Problem 18 - Calculus for Biology and Medicine

Question

Suppose that you follow the size of a population over time. When you plot the size of the population versus time on a semilog plot (i.e., the horizontal axis, representing time, is on a linear scale, whereas the vertical axis, representing the size of the population, is on a logarithmic scale), you find that your data fit a straight line which intercepts the vertical axis at 1 (on the log scale) and has slope \(-0.43 .\) Find a differential equation that relates the growth rate of the population at time \(t\) to the size of the population at time \(t\).

Step-by-Step Solution

Answer

The differential equation is \( \frac{dP}{dt} = -0.989P \).

Step 1: Understand the Relationship from a Semilog Plot

In a semilog plot where time is on a linear scale and population size is on a logarithmic scale, a straight line indicates an exponential relationship between time and the population. In this case, the line intercepts the log scale at 1, meaning that when time is zero, the log of the population size is 1, which translates to the initial population size being 10 (since \( \log_{10}(10) = 1 \)). The slope of the line represents the rate of exponential growth, which is \(-0.43\).

Step 2: Write the Exponential Growth Model

The exponential relationship can be expressed as: \( \log_{10}(P(t)) = -0.43t + 1 \), where \( P(t) \) is the population at time \( t \). This can be rearranged to express population size in terms of time: \( P(t) = 10^{(-0.43t + 1)} = 10 \cdot 10^{-0.43t} \). Simplifying further using natural logarithms, we obtain \( P(t) = 10e^{-0.989t} \).

Step 3: Differentiate the Exponential Model

To find a differential equation relating the growth rate of the population to its size, take the derivative of the exponential model. Differentiating \( P(t) = 10e^{-0.989t} \) with respect to \( t \) gives \( \frac{dP}{dt} = 10 \cdot (-0.989)e^{-0.989t} \).

Step 4: Relate the Derivative to Population Size

The derivative \( \frac{dP}{dt} = -9.89e^{-0.989t} \) can be rewritten in terms of \( P(t) \) itself. Since \( P(t) = 10e^{-0.989t} \), it follows that \( e^{-0.989t} = \frac{P}{10} \). Therefore, the differential equation becomes \( \frac{dP}{dt} = -0.989P \).

Step 5: Conclude with the Differential Equation

The differential equation that describes the growth rate of the population with respect to the size of the population is \( \frac{dP}{dt} = -0.989P \). This indicates that the population decreases exponentially over time, as expected from the negative slope on the semilog plot.

Key Concepts

Exponential Growth

Exponential growth describes a process where the growth rate of a quantity, such as a population, is proportional to its current size. This means that as the population increases, the rate at which it grows also becomes faster.
Imagine a situation where every member of a population can reproduce, thereby doubling the population in a constant unit of time. This describes a scenario of pure exponential growth.
Mathematically, exponential growth can be modeled by the formula:

• \( P(t) = P_0 e^{rt} \)

where:

• \( P(t) \) is the population size at time \( t \),
• \( P_0 \) is the initial population size,
• \( e \) is the base of the natural logarithm (approximately equal to 2.71828), and
• \( r \) is the growth rate.

A positive \( r \) value indicates growth, while a negative \( r \) suggests a decline.
In our example, the population is decreasing over time because our \( r \) value is negative (\( -0.989 \)). This exemplifies exponential decay, which is just the reverse process of exponential growth.

Population Dynamics

Population dynamics explores how and why the population size changes over time. This field of study encompasses a variety of factors influencing growth and decline, including birth rates, death rates, immigration, and emigration.
When modeling population dynamics, differential equations are often used. These equations relate the change in population size to current size, growth rate, and time. The key to understanding population dynamics is the realization that populations rarely remain static; they typically exhibit influxes and declines due to the mentioned factors.
In the exercise provided, the differential equation \( \frac{dP}{dt} = -0.989P \) gives us insight into how the population is changing over time.
This simple equation tells us:

• \( \frac{dP}{dt} \): The rate of change of the population over time.
• \( -0.989 \) represents the proportionality constant describing how quickly the population size decreases.
• \( P \) or \( P(t) \): The current size of the population at time \( t \).

By observing this model, we can predict future population sizes based on the current observed rate of change.

Semilog Plot

A semilog plot is a valuable visualization tool, especially used when data spans several orders of magnitude, or where exponential growth or decay needs to be analyzed.
On a semilog plot, the horizontal axis is linear, whereas one or both of the vertical axes are logarithmic.
This scaling method helps to equate exponential relationships as straight lines, making them easier to interpret.
In the exercise, plotting the population size on a semilog plot showed a straight line with a negative slope. This slope told us that the population is decreasing exponentially over time, with a specific rate represented by the slope's value.

• The vertical intercept on the log scale corresponds to the initial condition or baseline value of the given dataset.
• The slope indicates the growth rate: positive for growth, negative for decline.

Learning to interpret semilog plots is crucial because they distill complex, non-linear relations into simple, linear trends, providing an efficient way to analyze exponential phenomena. For population studies, such plots become unlockers of insights that might otherwise remain buried in complex calculations.