Question

A forest has four different tree species (A, B, C, and D) and their numbers are: \( A = 60 \); \( B = 20 \); \( C = 10 \); and \( D = 10 \). The Shannon biodiversity index of the trees in this forest is ................... (rounded off to 2 decimals)

Hint:

The Shannon biodiversity index measures the diversity of a community by considering both the number of species and the evenness of their distribution. A higher index indicates greater diversity.

Answer 1

Step 1: Formula for Shannon Biodiversity Index The Shannon biodiversity index (\(H'\)) is given by the formula: \[ H' = - \sum_{i=1}^{S} p_i \ln p_i \] Where:
- \( S \) = Total number of species in the ecosystem
- \( p_i \) = Proportion of the \(i\)-th species in the community, calculated as \(\frac{N_i}{N_{total}}\), where \( N_i \) is the number of individuals of species \(i\) and \( N_{total} \) is the total number of individuals in all species.
Step 2: Calculating the total number of trees
The total number of trees in the forest is: \[ N_{total} = 60 + 20 + 10 + 10 = 100 \] Step 3: Calculating the proportion of each species
- \( p_A = \frac{60}{100} = 0.60 \)
- \( p_B = \frac{20}{100} = 0.20 \)
- \( p_C = \frac{10}{100} = 0.10 \)
- \( p_D = \frac{10}{100} = 0.10 \)
Step 4: Applying the formula \[ H' = - \left( 0.60 \ln 0.60 + 0.20 \ln 0.20 + 0.10 \ln 0.10 + 0.10 \ln 0.10 \right) \] Calculating the logarithms: \[ \ln 0.60 = -0.5108, \quad \ln 0.20 = -1.6094, \quad \ln 0.10 = -2.3026 \] Substituting these into the equation: \[ H' = - \left( 0.60 \times (-0.5108) + 0.20 \times (-1.6094) + 0.10 \times (-2.3026) + 0.10 \times (-2.3026) \right) \] \[ H' = - \left( -0.3065 - 0.3219 - 0.2303 - 0.2303 \right) = 1.0889 \] Final Answer: \[ \boxed{1.09} \]