Problem 67 - Calculus for Biology and Medicine

Question

Species Diversity Chapter 3 introduced the Shannon diversity index for the diversity of a habitat. If a population contains two different species of animals, and they are present in the proportions \(p\) and \(1-p\) respectively, then the Shannon diversity index for the population is given by the formula: $$ H(p)=-p \ln p-(1-p) \ln (1-p), \quad 0

Step-by-Step Solution

Answer

The function is continuous at \(p=0\) and \(p=1\); maximum diversity occurs at \(p=1/2\).

Step 1: Understand the limits at endpoints

To show the function is continuous at the endpoints, we need to find the limits of the Shannon index, \(H(p)\), as \(p\) approaches 0 from the right and 1 from the left. We are given that \(H(0) = 0\) and \(H(1) = 0\). Therefore, if we can show that: \[ \lim_{p \to 0^+} H(p) = 0 \] and \[ \lim_{p \to 1^-} H(p) = 0 \]then \(H(p)\) is continuous at these points.

Step 2: Apply L'Hopital's Rule for \(\lim_{p \to 0^+}\)

Consider \( \lim_{p \to 0^+} H(p) = \lim_{p \to 0^+} [-p \ln p - (1-p) \ln(1-p)] \).As \(p \to 0^+\), \(-p \ln p\) is an indeterminate form of type \(0 \times (-\infty)\). Rewriting \(-p \ln p\) as \(\frac{-\ln p}{1/p}\), we can apply L'Hopital's Rule. The derivative of the numerator, \(-\ln p\), is \(-1/p\), and the derivative of the denominator, \(1/p\), is \(-1/p^2\). Applying L'Hopital's Rule:\[\lim_{p \to 0^+} \frac{-\ln p}{1/p} = \lim_{p \to 0^+} \frac{-1/p}{-1/p^2} = \lim_{p \to 0^+} p = 0\]Additionally, since \( (1-p) \ln(1-p) \approx 0\) as \(p \to 0^+\), the overall limit is zero:

Step 3: Apply L'Hopital's Rule for \(\lim_{p \to 1^-}\)

Consider \( \lim_{p \to 1^-} H(p) = \lim_{p \to 1^-} [-p \ln p - (1-p) \ln(1-p)] \).As \(p \to 1^-, (1-p) \ln(1-p)\) is also the form \(0 \times (-\infty)\). Rewriting \((1-p) \ln(1-p)\) as \(\frac{\ln(1-p)}{1/(1-p)}\), and applying L'Hopital's Rule as done previously:\[\lim_{p \to 1^-} \frac{\ln(1-p)}{1/(1-p)} = \lim_{p \to 1^-} (1-p) = 0\]Since \(-p \ln p\) also approaches zero as \(p \to 1^-\), the overall limit is zero:

Step 4: Find the derivative of \(H(p)\)

To find where the maximum occurs, calculate the derivative of \(H(p)\) with respect to \(p\):\[ H'(p) = \frac{d}{dp} \left[-p \ln p - (1-p) \ln(1-p)\right] \]Differentiating term by term:\[ H'(p) = -\ln p - 1 - \ln(1-p) + 1 = -\ln p + \ln(1-p) \]Set the derivative equal to zero to find critical points:\[-\ln p + \ln(1-p) = 0 \]Simplifying gives: \(-\ln p = -\ln(1-p)\). Thus, \(p = 1-p\), which solves to:

Step 5: Solve for critical point

From the equation \(p = 1-p\), solving gives:\[ 2p = 1 \]\[ p = \frac{1}{2} \]This is the only critical point in the interval \(0 < p < 1\).

Step 6: Verify maximum at critical point

Substitute \(p = \frac{1}{2}\) to find the maximum Shannon index value:\[ H\left(\frac{1}{2}\right) = -\frac{1}{2} \ln\left(\frac{1}{2}\right) - \frac{1}{2} \ln\left(\frac{1}{2}\right) = -\ln\left(\frac{1}{2}\right) = \ln(2) \]\(H\left(\frac{1}{2}\right) = \ln(2)\) is the maximum value, confirming that the maximum diversity occurs at \(p = \frac{1}{2}\).

Key Concepts

Species Diversity

Species diversity is an essential concept in ecology, describing the variety within a habitat. It takes into account the number of different species and their relative abundance. A higher species diversity means that there are many different species and a balanced number among them, creating a stable and resilient ecosystem.
The Shannon diversity index is a popular method for measuring species diversity. It accounts for both richness (the number of species) and evenness (how evenly the species are distributed). This index can quantify the uncertainty in predicting the species of a randomly selected individual.

• If the index value is higher, it indicates greater diversity.
• A value of 0 means that the habitat is entirely dominated by one species.

Understanding species diversity can help ecologists assess the health of ecosystems and guide conservation efforts.

Continuity at Endpoints

In calculus, continuity is a property of a function where the graph can be drawn without lifting the pencil. When evaluating the continuity at endpoints, especially for functions defined on a closed interval, we check the limits at these points.
For the Shannon diversity index, ensuring that the function is continuous at the endpoints involves showing:

• extreme precision when approaching 0 and 1, right and left respectively.
• the values of the function agree with the endpoint limits.

To establish continuity at the endpoints, the limit of the Shannon index as it approaches both 0 and 1 should equate with the actual value at those endpoints, which is zero in this case.

L'Hopital's Rule

L'Hopital's Rule is a valuable tool in calculus used to find limits of indeterminate forms like \(0/0\) or \(\infty/\infty\). It works by differentiating the numerator and denominator separately, then taking the limit.
This rule is particularly useful when dealing with the Shannon diversity index because both components of the function approach indeterminate forms as \(p\) approaches 0 or 1.

• When \(p\) is near 0, the term \(-p \ln p\) needs L'Hopital's Rule to find its limit.
• Similarly, near 1, \((1-p) \ln(1-p)\) requires the rule to resolve its indeterminate nature.

By applying L'Hopital's Rule, we manage to confirm the continuity of the Shannon diversity index at its endpoints, reinforcing its mathematical validity.

Derivative Analysis

Analyzing derivatives is a fundamental part of calculus, often used to find where functions reach minimum or maximum values—known as critical points. In the context of the Shannon diversity index:

• The derivative, \(H'(p)\), helps identify where the diversity index reaches its peak.
• This involves calculating \(H'(p)\), then solving \(H'(p) = 0\) to find the critical point.

For the Shannon diversity index, solving this leads to discovering that maximum diversity occurs at \(p = \frac{1}{2}\). This means having equal proportions of two species results in the highest diversity measure according to the index. Derivative analysis thus provides critical insights into the function’s behavior across its domain.