Problem 53
Question
In a case study by Taylor et al. (1980) in which the maximal rate of oxygen consumption (in \(\mathrm{ml} \mathrm{s}^{-1}\) ) for nine species of wild African mammals was plotted against body mass (in \(\mathrm{kg}\) ) on a log-log plot, it was found that the data points fall on a straight line with slope approximately equal to \(0.8 .\) Find a differential equation that relates maximal oxygen consumption to body mass.
Step-by-Step Solution
Verified Answer
The differential equation is \(\frac{dy}{dx} = 0.8a x^{-0.2}\).
1Step 1: Introduction to Log-Log Plots
A log-log plot is a type of plot where both axes are on a logarithmic scale. When data falls on a straight line on this type of plot, it often means there is a power-law relationship between the variables. In this question, the variables are maximal oxygen consumption (\(y\)) and body mass (\(x\)).
2Step 2: Understanding the Power Law Relationship
A linear relationship on a log-log plot suggests the power law relation \(y = ax^b\). The given slope \(b\) is 0.8, indicating that the formula describing the relationship between the maximal rate of oxygen consumption and body mass can be expressed as \(y = ax^{0.8}\).
3Step 3: Formulating the Differential Equation
To find a differential equation, differentiate the power law relation \(y = ax^{0.8}\) with respect to \(x\). The differential form is acquired by applying the derivative: \(\frac{dy}{dx} = 0.8ax^{-0.2}\).
4Step 4: Rearrange to Standard Differential Equation
Multiply both sides by \(x^{0.2}\) to remove the negative exponent on the right-hand side: \(x^{0.2}\frac{dy}{dx} = 0.8a\). This can be rewritten as \(\frac{dy}{dx} = 0.8a x^{-0.2}\) to form the differential equation that describes how maximal oxygen consumption changes with body mass.
Key Concepts
Log-Log PlotsPower Law RelationshipUnderstanding Oxygen Consumption
Log-Log Plots
A log-log plot is a visual tool that transforms both the x-axis and the y-axis to a logarithmic scale. This type of plot is particularly useful when investigating relationships where the data spans several orders of magnitude, such as in biological cases, economics, and physics.
Logarithmic scales help in linearizing exponential relationships, making it easier to discern patterns that may not be obvious in a standard plot.
When data is plotted on a log-log plot and appears as a straight line, it serves as an indicator that the variables are related through a power law. This means the relationship between the two variables can be described by an equation of the form \(y = ax^b\), where \(a\) is a constant and \(b\) is the slope of the line on the log-log plot.
Logarithmic scales help in linearizing exponential relationships, making it easier to discern patterns that may not be obvious in a standard plot.
When data is plotted on a log-log plot and appears as a straight line, it serves as an indicator that the variables are related through a power law. This means the relationship between the two variables can be described by an equation of the form \(y = ax^b\), where \(a\) is a constant and \(b\) is the slope of the line on the log-log plot.
- Logarithmic scales transform multiplicative processes into additive processes.
- Straight lines on a log-log plot reveal power law relationships.
Power Law Relationship
A power law relationship describes a functional connection between two quantities, where one quantity varies as a power of another. Mathematically, this is expressed as \(y = ax^b\).
In the context of the exercise, maximal oxygen consumption \(y\) and body mass \(x\) follow a power law relationship which on a log-log plot forms a straight line with slope \(b = 0.8\).
In the context of the exercise, maximal oxygen consumption \(y\) and body mass \(x\) follow a power law relationship which on a log-log plot forms a straight line with slope \(b = 0.8\).
- \(y = ax^b\) represents the power law equation, where \(b\) is a critical exponent.
- The slope of the line represents how rapidly \(y\) changes with respect to \(x\), based on this power relationship.
Understanding Oxygen Consumption
Oxygen consumption is a critical measure in physiology, reflecting how living organisms use energy and sustain their biological processes. Specifically, maximal oxygen consumption is an important parameter indicating an organism's aerobic capacity, often associated with physical fitness in animals and humans.
In the exercise, maximal oxygen consumption is scrutinized in correlation to body mass across various mammals using a log-log plot, which reveals a linear power law with a slope of 0.8.
In the exercise, maximal oxygen consumption is scrutinized in correlation to body mass across various mammals using a log-log plot, which reveals a linear power law with a slope of 0.8.
- This suggests that larger animals consume oxygen at a rate that increases predictably with body size, though not directly proportional.
- The exponent 0.8 indicates a sublinear scaling; the increase in oxygen consumption is less than the increase in body mass.
Other exercises in this chapter
Problem 51
Solve each differential equation with the given initial condition. \(\frac{d r}{d t}=r e^{-t}\), with \(r_{0}=1\) if \(t_{0}=0\)
View solution Problem 52
Solve each differential equation with the given initial condition. \(\frac{d x}{d y}=\frac{1}{2} \frac{x}{y}\), with \(x_{0}=2\) if \(y_{0}=3\)
View solution Problem 54
Consider the following differential equation, which is important in population genetics: $$a(x) g(x)-\frac{1}{2} \frac{d}{d x}[b(x) g(x)]=0$$ Here, \(b(x)>0\).
View solution Problem 55
When phosphorus content in Daphnia was plotted against phosphorus content of its algal food on a log-log plot, a straight line with slope \(1 / 7.7\) resulted.
View solution