rt} = \frac{N_0}{K-N_0} \]\[ t = -\frac{1}{r} \ln \left(\frac{N_0}{K-N_0}\right) \]Plug in known values of \(N_0\), \(K\), and \(r\) to get \(t\).

05

Calculate the Year When Population Reaches Half Carrying Capacity

Add the computed time \(t\) from Step 4 to the initial year (1968) to determine the year. This requires the values from the regression model which are often generated computationally.

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Key Concepts

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

Elk Population Study

Elk populations provide a fascinating insight into wildlife ecology and managing such populations is crucial for ecological balance. In the northern range of Yellowstone National Park, an intriguing study focused on the elk population from the late 1960s to early 1970s. During this period, a moratorium on elk hunting was put in place, which allowed researchers to observe a more natural growth pattern of the elk.​
The study aimed to understand how the population grew over several years without the influence of hunting. With wolves absent as a major predator during this time, the study emphasized environmental factors like food supply as primary population regulators.
This setting provided a perfect natural laboratory to explore how elk populations grow over time, emphasizing the roles of different ecological factors.

Regression Analysis

To accurately describe how the elk population changed over time, scientists often turn to the technique of regression analysis. It's a statistical method that helps build a model to predict a particular outcome based on one or more predictor variables.
In this scenario, researchers used regression analysis to fit the logistic growth model to their observed data. The logistic growth model helps predict population growth in a scenario where resources (like food) are limited. Population growth initially accelerates, then slows down as it nears a carrying capacity. This is where regression becomes key in estimating important parameters like the initial population (\(N_0\)), the growth rate (\(r\)), and the carrying capacity (\(K\)).
With modern tools like Excel or Python libraries, researchers can perform these analyses, transforming observed data into a comprehensive model that predicts future population changes.

Carrying Capacity

Remember, carrying capacity (\(K\)) is a crucial concept in understanding population dynamics. It refers to the maximum number of individuals an environment can support sustainably over time. For the elk in Yellowstone, this would be the point where the food resources and other ecological factors could consistently support a stable number of elk without degradation of their habitat.
The logistic growth model incorporates carrying capacity to ensure that population predictions are realistic. As the elk population numbers approach the carrying capacity, the growth rate slows, and the population stabilizes. Estimating the carrying capacity helps wildlife managers make decisions regarding habitat conservation and population control measures.
In the elk study, carrying capacity is intimately linked to both the availability of food and the absence of predators, providing an ideal case study for how different ecological pressures balance one another.

Population Dynamics

Population dynamics involves studying how and why populations change over time and space. It's essential for managing wildlife effectively and ensuring biodiversity is maintained. This concept considers birth rates, death rates, and migration patterns, along with environmental factors like food availability and predation.
In the exercise about Yellowstone's elk, the population dynamics were observed in a situation with natural growth influenced primarily by food supply due to the absence of major predators. By exploring how the elk population dynamics evolved over the study period, scientists gained invaluable insights into how populations grow and stabilize under different environmental pressures.
Understanding these dynamics allows researchers to make informed predictions about future changes, design effective wildlife management strategies, and promote ecological health through sustainable practices.