Problem 73 - Calculus: Early Transcendental Functions

Question

For the logistic equation \(y^{\prime}(t)=k y(M-y),\) show that a graph of \(\frac{1}{y} y^{\prime}\) as a function of \(y\) produces a linear graph. Given the slope \(m\) and intercept \(b\) of this line, explain how to compute the model parameters \(k\) and \(M\)

Step-by-Step Solution

Answer

A graph of \(\frac{1}{y} y^{\prime}\) as a function of \(y\) produces a linear graph, with the model parameters given by the slope and intercept of the line. Specifically, \(k = m\) and \(M = b\), where \(m\) is the slope and \(b\) is the intercept.

Step 1: Simplify \(\frac{1}{y} y^{\prime}\)

From the logistic equation \(y^{\prime}(t)=k y(M-y)\), we get \(\frac{1}{y} y^{\prime} = k(M- y)\). This is the equation we will plot against \(y\). Therefore, \(\frac{1}{y} y^{\prime}\) is equivalent to \(k(M- y)\).

Step 2: Determine Linearity of the Graph

To demonstrate linearity, we have to show that this equation resembles a linear equation, which has the form \(y = mx + b\). Here, comparing \(k(M- y)\) with \(y = mx + b\), we find that \(k\) is the slope \(m\) and \(M\) is the equivalent of the y-coordinate when \(x = 0 (b)\). Hence, this verifies that a graph of \(\frac{1}{y} y^{\prime}\) as a function of \(y\) produces a linear graph.

Step 3: Determine the model parameters \(k\), \(M\)

From step 2, we can see that the slope \(m\) of the linear graph is equivalent to \(k\) and the intercept \(b\) is \(M\). Thus, \(k = m\) and \(M = b\). Hence, the model parameters can be computed directly from the slope and y-intercept of the line.

Key Concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They serve as a cornerstone in modeling the rate of change in various scientific disciplines such as physics, engineering, and biology. In a differential equation, the unknown is a function that depends on one or more variables, and the equation prescribes the rate at which the function changes.

When dealing with logistic growth models, differential equations are especially handy, as they allow us to capture the dynamic process of growth which is affected by the environment's carrying capacity. The logistic equation in the form of \( y' = ky(M-y) \) is an example of such a differential equation, where \( y \) represents the population at time \( t \) and \( y' \) indicates its rate of change over time. Parameter \( k \) is the relative growth rate, and \( M \) is the maximum population size, also known as the carrying capacity, that the environment can sustain.

Understanding differential equations is vital as it allows students to interpret how the change in one variable affects another over time, which is a fundamental concept in understanding complex systems.

Linear Graph

A linear graph is a visual representation of a linear equation, typically in two dimensions. It characterizes a relationship between two variables with a straight line when plotted on a coordinate plane. Each point on the line matches the equation of the line, usually in the form of \( y = mx + b \) where \( m \) represents the slope of the line and \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

In our logistic growth scenario, by plotting \( \frac{1}{y} y' \) versus \( y \) and obtaining a linear graph, it helps to simplify the complexity of the arrangement. By doing so, the equation's visual simplicity enables easier comprehension and computation of model parameters. This linearity is useful because it tells us that the rate of change of the population (per individual), \( \frac{1}{y} y' \) is decreasing linearly as \( y \) approaches the carrying capacity \( M \)—a key characteristic of the logistic model.

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a straight line. It follows the format \( y = mx + b \) with \( m \) indicating the slope and \( b \) representing the y-intercept. The slope \( m \) measures how much \( y \) changes for a unit change in \( x \) — it's the tilt or steepness of the line. The y-intercept \( b \) indicates where the line crosses the y-axis. As \( x \) takes the value of zero, \( y \) will equal \( b \) showing us the starting point of \( y \) on the graph.

When interpreting the logistic growth equation in slope-intercept form, we identify \( k \) as the slope and \( M \) as the y-intercept in the converted linear expression \( \frac{1}{y} y' = k(M - y) \) which simplifies comparison between the logistic equation and a linear equation. By rearranging the equation to align with the slope-intercept form, it becomes more manageable to extract the values of \( k \) and \( M \)—critical parameters describing the growth model.