Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The population of wildebeest in the Serengeti was decimated by a rinderpest plague in the \(1950 \mathrm{~s}\). In 1961 the Serengeti supported a population of a quarter of a million wildebeest. By 1978 the wildebeest population was \(1.5\) million and by 1991 it had reached 2 million. (Craig Packer Into Africa, Chicago, The University of Chicago Press, 1996 p. \(250 .\) ) Given this data, would you be more inclined to model the growth of the wildebeest population using an exponential growth model or using a logistic growth model? Explain your reasoning.

Step-by-Step Solution

Answer

The logistic growth model would be more appropriate to use as it accounts for environmental limitations that a species encounters as its population approaches the carrying capacity of its ecosystem. The decreasing rate of growth observed in the data supports this.

Step 1: Understand the Concepts of Exponential and Logistic Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. In other words, as the population increases, the rate of growth also increases. Linear growth, on the other hand, refers to steady growth at a constant rate. Logistic growth is S-shaped growth with a period of exponential growth and then the growth slows down due to environmental factors.

Step 2: Interpret the Data

From the given information, the population of wildebeest increased from 250,000 in 1961 to 1,500,000 in 1978 and then to 2,000,000 in 1991. Notice that the rate of increase between 1961 and 1978 suggests a relatively high rate of growth(6 times), then from 1978 to 1991 the growth rate falls (as the population only increased by a third). This changing rate of growth suggests we could be dealing with logistic growth.

Step 3: Make Decision

Given the decreasing rate of growth as shown in the data argued in Step 2, it is more appropriate to use a logistic growth model to describe the population growth of wildebeests in the Serengeti as it likely accounts for environmental limitations to sustained exponential growth.

Key Concepts

Exponential Growth

Exponential growth is a concept where the rate of population growth is directly proportional to the current population size. This means the larger the population, the faster it grows. Exponential growth can be described by the mathematical equation: \[ P(t) = P_0 e^{rt} \] where:

• \( P(t) \) is the population at time \( t \)
• \( P_0 \) is the initial population
• \( r \) is the growth rate
• \( t \) is time

This model is suitable in environments where resources are unlimited, and there are no significant predators or diseases to control the population size. It resembles a J-shaped curve on a graph.
The wildebeest population initially showed signs of exponential growth, especially with the dramatic increase from 250,000 to 1,500,000 between 1961 and 1978. This kind of rapid increase is typical of exponential growth when environmental conditions are ideal.

Logistic Growth

Logistic growth is characterized by an S-shaped curve and occurs when a population's growth rate slows as it approaches the carrying capacity of its environment. This carrying capacity is the maximum population size that the environment can sustain indefinitely. The logistic growth model can be represented by the equation:\[ P(t) = \frac{K}{1 + \left( \frac{K-P_0}{P_0} \right) e^{-rt}} \]where:

• \( P(t) \) is the population at time \( t \)
• \( P_0 \) is the initial population
• \( r \) is the growth rate
• \( K \) is the carrying capacity
• \( t \) is time

In the case of the wildebeest, the growth rate slowed between 1978 and 1991, indicating the approach of the carrying capacity. This shift in the growth pattern from a rapid increase to a slower rate supports the logistic model hypothesis. The environmental limitations of the Serengeti, such as space, food, and water, likely played a role in this growth pattern.

Environmental Limitations

Environmental limitations refer to factors in an ecosystem that can limit the growth, abundance, or distribution of an organism's population. These factors include:

• Resources: Availability of food, water, and shelter.
• Predators: Natural predators that can control population size.
• Disease: Outbreaks that can decimate populations.
• Space: Physical space to live and reproduce.

When a population like the wildebeest initially increases rapidly, it may seem that resources are plentiful. However, as the population grows, these environmental limitations begin to have a noticeable impact, resulting in a slowed growth rate as seen in logistic growth models. In the Serengeti, these factors helped regulate the wildebeest population as it progressed from exponential growth towards a stabilized state in a more realistic representation of nature's balance.