Problem 84 - Calculus for Biology and Medicine

Question

Suppose that you follow a population over time. When you plot your data on a semilog plot, a straight line with slope \(0.03\) results. Furthermore, assume that the population size at time 0 was 20\. If \(N(t)\) denotes the population size at time \(t\), what function best describes the population size at time \(t\) ?

Step-by-Step Solution

Answer

The population size at time \( t \) is described by \( N(t) = 20 e^{0.03t} \).

Step 1: Understand the Semilog Plot Interpretation

In a semilog plot, the vertical axis is logarithmic while the horizontal axis remains linear. Thus, if the plot of the population size as a function of time results in a straight line, the relationship between the variables is exponential. The slope of the line, in this case, indicates the growth rate of the population.

Step 2: Recognize the Exponential Growth Model

In exponential growth, the population size is given by the function \( N(t) = N_0 e^{kt} \), where \( N_0 \) is the initial population size and \( k \) is the growth rate. From the problem, we have \( N_0 = 20 \) and the slope of the log plot, which represents \( k \), is given as 0.03.

Step 3: Substitute Known Values Into the Exponential Function

Substitute the values \( N_0 = 20 \) and \( k = 0.03 \) into the general exponential function, yielding the specific function \( N(t) = 20 e^{0.03t} \). This function describes how the population size changes over time.

Key Concepts

Semilog Plot

A semilog plot is an invaluable tool for visualizing exponential relationships. It uses a logarithmic scale on one axis, usually the vertical one, and a linear scale on the other axis, typically time on the horizontal axis.
In the context of population growth, if a semilog plot produces a straight line, it suggests that the population is undergoing exponential growth. This is because the logarithm of an exponential function results in a linear function. Therefore, in our scenario, the straight line with a slope of 0.03 tells us the rate at which the population is growing.
Key benefits of the semilog plot include:

• Ease of identification of exponential growth patterns.
• The ability to interpret the slope as the growth rate directly.
• Simplification of complex data into understandable linear trends.

Being familiar with semilog plots can make it much easier to quickly identify and understand growth trends in population data or any other set of data that grows exponentially over time.

Population Growth Model

The Population Growth Model is a mathematical representation of how populations increase in size. One of the simplest and most common models is the exponential growth model, which fits well for populations with constant rates of growth.
This model is expressed as:
\[ N(t) = N_0 e^{kt} \]
Where:

• \(N(t)\) is the population size at time \(t\).
• \(N_0\) is the initial population size.
• \(k\) is the growth rate.

Through this exercise, we observed that a population with an initial size of 20 and a growth rate of 0.03 can be modeled by this function.
In practical terms, the exponential growth model is ideal when:

• The population experiences unlimited resources.
• Competing factors such as disease or predation are minimal.
• The growth is measured over a relatively short time span where limiting factors don't drastically alter the growth behavior.

Understanding this model helps researchers and students to predict how a population might change over different time periods when conditions are favorable.

Logarithmic Scale

A logarithmic scale is used to represent data that covers a wide range of values. Specifically, one step on the scale represents a tenfold change, which makes it particularly useful for visualizing exponential growth on semilog plots.
When using logarithmic scales:

• Exponential growth appears as a straight line.
• It compresses large numerical ranges into more manageable visual distances.
• Proportional growth rates are easier to interpret across broader scales.

The logarithmic scale shines in various fields where exponential growth occurs, such as in population biology, financial markets, and even sound intensity. By transforming the exponential data into linear data, it simplifies the interpretation and analysis of trends that would be overwhelming or obscured on a standard linear graph.
Overall, using a logarithmic scale is a powerful method to deal with data spanning several orders of magnitude, keeping the analysis straightforward and insightful.