Problem 36
Question
In the 1930 s, the Soviet ecologist G. F. Gause \(^{22}\) studied the population growth of yeast. Fit a logistic curve, \(d P / d t=k P(1-P / L),\) to his data below using the method outlined below. $$\begin{array}{l|c|c|c|c|c|c|c}\hline \text { Time (hours) } & 0 & 10 & 18 & 23 & 34 & 42 & 47 \\ \hline \text { Yeast pop } & 0.37 & 8.87 & 10.66 & 12.50 & 13.27 & 12.87 & 12.70 \\\\\hline\end{array}$$(a) Plot the data and use it to estimate (by eye) the carrying capacity, \(L\) (b) Use the first two pieces of data in the table and your value for \(L\) to estimate \(k\) (c) On the same axes as the data points, use your values for \(k\) and \(L\) to sketch the solution curve \(P=\frac{L}{1+A e^{-k t}} \quad\) where \(\quad A=\frac{L-P_{0}}{P_{0}}\)
Step-by-Step Solution
Verified Answer
Carrying capacity \( L \) is approximately 13.5, and grow rate \( k \) can be estimated by fitting the logistic model to starting data.
1Step 1: Plot the Data and Estimate Carrying Capacity (L)
First, plot the given yeast population data against time to visualize the logistic growth. Observing the data, the yeast population seems to initially grow and then level out slightly above 13. It appears that beyond the time of 34 hours, the population doesn't significantly increase, suggesting the population has reached its carrying capacity. By eye, estimate the carrying capacity, \( L \), as approximately 13.5.
2Step 2: Use Initial Data to Estimate Growth Rate (k)
With the carrying capacity \( L \) estimated as 13.5, and using the initial population data points (Time 0 and Time 10), substitute into the logistic growth formula: \( P = \frac{L}{1 + Ae^{-kt}} \). Solve for \( k \) using the initial condition \( P_0 = 0.37 \) at \( t = 0 \) to find \( A = \frac{13.5 - 0.37}{0.37} \). Next, substitute \( t = 10 \) with \( P = 8.87 \) into the formula and solve for \( k \).
3Step 3: Obtain Expression for Solution Curve
Substitute \( L = 13.5 \), the calculated \( A \), and \( k \) into the logistic growth solution formula \( P = \frac{L}{1 + Ae^{-kt}} \). This provides an equation specific to the yeast population data.
4Step 4: Sketch the Logistic Curve
Plot the logistic curve obtained from Step 3 on the same graph with the original data points. Ensure the curve begins close to \( P_0 = 0.37 \) at \( t = 0 \), rises logging following the data trend, and approaches the carrying capacity \( L = 13.5 \) as time increases, matching closely to the given data points.
Key Concepts
Population GrowthCarrying CapacityGrowth Rate EstimationLogistic Curve Fitting
Population Growth
Population growth refers to the change in the size of a population over time.
This change can be influenced by various factors such as birth rates, death rates, immigration, and emigration.
In the case of yeast studied by G. F. Gause, we are observing how the size of the yeast population increases over time until it stabilizes.
Population growth is often represented through mathematical models that help predict and understand real-world scenarios.
The logistic growth model is one such model, which is particularly useful for populations growing in environments with limited resources.
It is expressed by the differential equation \( \frac{dP}{dt} = kP(1 - \frac{P}{L}) \), where \(k\) is the growth rate and \(L\) is the carrying capacity.
In this model, at the beginning, the population grows rapidly because resources are abundant.
As the population approaches its carrying capacity, the growth rate decreases, and eventually, the population size levels out.
This pattern results in an S-shaped curve when population size is plotted against time.
This change can be influenced by various factors such as birth rates, death rates, immigration, and emigration.
In the case of yeast studied by G. F. Gause, we are observing how the size of the yeast population increases over time until it stabilizes.
Population growth is often represented through mathematical models that help predict and understand real-world scenarios.
The logistic growth model is one such model, which is particularly useful for populations growing in environments with limited resources.
It is expressed by the differential equation \( \frac{dP}{dt} = kP(1 - \frac{P}{L}) \), where \(k\) is the growth rate and \(L\) is the carrying capacity.
In this model, at the beginning, the population grows rapidly because resources are abundant.
As the population approaches its carrying capacity, the growth rate decreases, and eventually, the population size levels out.
This pattern results in an S-shaped curve when population size is plotted against time.
Carrying Capacity
Carrying capacity is a central concept in the study of population growth.
It represents the maximum population size that an environment can sustain indefinitely.
This capacity is dependent upon the availability of resources such as food, water, and living space.
In Gause's yeast experiment, the carrying capacity \(L\) represents the maximum number of yeast cells that can be supported by their environment.
From the data, it can be observed that the yeast population levels off at around 13.5 units, suggesting this environment’s carrying capacity.
Understanding the carrying capacity is crucial as it helps scientists and researchers predict how a population will grow and when it might stabilize.
For any given population, if the carrying capacity is exceeded, it could lead to resource depletion, causing the population to decline.
Therefore, maintaining balance with the carrying capacity helps ensure the health and sustainability of a population.
It represents the maximum population size that an environment can sustain indefinitely.
This capacity is dependent upon the availability of resources such as food, water, and living space.
In Gause's yeast experiment, the carrying capacity \(L\) represents the maximum number of yeast cells that can be supported by their environment.
From the data, it can be observed that the yeast population levels off at around 13.5 units, suggesting this environment’s carrying capacity.
Understanding the carrying capacity is crucial as it helps scientists and researchers predict how a population will grow and when it might stabilize.
For any given population, if the carrying capacity is exceeded, it could lead to resource depletion, causing the population to decline.
Therefore, maintaining balance with the carrying capacity helps ensure the health and sustainability of a population.
Growth Rate Estimation
Estimating the growth rate of a population is an important task in understanding its dynamics over time.
In logistic growth, the growth rate parameter \(k\) dictates how quickly a population approaches its carrying capacity.
For estimating \(k\) using data, we use the initial and subsequent population sizes at known time intervals.
For example, using Gause's data, we start with a yeast population size \(P_0 = 0.37\) at \(t = 0\).
At \(t = 10\), the population is 8.87.
This information allows us to use the logistic growth formula \(P = \frac{L}{1 + Ae^{-kt}}\) to solve for \(k\).
Given \(L\) (carrying capacity) previously estimated and \(A = \frac{L - P_0}{P_0}\), we solve for \(k\) with these values and observed data points.
Accurate estimation of \(k\) is vital as it affects predictions made by mathematical models regarding population behavior, and it helps tailor conservation and management strategies.
In logistic growth, the growth rate parameter \(k\) dictates how quickly a population approaches its carrying capacity.
For estimating \(k\) using data, we use the initial and subsequent population sizes at known time intervals.
For example, using Gause's data, we start with a yeast population size \(P_0 = 0.37\) at \(t = 0\).
At \(t = 10\), the population is 8.87.
This information allows us to use the logistic growth formula \(P = \frac{L}{1 + Ae^{-kt}}\) to solve for \(k\).
Given \(L\) (carrying capacity) previously estimated and \(A = \frac{L - P_0}{P_0}\), we solve for \(k\) with these values and observed data points.
Accurate estimation of \(k\) is vital as it affects predictions made by mathematical models regarding population behavior, and it helps tailor conservation and management strategies.
Logistic Curve Fitting
Logistic curve fitting is a process used to model how populations grow over time, following a logistic growth pattern.
This involves fitting the logistic growth curve to actual data by determining the best values for parameters such as \(L\) (carrying capacity) and \(k\) (growth rate).
Once these parameters are estimated, they are used in the logistic growth equation \(P = \frac{L}{1 + Ae^{-kt}}\) to describe the entire growth process accurately.
This logistics curve starts close to the initial population size, rises smoothly, and levels off at the carrying capacity, creating an "S" shape.
The fitted logistic curve gives insights into the growth dynamics of the population and can be visually compared with actual data.
This makes it a powerful tool for scientists and researchers, as, by matching the curve closely with observed data, we are able to confirm the validity of the estimated growth parameters.
It's also useful in predicting future population trends, which is essential for planning resource allocation and understanding ecological impacts.
This involves fitting the logistic growth curve to actual data by determining the best values for parameters such as \(L\) (carrying capacity) and \(k\) (growth rate).
Once these parameters are estimated, they are used in the logistic growth equation \(P = \frac{L}{1 + Ae^{-kt}}\) to describe the entire growth process accurately.
This logistics curve starts close to the initial population size, rises smoothly, and levels off at the carrying capacity, creating an "S" shape.
The fitted logistic curve gives insights into the growth dynamics of the population and can be visually compared with actual data.
This makes it a powerful tool for scientists and researchers, as, by matching the curve closely with observed data, we are able to confirm the validity of the estimated growth parameters.
It's also useful in predicting future population trends, which is essential for planning resource allocation and understanding ecological impacts.
Other exercises in this chapter
Problem 36
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