Problem 40 - Calculus for Biology and Medicine

Question

(Adapted from Reiss, 1989 ) Suppose that the rate at which body weight \(W\) changes with age \(x\) is $$ \frac{d W}{d x} \propto W^{a} $$ where \(a\) is some species-specific positive constant. (a) The relative growth rate (percentage weight gained per unit of time ) is defined as $$ \frac{1}{W} \frac{d W}{d x} $$ What is the relationship between the relative growth rate and body weight? For which values of \(a\) is the relative growth rate increasing, and for which values is it decreasing? (b) As fish grow larger, their weight increases each day but the relative growth rate decreases. If the rate of growth is described by (5.7), what values of \(a\) can you exclude on the basis of your results in (a)? Explain how the increase in percentage weight (relative to the current body weight) differs for juvenile fish and for adult fish.

Step-by-Step Solution

Answer

For (a), relative growth rate decreases for \( a < 1 \). For (b), exclude \( a \geq 1 \).

Step 1: Understand the Proportionality

We are given that \( \frac{dW}{dx} \propto W^a \). This means that the derivative of weight with respect to age is proportional to \( W^a \), where \( a \) is a positive constant.

Step 2: Expressing Proportionality as an Equation

We can express the proportionality as \( \frac{dW}{dx} = kW^a \), where \( k \) is some constant of proportionality.

Step 3: Define the Relative Growth Rate

The relative growth rate is given by \( \frac{1}{W} \frac{dW}{dx} \). Substituting \( \frac{dW}{dx} = kW^a \), we get the relative growth rate as \( kW^{a-1} \).

Step 4: Analyze the Relative Growth Rate

The relative growth rate \( kW^{a-1} \) will increase with \( W \) if \( a-1 > 0 \), i.e., \( a > 1 \). It will decrease with \( W \) if \( a-1 < 0 \), i.e., \( a < 1 \).

Step 5: Determine Relevant Values of 'a'

From (b), we know that as fish grow larger, the relative growth rate decreases. This implies that the value of \( a \) must be less than 1. Thus, we can exclude values of \( a \) greater than or equal to 1 based on this model.

Step 6: Explain Growth Differences

When \( a < 1 \), the relative growth rate \( kW^{a-1} \) decreases as weight increases, explaining that juvenile fish have a higher relative growth rate compared to adult fish due to their smaller size. Adult fish's slower relative growth is due to their larger size.

Key Concepts

Relative Growth Rate

The relative growth rate is a crucial concept in understanding how an organism's weight changes in relation to its current size. It's defined as the percentage change in weight per unit time, given by \( \frac{1}{W} \frac{dW}{dx} \). This formula calculates how much weight an organism gains in relation to its existing weight.

By substituting the expression \( \frac{dW}{dx} = kW^a \) into the relative growth rate formula, we find it simplifies to \( kW^{a-1} \).
This expression shows that the relative growth rate is clearly dependent on the value of \( a \).

• If \( a > 1 \), the term \( W^{a-1} \) becomes larger as \( W \) increases, meaning the relative growth rate increases with weight.
• If \( a < 1 \), the exponential term decreases as \( W \) increases, indicating that the relative growth rate decreases with increasing weight.
• When \( a = 1 \), the growth rate remains constant as it does not change with weight.

This tells us that smaller organisms or those at earlier growth stages will typically exhibit high relative growth rates, while larger or more mature organisms will show lower rates.

Differential Equations

Differential equations play a pivotal role in modeling biological growth processes. A differential equation is an equation that relates a function to its derivatives, showing how a quantity changes over time.

In the exercise, the differential equation \( \frac{dW}{dx} = kW^a \) describes how weight \( W \) changes with age \( x \). Here, the constant \( k \) represents the rate at which this change occurs depending on the species.

This type of equation captures the dynamic nature of biological systems, where changes aren't always constant but depend on current states like body weight.

• By integrating these equations, we can predict changes over time, a process crucial in fields such as epidemiology, population dynamics, and ecology.
• Solutions to differential equations can provide insights into the long-term behavior of the system, like predicting eventual size or lifespan.

Thus, differential equations provide a powerful tool for understanding complex biological processes, like growth, in a quantitative manner.

Proportionality in Growth Models

Proportionality in growth models helps us understand how growth factors relate to the size of biological entities. When we say \( \frac{dW}{dx} \propto W^a \), it means the rate of weight change is proportional to the weight raised to some power \( a \).

This concept of proportionality is fundamental in growth models because:

• It reflects biological realities where organisms grow until the energy burned equals energy absorbed.
• It allows us to generalize growth dynamics across different species or environments just by changing \( a \) or \( k \).
• Through proportionality constants \( k \) and \( a \), we can adjust models to better fit empirical data from experimental observations.

Proportionality also highlights why certain organisms grow rapidly when small and why growth rates diminish as organisms become larger. Therefore, understanding how proportionality works in growth models helps biologists develop accurate predictions on organism development and energy needs.