Problem 28 - Calculus for Biology and Medicine

Question

Tree Growth Rate Sperry et al. (2012) studied how the growth rates \(G\) of trees depend upon their body mass, \(M\). They argued that \(G=C M^{0.7}\) for some constant \(C .\) As the tree grows, \(M\) changes. (a) Show how \(d G / d t\) is related to \(d M / d t\). (b) Show how the fractional rate of increase of \(G, \frac{1}{G} \frac{d G}{d t}\), is related to the fractional rate of growth \(\frac{1}{M} \frac{d M}{d t}\).

Step-by-Step Solution

Answer

(a) \( \frac{dG}{dt} = 0.7C M^{-0.3} \frac{dM}{dt} \); (b) \( \frac{1}{G} \frac{dG}{dt} = 0.7 \frac{1}{M} \frac{dM}{dt} \).

Step 1: Differentiating Growth Rate with Respect to Time

Given the equation for the growth rate of trees, \( G = C M^{0.7} \), we need to show how \( \frac{dG}{dt} \) is related to \( \frac{dM}{dt} \). First, differentiate \( G \) with respect to time \( t \) using the chain rule. This gives us:\[\frac{dG}{dt} = \frac{d}{dt} (C M^{0.7}) = C \cdot 0.7 \cdot M^{0.7 - 1} \cdot \frac{dM}{dt} = 0.7C M^{-0.3} \frac{dM}{dt}.\]

Step 2: Expression for Fractional Growth Rate of G

Now, we express \( \frac{1}{G} \frac{dG}{dt} \) in terms of \( \frac{1}{M} \frac{dM}{dt} \). Start by using \( \frac{dG}{dt} = 0.7C M^{-0.3} \frac{dM}{dt} \) and the formula for \( G \):\[ G = C M^{0.7}. \] The fractional growth rate becomes:\[ \frac{1}{G} \frac{dG}{dt} = \frac{1}{C M^{0.7}} \cdot 0.7C M^{-0.3} \frac{dM}{dt}. \]

Step 3: Simplifying the Fractional Growth Rate Expression

Simplify the expression for the fractional growth rate:\[\frac{1}{G} \frac{dG}{dt} = 0.7 \cdot \frac{1}{M^{0.7}} \cdot M^{-0.3} \cdot \frac{dM}{dt}.\]Since \( M^{-0.3} \times M^{-0.7} = M^{-1} \), this becomes:\[ \frac{1}{G} \frac{dG}{dt} = 0.7 \frac{1}{M} \frac{dM}{dt}. \]

Step 4: Concluding the Fractional Growth Rate Relationship

Therefore, the fractional rate of increase of \( G \) is directly proportional to the fractional rate of increase of \( M \):\[ \frac{1}{G} \frac{dG}{dt} = 0.7 \frac{1}{M} \frac{dM}{dt}. \]

Key Concepts

Tree Growth Rate

Trees exhibit different rates of growth as they increase in size. Understanding these rates is crucial for studying ecological dynamics and forest management. The growth rate of a tree, denoted as \( G \), often depends on the tree's body mass \( M \). According to Sperry et al. (2012), the relationship between the growth rate \( G \) and mass \( M \) can be described by the equation: \[ G = C M^{0.7} \] Where:

• \( C \) is a constant specific to the species or type of tree studied.
• \( M^{0.7} \) signifies that growth rate scales with mass.

This equation indicates that as the mass of the tree increases, its growth rate does not increase linearly but instead follows a power-law relationship. That is, larger trees grow slower relative to their size, compared to smaller trees.

Differentiation in Biology

Differentiation is a fundamental concept in calculus that helps us understand how things change over time. In the biology of tree growth, differentiation allows us to find the rate of change of growth \( G \) with respect to time \( t \). By differentiating the growth rate equation \( G = C M^{0.7} \) with respect to time, we can explore how the growth rate changes as the mass changes. To differentiate \( G = C M^{0.7} \) with respect to time \( t \), we use the chain rule: \[ \frac{dG}{dt} = \frac{d}{dt}(C M^{0.7}) = 0.7C M^{-0.3} \frac{dM}{dt} \] This expression shows the relationship between the rate of change of growth \( \frac{dG}{dt} \) and the rate of change of mass \( \frac{dM}{dt} \). Essentially, the change in growth over time is a function of both the current mass and the rate at which the mass is changing. Differentiation in this context helps biologists understand dynamic growth patterns and how trees respond to environmental changes.

Fractional Growth Rate

The fractional growth rate provides insight into how fast a tree's growth rate is increasing relative to its current size. This concept is vital for understanding proportional changes and scaling in biological systems. In our context, the fractional growth rate of the tree's growth \( G \) can be expressed as:\[ \frac{1}{G} \frac{dG}{dt} \]Using the equation \( G = C M^{0.7} \) and our previous differentiation results:\[ \frac{1}{G} \frac{dG}{dt} = \frac{1}{C M^{0.7}} \cdot 0.7C M^{-0.3} \frac{dM}{dt} \]This equation simplifies to:\[ \frac{1}{G} \frac{dG}{dt} = 0.7 \frac{1}{M} \frac{dM}{dt} \]The fractional growth rate of \( G \) is directly proportional to the fractional growth rate of \( M \) but scaled by a factor of 0.7. This relationship indicates that for a given rate of mass increase, the proportional growth rate of the tree is only a fraction of that, specifically 70%. These calculations are essential for predicting tree growth responses to environmental factors and assessing forestry practices.