Problem 45 - Calculus for Biology and Medicine

Question

Diversity Index The diversity of a population that contains two species can be measured by the Gini-Simpson diversity index. If the fraction of organisms from species 1 is \(p\), then the diversity is given by: $$ H(p)=2 p(1-p) $$ Use the product rule to calculate \(H^{\prime}(p) .\) For what value of \(p\) does \(H^{\prime}(p)=0 ?\)

Step-by-Step Solution

Answer

\(H'(p) = 2 - 4p\). \(H'(p) = 0\) when \(p = \frac{1}{2}\).

Step 1: Understand the Function

The given diversity function is \( H(p) = 2p(1-p) \). The goal is to find the derivative \( H'(p) \) using the product rule.

Step 2: Apply the Product Rule

The product rule states that if you have a function \( u(p)v(p) \), then its derivative \( (uv)'(p) = u'(p)v(p) + u(p)v'(p) \). Identify \( u(p) = 2p \) and \( v(p) = (1-p) \).

Step 3: Differentiate Each Part

Differentiate \( u(p) = 2p \), which gives \( u'(p) = 2 \). Differentiate \( v(p) = (1-p) \), giving \( v'(p) = -1 \).

Step 4: Use the Product Rule Formula

Substitute \( u(p) \), \( v(p) \), \( u'(p) \), and \( v'(p) \) into the product rule formula: \( H'(p) = u'(p)v(p) + u(p)v'(p) = 2(1-p) + 2p(-1) \).

Step 5: Simplify the Derivative

Simplify the expression for \( H'(p) \): \( H'(p) = 2 - 2p - 2p = 2 - 4p \). The derivative is \( H'(p) = 2 - 4p \).

Step 6: Solve for When \( H'(p) = 0 \)

Set \( H'(p) = 0 \) to find critical points: \( 2 - 4p = 0 \). Solve for \( p \): \( 4p = 2 \) leads to \( p = \frac{1}{2} \).

Key Concepts

Gini-Simpson Index

The Gini-Simpson diversity index is a mathematical formula used to measure biodiversity within a population. It's particularly useful when analyzing a community that consists of two distinct species. If you denote the proportion of species 1 as \( p \), then the formula to calculate the diversity index becomes \( H(p) = 2p(1-p) \). This equation represents the probability that two randomly selected individuals from the population belong to different species.

This index has a value between 0 and 1. An index of 0 indicates no diversity, meaning all individuals belong to the same species. In contrast, an index closer to 1 suggests high diversity, implying a more even mix of species. When studying such measures, it is crucial to note that the degree of biodiversity gives insights into the health and stability of an ecosystem.

In this context, understanding how the diversity changes with respect to \( p \) is essential and requires calculus tools such as differentiation. Differentiation helps us find the rate at which the diversity changes as the proportion of species changes.

Product Rule

The product rule is a critical tool in calculus when dealing with functions that are products of two or more separate functions. If you have a function defined as the product of two differentiable functions \( u(p) \) and \( v(p) \), the product rule allows you to find its derivative efficiently. The rule is expressed as:

• \( (uv)'(p) = u'(p)v(p) + u(p)v'(p) \)

In our exercise, we need to differentiate the function \( H(p) = 2p(1-p) \).

First, we identify the components:

• Let \( u(p) = 2p \).
• Let \( v(p) = (1-p) \).

The next step is to calculate the derivative of these components separately. Differentiating \( u(p) \) gives \( u'(p) = 2 \), and differentiating \( v(p) \) gives \( v'(p) = -1 \).

Now substitute these derivatives back into the product rule formula to find \( H'(p) \). This systematic approach shows how two simpler derivatives can be combined to solve more complex derivative problems.

Derivative Calculation

Differentiating a function helps us understand how the function's output changes as its input changes. After identifying the components of \( H(p) = 2p(1-p) \) for the product rule, we use the calculated derivatives \( u'(p) = 2 \) and \( v'(p) = -1 \). Substituting these back into the product rule gives an expression for the derivative of the diversity index:

\[H'(p) = u'(p)v(p) + u(p)v'(p) = 2(1-p) + 2p(-1)\]

Expanding and simplifying this expression is the key to finding the rate of change of the diversity index. The calculation simplifies to:

• \( H'(p) = 2 - 2p - 2p \)
• which further reduces to \( H'(p) = 2 - 4p \)

This derivative function \( H'(p) = 2 - 4p \) tells us how the diversity index changes as the proportion \( p \) of species 1 varies.

To fully grasp these concepts, it's important to frequently practice differentiation using the product rule and other calculus tools.

Critical Points

In calculus, critical points of a function are where the function's derivative is zero, which may indicate potential maximum, minimum, or inflection points. Identifying these points helps us understand key features about the function’s behavior. In the context of the diversity index, solving \( H'(p) = 2 - 4p = 0 \) allows us to find such a critical point.

By setting the derivative equal to zero, we solve:

• \( 2 - 4p = 0 \)
• Rearrange to find \( 4p = 2 \)
• Solve for \( p \) to get \( p = \frac{1}{2} \)

This solution tells us that at \( p = \frac{1}{2} \), the rate of change of our diversity index reaches a pivotal point. At \( p = \frac{1}{2} \), the diversity is potentially balanced between the two species, reflecting the maximum diversity in this setup where each species equally contributes to the population.

A thorough understanding of critical points aids significantly in optimizing and analyzing different mathematical and real-world scenarios.