Problem 17 - Applied Calculus

Question

The number of species of lizards, \(N\), found on an island off Baja California is proportional to the fourth root of the area, \(A\), of the island \({ }^{66}\) Write a formula for \(N\) as a function of \(A\). Graph this function. Is it increasing or decreasing? Is the graph concave up or concave down? What does this tell you about lizards and island area?

Step-by-Step Solution

Answer

The function \(N(A) = k \cdot A^{1/4}\) is increasing and concave down, indicating larger areas support more species, but the increase slows with larger areas.

Step 1: Identify Proportional Relationship

The problem states that the number of species of lizards, \(N\), is proportional to the fourth root of the area, \(A\), of the island. This means \(N = k \cdot A^{1/4}\) where \(k\) is a constant of proportionality.

Step 2: Formulate the Function

Given the relationship, the function can be written as \(N(A) = k \cdot A^{1/4}\). This is the formula expressing \(N\) as a function of the area \(A\).

Step 3: Behavior of the Function

Since the function \(N(A) = k \cdot A^{1/4}\) involves a positive power of \(A\) greater than zero, as \(A\) increases, \(N\) will also increase. Therefore, the function is increasing.

Step 4: Concavity of the Function

To determine concavity, we need the second derivative of \(N\). First, find the first derivative: \(N'(A) = \frac{k}{4} \cdot A^{-3/4}\). Then, the second derivative is \(N''(A) = -\frac{3k}{16} \cdot A^{-7/4}\). Since the second derivative is negative, the function is concave down.

Step 5: Interpret the Graph

The function \(N(A) = k \cdot A^{1/4}\) is increasing and concave down. An increasing function means larger islands support more lizard species, and because the graph is concave down, the rate of increase in species number decreases as island area grows.

Key Concepts

Lizard Species

Lizards are a group of reptiles with a rich diversity of species. In island ecosystems, the number of lizard species tends to rely on the size of the island. This is due to several reasons:

• Larger islands often provide more diverse habitats and resources.
• More space means less competition, allowing more species to coexist.
• A greater variety of microclimates may exist, supporting varied lizard species.

The formula for the number of lizard species, \(N\), can be given by \(N = k \cdot A^{1/4}\), where \(A\) is the area of the island and \(k\) is a constant specific to the environment. This mathematical relationship helps in understanding how biodiversity can be sustained based on land area. The notion that larger islands host more species is a key aspect of island biogeography, indicating that space is crucial for sustaining species diversity.

Island Biogeography

Island biogeography is a field in ecology that explores how species diversity and island characteristics interrelate. It looks at factors like island size, distance from the mainland, and climate.

• Island Size: Larger islands usually support more species because they provide more habitats and resources.
• Isolation: The distance an island is from the mainland can affect the species it harbors, influencing both the ease for species to arrive and their survival chances. Islands closer to the mainland generally have more species.
  

The study of island biogeography also involves understanding the natural balance of colonization and extinction rates. It provides insights into how species adapt and evolve in island environments, reinforcing the idea that the area itself is crucial in sustaining biological diversity. The proportional relationship between the number of lizard species and the fourth root of the area underscores the importance of island area in the development of biodiversity.

Concavity of Functions

Understanding the concavity of a function is fundamental in determining the nature of its graph. Concavity tells us how a rate of change itself is changing, which is crucial for interpreting various behaviors in mathematical and real-world contexts.

• Concave Up: When a function is concave up, it means that the rate of increase of the function is accelerating. Graphically, this resembles a cup or a smiley face.
• Concave Down: Conversely, concavity down indicates that the increase in the function's value is decelerating. This results in a graph that looks like a frown.
  

For the lizard species function \(N(A) = k \cdot A^{1/4}\), the second derivative \(N''(A) = -\frac{3k}{16} \cdot A^{-7/4}\) is negative, which signifies the graph is concave down. This means as the island area increases, the growth rate of the lizard population becomes slower. Understanding concavity allows us to predict and explain trends in species populations as influenced by variables like island size.