Problem 47 - Calculus for Biology and Medicine

Question

Chewing in Mammals Druzinsky (1993) showed that chewing frequency (i.e., the number of mass an animal chews in one minute) is proportional to its body mass raised to the power of \(-0.128\), that is, if \(M\) is the body mass and \(c\) the chewing frequency. then, $$ c=k M^{-0.128} $$ for some positive constant \(k\). (a) Assume that the body mass of a particular mammal is given by a formula \(M(t)=1+2 \sqrt{t} .\) Calculate \(d c / d t\). (b) Druzinsky also found that the jaw length of the animal, \(L\), is proportional to its body mass raised to the \(0.312\) power, i.e., \(L=r M^{0.312}\) where \(r\) is another positive constant.

Step-by-Step Solution

Answer

(a) \(\frac{dc}{dt} = -0.128k(1 + 2\sqrt{t})^{-1.128} \times \frac{1}{\sqrt{t}}\). (b) Jaw length \(L\) is expressed as \(L = rM^{0.312}\).

Step 1: Expressing Chewing Frequency as a Function of Time

Given the formula for body mass as a function of time, \(M(t) = 1 + 2\sqrt{t}\), the chewing frequency is \(c(t) = k (1 + 2\sqrt{t})^{-0.128}\). This expression allows us to examine the rate of change of chewing frequency over time.

Step 2: Calculating the Derivative \(\frac{dc}{dt}\)

To find the derivative \(\frac{dc}{dt}\), apply the chain rule on \(c(t) = k (1 + 2\sqrt{t})^{-0.128}\). First, find \(\frac{d}{dt}(1 + 2\sqrt{t})^{-0.128}\) by using the chain rule:\[\frac{dc}{dt} = -0.128k(1 + 2\sqrt{t})^{-0.128 - 1} \times \frac{d}{dt}(1 + 2\sqrt{t})\]Next, compute \(\frac{d}{dt}(1 + 2\sqrt{t}) = \frac{2}{\sqrt{t}}\). So,\[\frac{dc}{dt} = -0.128k(1 + 2\sqrt{t})^{-1.128} \times \frac{1}{\sqrt{t}}\]

Step 3: Expressing Jaw Length as a Function of Body Mass

The formula for jaw length in terms of body mass \(M\) is \(L = rM^{0.312}\). This indicates that jaw length is a power function of body mass.

Key Concepts

Chewing Frequency

Chewing frequency in mammals refers to how many times an animal chews in a minute. Interestingly, it does not increase linearly with body size. Instead, it follows a specific mathematical relationship involving body mass. Researchers like Druzinsky have shown that this frequency is in fact proportional to the body mass raised to the power of \(-0.128\). This means, larger animals do not chew proportionally more than smaller ones. Consider the formula:

• \(c = k M^{-0.128}\)

where \(c\) is the chewing frequency and \(M\) is the body mass. Here, \(k\) is a constant specific to each animal or species.This model allows us to predict how changes in body mass could influence an animal's chewing rate. By integrating body mass over time, we can gain further insights into how this frequency might evolve as the animal grows.

Body Mass

Body mass, in the context of biomathematics, is a crucial factor influencing various biological traits. It is often expressed in a function relative to time, like in our exercise:

• \(M(t) = 1 + 2\sqrt{t}\)

This shows how body mass changes as the animal ages. The formula suggests that the body mass is not constant but increases as the animal grows older.Understanding the relationship between body mass and chewing frequency helps us comprehend the adaptive strategies animals develop for feeding. It also highlights the importance of body mass in determining other biological characteristics such as energy needs and life expectancy.

Derivative Calculation

Derivatives in biomathematics often help us understand how biological functions change over time. Here, we are tasked with finding \(\frac{dc}{dt}\), the rate of change of chewing frequency with respect to time. This involves applying the chain rule in calculus, particularly because we are dealing with a composite function of time.To find the derivative:

• Start with the chewing function: \(c(t) = k (1 + 2\sqrt{t})^{-0.128}\)
• Apply the chain rule: \[\frac{dc}{dt} = -0.128k(1 + 2\sqrt{t})^{-1.128} \times \frac{1}{\sqrt{t}}\]

This calculation tells us how quickly the rate of chewing frequency changes as time progresses, providing insights into the overall development pattern of the mammal.

Jaw Length

Jaw length is another biological parameter that is closely linked to body mass. According to Druzinsky's findings, jaw length is proportional to the body mass raised to the power of \(0.312\). This scaling reveals how structural characteristics like jaw size grow with respect to body mass in a non-linear fashion. Using the formula:

• \(L = rM^{0.312}\)

where \(L\) is the jaw length, \(r\) is a constant, and \(M\) is body mass, we can understand how larger body mass often results in proportionally larger jaws. This is vital for understanding feeding strategies, as the jaw size must support the necessary mechanical actions required for the animal's dietary needs.