Problem 9 - Campbell Biology

Question

Community 1 contains 100 individuals distributed among four species: \(5 \mathrm{~A}, 5 \mathrm{~B}, 85 \mathrm{C},\) and \(5 \mathrm{D}\). Community 2 contains 100 individuals distributed among three species: \(30 \mathrm{~A}, 40 \mathrm{~B}\) and \(30 \mathrm{C}\). Calculate the Shannon diversity index \((H)\) for each community. Which community is more diverse?

Step-by-Step Solution

Answer

Community 1: 0.5875, Community 2: 1.089. Community 2 is more diverse.

Step 1 - Understanding Shannon Diversity Index formula

The Shannon Diversity Index (H) is calculated using the formula: \[ H = -\sum (p_i \times \text{ln}(p_i)) \] where \(p_i\) is the proportion of individuals that belong to the \(i\)-th species in the community.

Step 2 - Calculate proportions for Community 1

In Community 1, there are 100 individuals. Calculate the proportions for each species: \( p_A = \frac{5}{100} = 0.05 \)\( p_B = \frac{5}{100} = 0.05 \)\( p_C = \frac{85}{100} = 0.85 \)\( p_D = \frac{5}{100} = 0.05 \).

Step 3 - Calculate Shannon Index for Community 1

Using the proportions calculated, we find: \[ H_1 = -(0.05 \times \text{ln}(0.05) + 0.05 \times \text{ln}(0.05) + 0.85 \times \text{ln}(0.85) + 0.05 \times \text{ln}(0.05)) \] Calculate each term individually and sum up: \[ H_1 = -(0.05 \times (-2.9957) + 0.05 \times (-2.9957) + 0.85 \times (-0.1625) + 0.05 \times (-2.9957)) \] \[ H_1 = -(-0.14978 - 0.14978 - 0.138125 - 0.14978) \] \[ H_1 = 0.5875 \].

Step 4 - Calculate proportions for Community 2

In Community 2, there are 100 individuals. Calculate the proportions for each species: \( p_A = \frac{30}{100} = 0.30 \)\( p_B = \frac{40}{100} = 0.40 \)\( p_C = \frac{30}{100} = 0.30 \).

Step 5 - Calculate Shannon Index for Community 2

Using the proportions calculated, we find: \[ H_2 = -(0.30 \times \text{ln}(0.30) + 0.40 \times \text{ln}(0.40) + 0.30 \times \text{ln}(0.30)) \] Calculate each term individually and sum up: \[ H_2 = -(0.30 \times (-1.204) + 0.40 \times (-0.9163) + 0.30 \times (-1.204)) \] \[ H_2 = -(-0.3612 - 0.36652 - 0.3612) \] \[ H_2 = 1.089 \].

Step 6 - Compare the diversities of the two communities

Community 1 has a Shannon Diversity Index of 0.5875, and Community 2 has a higher Shannon Diversity Index of 1.089. Therefore, Community 2 is more diverse.

Key Concepts

biodiversity measurement

Biodiversity measurement is an essential aspect of ecology that helps scientists understand the variety of life in different ecosystems. It’s a way to quantify the diversity within a community, without having to tally each individual species. One popular way to measure biodiversity is through the Shannon Diversity Index. This index accounts for both the number of species present (species richness) and their relative abundance. The result is a more nuanced picture of diversity than simply counting species.

species richness

Species richness is one of the simplest ways to measure biodiversity. It refers to the count of different species in a given community. For instance, a community with 10 different species has a higher species richness than one with only 5 species. However, species richness alone doesn’t account for how individuals are distributed among these species. For example, if one species dominates the population, the ecological diversity might still be low. In our given exercise, Community 1 has four species, and Community 2 has three. Even though Community 1 has a higher species richness, the sheer number of species doesn't always give the full picture of ecological diversity.

ecological diversity

Ecological diversity captures not just the number of different species (species richness), but also considers how evenly individuals are distributed across those species. This is where the Shannon Diversity Index is particularly useful. It takes the proportion of each species into account, offering a more comprehensive measure of community diversity. In our exercise, even though Community 1 has more species, Community 2 is more diverse because its species are more evenly distributed. The index values reflect this: Community 1 has a Shannon Diversity Index of 0.5875, while Community 2 scores higher at 1.089. Thus, ecological diversity provides deeper insight into the health and complexity of an ecosystem.