Problem 29 - Calculus for Biology and Medicine

Question

In Example 4 we discussed the GiniSimpson diversity index, which we defined to be the probability that two individuals randomly chosen from a population of two cell types are genetically different. We gave a formula for calculating Gini-Simpson diversity index, \(H\) : $$ H(p)=2 p(1-p) \quad p \in[0,1]. $$ Here \(p\) is the proportion of red individuals in the population. Conversely, we could ask what is the probability that the two individuals are genetically identical. Call this probability \(I(p)\). It is given by $$ I(p)=2 p^{2}-2 p+1. $$ (a) The function \(I(p)\) is known as the Simpson index. Explain why the domain of \(I\) is \(p \in[0,1]\). (b) Use a graphing calculator to plot \(I(p)\). (c) What is the range \(I([0,1])\) ?

Step-by-Step Solution

Answer

(a) The domain of \( I \) is \([0,1]\) because \( p \) is a proportion. (b) Graph \( I(p) \) over \([0,1]\) using a calculator. (c) The range is \([0,1]\).

Step 1: Understand the Domain

The function \( I(p) = 2p^2 - 2p + 1 \) involves \( p \), representing the proportion of one type in a population, so it must be between 0 and 1 inclusive. This reflects a probability measure, capturing the realistic constraints that proportions can neither be negative nor exceed the whole of the population.

Step 2: Confirm the Domain

Since \( p \) represents a proportion and by definition must obey \( 0 \leq p \leq 1 \), the domain of \( I \) is \([0, 1]\). This captures all possible proportions of the red individuals in the population.

Step 3: Plot the Function

Using a graphing calculator, plot the quadratic function \( I(p) = 2p^2 - 2p + 1 \) over the domain \([0,1]\). This will visually display how \( I(p) \) changes as \( p \) varies from 0 to 1.

Step 4: Determine the Range

The quadratic function resembles an upward-opening parabola. At \( p = 0 \), \( I(0) = 1 \), and at \( p = 1 \), \( I(1) = 1 \). The vertex form of a parabola \( y = ax^2 + bx + c \) can help find extreme values, noting here that the parabola's symmetry will show it's minimum or maximum clearly. Hence, the range is \([0, 1]\).

Key Concepts

Simpson index

The Simpson index is a measure in ecology to quantify the diversity of a population. In this context, it is presented as a probability function, \[ I(p) = 2p^2 - 2p + 1 \] where \( p \) represents the proportion of a particular type within a population.It's crucial to understand the nature of \( I(p) \).

• The domain of \( I(p) \) is \([0,1]\). This is because \( p \) represents a proportion, and thus cannot be less than 0 or greater than 1.
• The function is quadratic, opening upwards, which informs us about the function's behavior as \( p \) changes within this interval.

When you plot \( I(p) \) on a graph, you will see a parabola that shows probabilities for genetic similarity, reflecting the likelihood that two randomly chosen individuals are genetically identical within the defined proportions. By understanding the parabolic nature, you observe that the minimum probability happens when the genetic makeup is entirely diverse.

genetic diversity

Genetic diversity within a population refers to the variety of genetic characteristics found among individuals in that group. This concept is important because higher genetic diversity reduces the likelihood of population decline due to diseases or environmental changes.Having a higher genetic diversity means there are more genetic differences between individuals, increasing the population's ability to adapt to changing conditions. In terms of mathematical representation

• The Gini-Simpson index, defined by \( H(p) = 2p(1-p) \), measures the probability that two randomly chosen individuals are genetically different.
• Diverse populations are less vulnerable to threats that could affect genetically similar individuals.

Therefore, the Gini-Simpson diversity index helps us quantify this important aspect of population genetics, directly correlating to higher health and resilience of populations.

quadratic function

A quadratic function is a polynomial function of degree two, generally represented by the form \[ ax^2 + bx + c \]where \( a, b, \) and \( c \) are constants, and \( a eq 0 \).In this exercise, the function \[ I(p) = 2p^2 - 2p + 1 \]is a perfect example of a quadratic function.

• The coefficient \( 2 \) attached to \( p^2 \) determines the shape and direction of the parabola. Here, the parabola opens upwards.
• The quadratic function's vertex can indicate a minimum or maximum value within the function's range.

Quadratic functions like this are not just abstract mathematics but have real-world applications in predicting behaviors and must be properly understood to interpret and apply resulting data effectively.