Chapter 58, Problem 2COQ

Summary Introduction

To determine: The Shannon diversity index of the given four communities.

Introduction: Shannon diversity index is a tool used by ecologists to study the species diversity of a community. This allows ecologists to correlate the species richness of an area with the relative abundance of each species.

Explanation of Solution

The Shannon diversity index of Community 1 may be calculated as follows:

The formula to be used is as follows:

${\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum {\text{p}}_{\text{i}}\text{ln}\left({\text{p}}_{\text{i}}\right)}$

The values given in the question are as follows:

$\begin{array}{c}{\text{p}}_1\text{ = 90/100}\\ {\text{p}}_2\text{ = 10/100}\end{array}$

Substituting the values in the given formula, the Shannon diversity index of Community 1 can be calculated as follows:

$\begin{array}{c}{\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum \left({\text{p}}_{\text{1}}{\text{ln (p}}_{\text{1}}{\text{) + p}}_{\text{2}}{\text{ln (p}}_{\text{2}}\text{)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{0}\text{.9 ln (0}\text{.9) + 0}\text{.1 ln (0}\text{.1)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{(-0}\text{.09) + (-0}\text{.23)}\right)}\\ =\text{-(-0}\text{.325)}\\ \text{=0}\text{.325}\end{array}$

Therefore, the value of the Shannon diversity index of Community 1 is 0.325.

The Shannon diversity index of Community 2 may be calculated as follows:

The formula to be used is as follows:

${\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum {\text{p}}_{\text{i}}\text{ln}\left({\text{p}}_{\text{i}}\right)}$

The values given in the question are as follows:

$\begin{array}{c}{\text{p}}_1\text{ = 50/100}\\ {\text{p}}_2\text{ = 50/100}\end{array}$

Substituting the values in the given formula, the Shannon diversity index of Community 2 can be calculated as follows:

$\begin{array}{c}{\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum \left({\text{p}}_{\text{1}}{\text{ln (p}}_{\text{1}}{\text{) + p}}_{\text{2}}{\text{ln (p}}_{\text{2}}\text{)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{0}\text{.5 ln (0}\text{.5) + 0}\text{.5 ln (0}\text{.5)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{(-0}\text{.346) + (-0}\text{.346)}\right)}\\ =\text{-(-0}\text{.693)}\\ \text{=0}\text{.693}\end{array}$

Therefore, the value of the Shannon diversity index of Community 2 is 0.693.

The Shannon diversity index of Community 3 may be calculated as follows:

The formula to be used is as follows:

${\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum {\text{p}}_{\text{i}}\text{ln}\left({\text{p}}_{\text{i}}\right)}$

The values given in the question are as follows:

$\begin{array}{c}{\text{p}}_1\text{ = 80/100}\\ {\text{p}}_2\text{ = 10/100}\\ {\text{p}}_3\text{ = 10/100}\end{array}$

Substituting the values in the given formula, the Shannon diversity index of Community 3 can be calculated as follows:

$\begin{array}{c}{\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum \left({\text{p}}_{\text{1}}{\text{ln (p}}_{\text{1}}{\text{) + p}}_{\text{2}}{\text{ln (p}}_{\text{2}}{\text{) + p}}_3{\text{ln (p}}_3\text{)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{0}\text{.8 ln (0}\text{.8) + 0}\text{.1 ln (0}\text{.1)+ 0}\text{.1 ln (0}\text{.1)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{(-0}\text{.178) + (-0}\text{.23) + (-0}\text{.23)}\right)}\\ =\text{-(-0}\text{.638)}\\ \text{=0}\text{.638}\end{array}$

Therefore, the value of the Shannon diversity index of Community 3 is 0.638.

The Shannon diversity index of Community 4 may be calculated as follows:

The formula to be used is as follows:

${\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum {\text{p}}_{\text{i}}\text{ln}\left({\text{p}}_{\text{i}}\right)}$

The values given in the question are as follows:

$\begin{array}{c}{\text{p}}_1\text{ = 33}\text{.3/99}\text{.9}\\ {\text{p}}_2\text{ = 33}\text{.3/99}\text{.9}\\ {\text{p}}_3\text{ = 33}\text{.3/99}\text{.9}\end{array}$

Substituting the values in the given formula, the Shannon diversity index of Community 4 can be calculated as follows:

$\begin{array}{c}{\text{H}}_{\text{s}}\text{= -}{\displaystyle \sum \left({\text{p}}_{\text{1}}{\text{ln (p}}_{\text{1}}{\text{) + p}}_{\text{2}}{\text{ln (p}}_{\text{2}}{\text{) + p}}_3{\text{ln (p}}_3\text{)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{0}\text{.333 ln (0}\text{.333) + 0}\text{.333 ln (0}\text{.333)+ 0}\text{.333 ln (0}\text{.333)}\right)}\\ =\text{-}{\displaystyle \sum \left(\text{(-0}\text{.366) + (-0}\text{.366) + (-0}\text{.366)}\right)}\\ =\text{-(-1}\text{.098)}\\ \text{=1}\text{.098}\end{array}$

Therefore, the value of the Shannon diversity index of Community 4 is 1.098.

Thus, the value of the Shannon diversity index of Community 1 is 0.325, Community 2 is 0.693, Community 3 is 0.638 and Community 4 is 1.098.

Summary Introduction

To determine: The community which has the highest diversity.

Introduction: Shannon diversity index is a tool used by ecologists to study the species diversity of a community. This allows ecologists to correlate the species richness of an area with the relative abundance of each species.

Explanation of Solution

The higher the value of the Shannon diversity index, the greater is the diversity. The value of the Shannon diversity index of Community 1 is 0.325, Community 2 is 0.693, Community 3 is 0.638 and Community 4 is 1.098. Therefore, the Shannon diversity index of Community 4 is the highest. Thus, it can be concluded that Community 4 has highest diversity.

Thus, Community 4 has highest diversity.

Summary Introduction

To determine: The maximum diversity that each community could have.

Introduction: Shannon diversity index is a tool used by ecologists to study the species diversity of a community. This allows ecologists to correlate the species richness of an area with the relative abundance of each species.

Explanation of Solution

Maximum possible diversity is calculated as the maximum relative abundance that a community may have and is equal for two communities having the same number of species.

Communities 1 and 2 have the same number of species; 100 species of Species 1 and Species 2 together. Maximum possible diversity is seen in Community 2 as it has a greater Shannon index value as compared to the Community 1 despite having the same number of species. Community 2 has a diversity of 0.693. Therefore, the maximum possible diversity of Community 1 and Community 2 is 0.693.

Similarly, the maximum possible diversity of Community 3 and Community 4 can be calculated. Community 3 has a diversity of 0.638 and Community 4 has a diversity of 1.098. Therefore, the maximum possible diversity of Community 3 and Community 4 is 1.098.

Thus, the maximum possible diversity of Community 1 and Community 2 is 0.693 and the maximum possible diversity of Community 3 and Community 4 is 1.098.