Problem 37 - Calculus

Question

The population data from another experiment on yeast by the ecologist G. F. Gause is given. $$\begin{array}{l|c|c|c|c|c|c|c}\hline \text { Time (hours) } & 0 & 13 & 32 & 56 & 77 & 101 & 125 \\\\\hline \text { Yeast pop } & 1.00 & 1.70 & 2.73 & 4.87 & 5.67 & 5.80 & 5.83 \\\\\hline\end{array}$$ (a) Do you think the population is growing exponentially or logistically? Give reasons for your answer. (b) Estimate the value of \(k\) (for either model) from the first two pieces of data. If you chose a logistic model in part (a), estimate the carrying capacity, \(L,\) from the data. (c) Sketch the data and the approximate growth curve given by the parameters you estimated.

Step-by-Step Solution

Answer

(a) Logistic growth; population levels off. (b) Growth rate \(k \approx 0.041\), carrying capacity \(L \approx 5.83\). (c) Data aligns with a logistic growth curve sketch.

Step 1: Analyze the Pattern of Growth

Observing the data, initially, the yeast population grows rapidly but then levels off as time progresses. Here, at time 0, the population is 1.00, rapidly increasing and showing diminishing growth after certain periods (e.g., at time 101 and 125 the values are 5.80 and 5.83, respectively). This pattern suggests a logistic growth.

Step 2: Estimate the Growth Rate from Data

The data at time 0 is 1.00 and at time 13 is 1.70. For exponential growth, the formula is \(P(t) = P_0 e^{kt}\). Solve for \(k\) using the population values at time 0 and 13:\[1.70 = 1.00 \, e^{13k} \e^{13k} = 1.7 \k \approx \frac{\ln(1.7)}{13} \approx 0.041 \\] This suggests \(k \approx 0.041\), assuming an initial examination consistent with the logistic model hypothesis where initial exponential growth occurs.

Step 3: Estimate the Carrying Capacity

The population levels off at around 5.8 from time 101 to 125, indicating the carrying capacity \(L\) for a logistic model is close to this value. Based on the data, the carrying capacity \(L\) can be estimated at approximately 5.83.

Step 4: Sketch the Data and Growth Curve

Plot the given data points on a graph with time on the x-axis and yeast population on the y-axis. Then, plot a logistic curve starting at 1.00, increasing rapidly initially, and leveling off near the carrying capacity, 5.83. The curve should follow the approximate population path to visualize the logistic growth behavior.

Key Concepts

Population Dynamics

Population dynamics refers to the way populations change over time. In the context of ecological studies, this involves tracking the number of individuals in a population and how they grow, reproduce, or die. It can be influenced by several factors such as food availability, predation, disease, and environmental conditions.
The yeast population data provided in the exercise is a great example to understand population dynamics. Here, the ecologist G. F. Gause's data shows changes in yeast population over time. Initially, there's a rapid increase, followed by a stabilization phase indicating the population reaches a steady state.
Population dynamics is an essential concept for understanding how populations interact with their environment and the consequences of these interactions over time. This knowledge is crucial for fields like conservation biology, resource management, and understanding human impacts on ecosystems.

• Rapid increase: This indicates a period where resources are abundant, and each individual is producing more offspring.
• Stabilization: As resources become limited, the population growth slows or stops.

Observing these patterns allows scientists to make predictions about future population changes and understand the underlying factors influencing population size.

Exponential Growth

Exponential growth describes a situation where the growth rate of a population is proportional to its current size. This means the bigger the population, the faster it grows, leading to a J-shaped curve. However, exponential growth cannot continue indefinitely because resources are limited.
In the provided exercise, the yeast population initially exhibits exponential growth. The population increases from a size of 1.00 at time 0 to 1.70 at time 13, following an exponential growth curve. This is determined by solving for the exponential growth constant "k" from the data using the formula:
\[P(t) = P_0 e^{kt}\]
Using the initial value and time interval data, the growth constant comes out as approximately 0.041. This suggests that the growth is initially fast due to ample resources.
Real-world populations often start with exponential growth when resources are plentiful. However, as resources dwindle, the growth rate slows down, indicating a shift from exponential to logistic growth.

Carrying Capacity

Carrying capacity is a crucial concept in population dynamics and logistic growth. It represents the maximum number of individuals in a population that an environment can sustain over the long term without degradation.
In the yeast population example, the carrying capacity is observed as the population levels off over time. From the data, we see the population size stabilizes at approximately 5.83, suggesting this is the carrying capacity. This leveling off occurs because of limitations in factors like resources or space, which prevent further growth.
The concept of carrying capacity is highlight important in understanding ecosystem balance. It helps scientists and ecologists predict the maximum sustainable population size for each species in a given environment.

• Limitation of Resources: As the population grows, resources become scarce, which affects growth.
• Environmental Capacity: Environment conditions can also limit the capacity for growth, such as space and nutrient availability.

Understanding carrying capacity is essential for managing wildlife populations, conserving biodiversity, and ensuring sustainable ecosystems.