Problem 30 - Calculus for Biology and Medicine

Question

Allometric Equations Suppose that two quantities, \(y\) and \(x\) are related by a power law: $$ y=k x^{a} $$ where \(k\) and \(a\) are both constants. \(x\) grows with time at a rate \(d x / d t\) (a) Explain why \(\frac{1}{x} \frac{d x}{d t}\) can be thought of as the relative rate of growth of \(x\). (b) Show that the relative rates of growth of \(y\) and \(x\) are related by an equation: $$ \frac{1}{y} \frac{d y}{d t}=\frac{a}{x} \frac{d x}{d t} $$

Step-by-Step Solution

Answer

(a) It's the normalized rate of change of \(x\). (b) The relative growth rate of \(y\) is \(a\) times that of \(x\): \(\frac{1}{y} \frac{d y}{d t} = \frac{a}{x} \frac{d x}{d t}\).

Step 1: Interpret the Relative Rate of Growth

The expression \( \frac{1}{x} \frac{d x}{d t} \) represents the relative rate of growth of \( x \) because it measures how \( x \) changes over time relative to its current size. It's the rate of change of \( x \) (\( \frac{d x}{d t} \)) normalized by \( x \) itself, illustrating the proportion by which \( x \) grows over time.

Step 2: Differentiate the Allometric Equation

Start with the given power law \( y = kx^a \). To find the growth rate of \( y \), differentiate \( y \) with respect to time \( t \). Using the chain rule, we have:\[ \frac{d y}{d t} = \frac{d}{d t} (kx^a) = k a x^{a-1} \frac{d x}{d t} \]

Step 3: Calculate Relative Growth Rate of y

We express the relative growth rate of \( y \) in a similar form to that of \( x \). This is accomplished by dividing \( \frac{d y}{d t} \) by \( y \):\[ \frac{1}{y} \frac{d y}{d t} = \frac{1}{kx^a} (k a x^{a-1} \frac{d x}{d t}) \]Cancelling terms:\[ = \frac{a}{x} \frac{d x}{d t} \]

Step 4: Conclude the Relationship

Thus, the expression \( \frac{1}{y} \frac{d y}{d t} = \frac{a}{x} \frac{d x}{d t} \) shows that the relative rate of growth of \( y \) is \( a \) times the relative rate of growth of \( x \). Therefore, as \( x \) grows, \( y \) grows at a rate proportional to \( x \)'s growth multiplied by the constant \( a \).

Key Concepts

Power Law

Power laws are mathematical relationships where one quantity varies as a power of another. In the context of allometric equations, a power law describes the relationship between two quantities, like how "y" is related to "x" using the formula \(y = kx^a\). Both "k" and "a" are constants here. Generally, power laws reveal how changes in one quantity affect another.

• "k" is a constant multiplier that scales the relationship.
• "a" determines how the dependent variable "y" scales with changes in "x".

Power laws can model a wide range of natural phenomena, from biology to economics, and in this exercise, it helps in modeling how growth rates of different variables are interconnected.

Relative Rate of Growth

Relative rate of growth measures how a quantity changes concerning its size. The formula \(\frac{1}{x} \frac{dx}{dt}\) simplifies the rate at which "x" changes over time. This rate is divided by "x" to show the proportion of change relative to its current value.

• It gives an insightful look into the dynamics of growth, avoiding absolute changes.
• This concept is crucial in judging efficiency, growth dynamics, and other performance metrics of systems.

In simpler terms, relative rate of growth allows us to understand not just the change but the significance of this change in relation to the original size of "x".

Chain Rule Differentiation

The chain rule is a core principle in calculus, used to find the derivative of compositions of functions. In the context of the power law equation, to differentiate \(y = kx^a\) with respect to time "t", the chain rule is utilised. This method provides a way to find how the rate of change of "y" depends on "x".

• First, take the derivative of the outer function \(kx^a\) with respect to "x".
• Multiply this by the derivative of "x" with respect to time \(\frac{dx}{dt}\).

This allows us to express the acceleration or growth rate of "y" based on that of "x", showing their interconnected change over time.

Proportional Growth

Proportional growth describes scenarios where changes in one quantity cause corresponding changes in another. Within the equation \(\frac{1}{y} \frac{dy}{dt} = \frac{a}{x} \frac{dx}{dt}\) from the original exercise, "y" grows at a rate that is scaled by the constant "a" multiplied by the rate of "x".

• The constant "a" influences how strongly "y" responds to shifts in "x".
• It indicates direct proportionality but with a specific multiplier which is critical in growth systems.

Proportional growth models help articulate clear relationships between variables, showing how they scale together when examined through their relative and differential changes.