Problem 37 - Calculus and its applications

Question

It has been shown that the home range, in hectares, of a carnivorous mammal weighing \(w\) grams can be approximated by $$ H(w)=0.11 w^{1.36} $$ (Source: Based on information in Emlen, J. M., Ecology: An Evolutionary Approach, p. \(216,\) Reading, MA: Addison-Wesley, 1973 ; and Harestad, A. S., and Bunnel, F. L., "Home Range and Body Weight-A Reevaluation," Ecology, Vol. 60, No. 2, pp. 405-418.) a) Find the average rate at which a carnivorous mammal's home range increases as the animal's weight grows from \(500 \mathrm{~g}\) to \(700 \mathrm{~g}\). b) Find \(\frac{H(300)-H(200)}{300-200}\). What does this rate represent?

Step-by-Step Solution

Answer

(a) The average rate is 0.4766 hectares/gram. (b) The rate 0.1858 hectares/gram indicates growth per gram from 200g to 300g.

Step 1: Define the Average Rate of Change Formula

To find the average rate of change of a function over a specific interval, use the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] where \( f(x) \) is the given function, and \( a \) and \( b \) are the endpoints of the interval.

Step 2: Calculate H(700) and H(500)

Use the formula provided for the home range: \( H(w) = 0.11 w^{1.36} \). First, calculate \( H(700) \): \[ H(700) = 0.11 \times 700^{1.36} \approx 210.95 \] Next, calculate \( H(500) \): \[ H(500) = 0.11 \times 500^{1.36} \approx 115.63 \]

Step 3: Find the Average Rate of Change from 500 to 700 grams

Substitute \( H(700) \) and \( H(500) \) into the average rate of change formula: \[ \frac{H(700) - H(500)}{700 - 500} = \frac{210.95 - 115.63}{200} = \frac{95.32}{200} \approx 0.4766 \] Thus, the average rate at which the home range increases from 500 grams to 700 grams is approximately 0.4766 hectares per gram.

Step 4: Calculate H(300) and H(200)

Calculate \( H(300) \) and \( H(200) \) using the function: \[ H(300) = 0.11 \times 300^{1.36} \approx 35.12 \] \[ H(200) = 0.11 \times 200^{1.36} \approx 16.54 \]

Step 5: Calculate Average Rate of Change from 200 to 300 grams

Use the formula: \[ \frac{H(300) - H(200)}{300 - 200} = \frac{35.12 - 16.54}{100} = \frac{18.58}{100} = 0.1858 \] This rate represents the average increase in home range per gram as the mammal's weight increases from 200 grams to 300 grams.

Key Concepts

Home Range Function

The home range function is a fascinating concept, particularly when applied to carnivorous mammals. In this context, a home range represents the area in which a mammal lives and roams, calculated in hectares. This concept is important for understanding animal behavior, ecology, and habitat requirements. In our exercise, the home range of a carnivorous mammal is described by the function \( H(w) = 0.11 w^{1.36} \), where \( w \) is the mammal's weight in grams. This mathematical model provides insights into how the living area that a mammal requires changes as its body weight increases. The power of \( 1.36 \) in the function suggests a non-linear relationship. This means the home range doesn't increase proportionally with weight, but rather at a different rate, reflective of a more complex ecological balance. This can help in strategizing conservation efforts and managing wildlife as we understand the space needed by these animals based on their size.

Calculus Application

Calculus serves as a powerful tool to assess how particular quantities change with respect to another. An essential calculus concept used in this context is the average rate of change. It's a way to comprehend how functions behave over specific intervals.The average rate of change formula \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] helps us understand the change in the home range between given weights. For the weights ranging from \( 500 \text{ g} \) to \( 700 \text{ g} \), we calculated the average rate of change to be approximately 0.4766 hectares per gram. This means that, on average, for every additional gram in body weight between these intervals, the required home range area increases by approximately 0.4766 hectares. By analyzing these changes, researchers and ecologists can design better wildlife management practices, ensuring enough space is available to support the physiological needs of the animal.

Carnivorous Mammals

Carnivorous mammals are a diverse group characterized by diets primarily consisting of meat. Understanding these creatures' home range requirements is crucial for their conservation. Carnivores like tigers, wolves, and lynxes demand vast areas for hunting and living. Their home range is influenced by factors such as food availability, territory, and mating habits. By assessing how weight affects their roaming area, ecologists and conservationists can predict how shrinking habitats or changing ecosystems might impact their populations. The exercise uses a mathematical model to relate weight with home range size. This model provides crucial insights into the spatial biological needs of carnivorous mammals and assists in ensuring healthy ecosystems by maintaining these animals' requirements. By recognizing these needs, efforts to preserve biodiversity can be more precisely focused, thereby supporting the delicate balance of ecosystems where these powerful hunters play a pivotal role.