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Chapter 2: Problem 19

Waterton Lakes National Park of Canada, where the Great Plains dramatically meet the Rocky Mountains in Alberta, has a migratory buffalo (bison) herd that spends falls and winters in the park. The herd is currently managed and so kept small; however, if it were unmanaged and allowed to grow, then the number \(N\) of buffalo in the herd could be estimated by the logistic formula $$ N=\frac{315}{1+14 e^{-0.23 t}} $$ Here \(t\) is the number of years since the beginning of 2002 , the first year the herd is unmanaged. a. Make a graph of \(N\) versus \(t\) covering the next 30 years of the herd's existence (corresponding to dates up to 2032 ). b. How many buffalo are in the herd at the beginning of 2002 ? c. When will the number of buffalo first exceed \(300 ?\) d. How many buffalo will there eventually be in the herd? e. When is the graph of \(N\), as a function of \(t\), concave up? When is it concave down? What does this mean in terms of the growth of the buffalo herd?

Short Answer

Expert verified
a) Graph the function from 2002 to 2032. b) 21 buffalo. c) Exceeds 300 in 2022. d) Eventually 315 buffalo. e) Concave up initially, then concave down near the carrying capacity, indicating initial rapid growth slowing to stabilize.

Step by step solution

01

Graphing the Logistic Function

To graph the function \(N(t) = \frac{315}{1 + 14 e^{-0.23 t}}\) over the next 30 years, we need to evaluate \(N(t)\) from \(t = 0\) to \(t = 30\). Use a graphing tool or software to plot this by calculating \(N\) at regular intervals, starting from \(t=0\) to \(t=30\). This will show how the buffalo population changes over time.
02

Initial Population in 2002

For the initial population at the start of 2002, set \(t = 0\) in the formula: \(N(0) = \frac{315}{1 + 14 e^{-0.23 \times 0}} = \frac{315}{1 + 14} = \frac{315}{15} = 21\). So, there are 21 buffalo at the beginning of 2002.
03

Population Exceeds 300

To find when the population first exceeds 300, solve \(N(t) > 300\): \(\frac{315}{1 + 14 e^{-0.23 t}} > 300\). Simplifying, \(1 + 14 e^{-0.23 t} < 1.05\). Thus, \(14 e^{-0.23 t} < 0.05\) leads to \(e^{-0.23 t} < \frac{0.05}{14}\). Taking natural logs, \(-0.23 t < \ln\left(\frac{0.05}{14}\right)\), \(t > \frac{\ln\left(\frac{0.05}{14}\right)}{-0.23}\). Calculate \(t\) to find \(t \approx 19.8\), meaning the population will exceed 300 in the 20th year (2022).
04

Maximum Population

The carrying capacity of the logistic function is \(315\), indicating this is the maximum population the herd can eventually reach, assuming no constraints arise other than those described in the logistic model.
05

Function Concavity

To determine when the graph is concave up or down, find the second derivative \(N''(t)\). In the interval where \(N''(t) > 0\), the graph is concave up, and where \(N''(t) < 0\), it is concave down. Concave up initially indicates an accelerating growth as resources are abundant, while concave down, as it nears its asymptote, signifies decelerating growth and nearing carrying capacity. The inflection point where concavity changes can be found, typically near the midpoint of the curve, but calculation complexity is high without computational tools.

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Key Concepts

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

Population Dynamics
Population dynamics is a field of study that focuses on how populations of organisms, like the buffalo herd in Waterton Lakes National Park, change over time. The logistic growth model is a common method for studying such dynamics. It takes into account how a population like the buffalo herd might grow rapidly when resources are plentiful, and slow down once those resources become limited. Logistic growth assumes that there is a maximum population size, known as the carrying capacity, which the environment can sustainably support. In our exercise, this carrying capacity is 315 buffalo. This type of growth pattern is sigmoidal or "S" shaped, initially showing rapid population growth, slowing down as it approaches the carrying capacity.

Important elements of population dynamics include:
  • Initial population size, which was 21 buffalo in this example at the start of 2002.
  • The growth rate, which affects how quickly the population increases. In this case, a rate of 0.23 is used in the formula.
  • The carrying capacity, which is the maximum population size the environment can support, identified as 315 buffalo.
Understanding these elements helps predict and manage wildlife populations for conservation and resource management.
Concavity
Concavity in a graph helps us understand how rapid or slow the growth of a population is at any given time. When a graph is concave up, it has a U-shape, meaning the rate of population increase is accelerating. Conversely, when the graph is concave down, it has an upside-down U-shape, indicating the rate of growth is decelerating.

For logistic growth models like the one described, the graph of population against time is initially concave up, reflecting an increasing growth rate due to the abundance of resources. As time progresses and the population grows closer to the carrying capacity, the graph becomes concave down, showing a decreasing growth rate. This change is due to limited resources as more individuals compete for the same resources.

The transition from concave up to concave down occurs at an inflection point, which represents the point of diminishing returns where the growth rate starts to decelerate. This understanding of concavity in logistic growth models is critical for predicting how quickly a population will reach its carrying capacity and is a vital tool for those managing animal populations.
Carrying Capacity
Carrying capacity is a central concept in the study of population dynamics, particularly in logistic growth models. It represents the maximum number of individuals that an environment can sustain indefinitely without significant degradation. In the exercise, the carrying capacity is given as 315 buffalo for the herd in the national park. This number is derived from the logistic growth formula, highlighting that the environment can provide enough resources to support up to 315 buffalo without exhaustion.

Carrying capacity is influenced by several factors, including:
  • Availability of resources like food, water, and space.
  • Competition between individuals within the population.
  • Predation, disease, and other environmental pressures.
Exceeding the carrying capacity may result in negative effects such as starvation, disease, and habitat damage, leading to a decline in the population. Thus, the concept of carrying capacity is crucial for wildlife management practices, helping in setting conservation policies and determining sustainable population sizes for different ecosystems.
Graphing Functions
Graphing functions is an essential skill for visualizing and understanding the behavior of relationships in mathematics, like the population of buffalo over time in this exercise. Creating a graph based on the logistic growth formula given allows us to see how the population changes year by year.
  • Start by setting the time variable \(t\) from 0 to 30, where \(t=0\) is the starting point in 2002.
  • Calculate \(N(t)\) for various points in time to get specific population values.
  • Plot these values to create a continuous curve showing population change from 2002 to 2032.
The shape of a logistic growth curve is particularly informative. Initially, it rises steeply, indicating rapid population growth. As it moves past the inflection point, the curve begins to flatten out as it nears the carrying capacity, showing slower growth rates.

Understanding how to graph functions and interpret their shapes equips us with the tools to both predict and illustrate population trends, a key aspect of ecological study and management.

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Most popular questions from this chapter

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