91Ó°ÊÓ

Ecologists have studied the relationship between the number \(S\) of species of a given taxonomic group within a given habitat (often an island) and the area \(A\) of the habitat. \({ }^{34}\) They have discovered a consistent relationship: Over similar habitats, \(S\) is approximately a power function of \(A\), and for islands the powers fall within the range \(0.2\) to \(0.4\). Table \(5.9\) gives, for some islands in the West Indies, the area in square miles and the number of species of amphibians and reptiles. a. Find a formula that models \(S\) as a power function of \(A\). b. Is the graph of \(S\) against \(A\) concave up or concave down? Explain in practical terms what your answer means. c. The species-area relation for the West Indies islands can be expressed as a rule of thumb: If one island is 10 times larger than another, then it will have _ times as many species. Use the homogeneity property of the power function you found in part a to fill in the blank in this rule of thumb. d. In general, if the species-area relation for a group of islands is given by a power function, the relation can be expressed as a rule of thumb: If one island is 10 times larger than another, then it will have _ times as many species. How would you fill in the blank? (Hint: The answer depends only on the power.) $$ \begin{array}{|l|r|c|} \hline \text { Island } & \text { Area } A & \begin{array}{c} \text { Number } S \\ \text { of species } \end{array} \\ \hline \text { Cuba } & 44,000 & 76 \\ \hline \text { Hispaniola } & 29,000 & 84 \\ \hline \text { Jamaica } & 4200 & 39 \\ \hline \text { Puerto Rico } & 3500 & 40 \\ \hline \text { Montserrat } & 40 & 9 \\ \hline \text { Saba } & 5 & 5 \\ \hline \end{array} $$

Step by step solution

01

Understanding the Power Function Model

The problem states that the number of species \(S\) is related to the area \(A\) of the habitat by a power function. This can be expressed as \(S = kA^n\), where \(k\) is a constant and \(n\) is the power. Our task is to determine this relationship using the given data. By calculating \(n\) and \(k\), we find an approximate formula.

02

Estimating the Power "n"

Using the data for larger islands like Cuba and Hispaniola where \(n\) tends to stabilize, estimate a suitable \(n\) within the range \(0.2\) to \(0.4\). Observing the data, we hypothesize \(n = 0.3\) as a reasonable estimate since it consistently fits the given range and test whether it helps satisfy the equation \(S = kA^n\).

03

Calculating the Constant "k"

Using the chosen \(n = 0.3\), pick a data pair, say for Cuba (\(A = 44,000\), \(S = 76\)), to find \(k\). Substitute these values into the power function equation: \(76 = k \cdot 44000^{0.3}\). Solving for \(k\) gives \(k \approx 1.74\).

04

Evaluating the Graph's Curvature

The graph of a power function \(S = kA^n\) where \(0 < n < 1\) is always concave down as the curve's slope decreases with an increase in \(A\). This means adding area results in a diminishing increase in species.

05

Applying the 10-Fold Increase Rule

According to the power property, if one island is 10 times larger, the number of species is \(10^n\) times more. With \(n = 0.3\), it becomes \(10^{0.3} \approx 2\). Thus, a 10-fold increase in area results in approximately twice as many species.

06

Generalizing the Rule of Thumb

With any power function, the expression \(10^n\) represents the increase in species with a tenfold increase in area. Hence for \(n\), the number of species will be \(10^n\) times more.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Species-Area Relationship

The species-area relationship is a concept in ecology that explores how the number of species (biodiversity) varies with the area of a habitat. In many ecological studies, scientists have observed that larger habitats tend to host more species. This relationship can often be modeled by a power function, which takes the form \( S = kA^n \), where \( S \) is the number of species, \( A \) is the area, \( k \) is a constant, and \( n \) is the exponent, typically between 0.2 and 0.4 for islands. For example, in the West Indies islands, data suggest that different islands have varying numbers of species largely depending on their size. Larger islands like Cuba and Hispaniola with greater area have more species compared to smaller islands such as Montserrat and Saba. This pattern is crucial for ecologists to understand, as it helps in conservation efforts and biodiversity planning.

Concavity in Graphs

Concavity is a term used to describe the way a curve bends. In the context of our species-area relationship, the graph of the function \( S = kA^n \), where \( 0 < n < 1 \), exhibits concave down curvature. This means the slope of the graph decreases as the area \( A \) increases. In practical terms, a concave down graph implies that while increasing the habitat area results in more species, each additional increment of area provides a lesser increase in the number of species than the previous one. Imagine you're expanding a habitat; the first few square miles add a significant number of species, but as you continue to add more area, the additional impact on species count diminishes. This understanding is vital for making informed decisions in habitat conservation and expansion. Recognizing the concavity helps us realize that there will be diminishing returns in species richness with continuous enlargement of habitat, underscoring the importance of optimizing available space.

Homogeneity Property

The homogeneity property of power functions reveals insightful patterns about scaling in ecological systems. If our power function \( S = kA^n \) is applied to scenarios where one habitat is a scaled version of another, this property helps predict changes in species richness. For any scalar multiplication \( c \) of the area (e.g., \( cA \)), the number of species becomes \( S = k(cA)^n = c^n \, kA^n \). This property effectively captures the idea that changes in size affect outcomes predictably by a multiply of the size factor raised to the power. In the species-area context, for example, if one island is 10 times larger than another, the species number increases by \( 10^n \). If \( n = 0.3 \) as hypothesized, then \( 10^{0.3} \approx 2 \). Thus, a tenfold increase in area results in the number of species approximately doubling. The homogeneity property is critical in generalizing the "rule of thumb" for species richness across ecological studies and can aid in predicting outcomes as habitats vary in size.

Mathematical Modeling

Mathematical modeling is a tool used in ecology to describe complex biological processes using mathematical equations. The species-area relationship is a prime example of such modeling, where ecological dynamics are expressed through a power function \( S = kA^n \). This model allows ecologists to translate observable data into a mathematical framework, providing predictions and insights into natural patterns. It aids in comprehending the interactions between species diversity and landscapes by simplifying real-world complexities into manageable equations. Models help in exploring hypothetical scenarios, like predicting species richness under different conservation strategies or understanding the effects of habitat loss. They also enable testing hypotheses about the importance of area on biodiversity without requiring exhaustive empirical experimentation. Thus, mathematical modeling serves as an essential bridge between theoretical ecology and practical applications, helping guide informed decision-making in conservation efforts.