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A community measure that takes both species abundance anc species richness into account is the Shannon diversity index $H$. To calculate $H$, the proportion $p_{i}$...

Question

A community measure that takes both species abundance anc species richness into account is the Shannon diversity index $H$. To calculate $H$, the proportion $p_{i}$ of species $i$ in the community used. Assume that the community consists of $S$ species. Then $$ H=-\left(p_{1} \ln p_{1}+p_{2} \ln p_{2}+\cdots+p_{S} \ln p_{S}\right) $$ (a) Assume that $S=5$ and that all species are equally abundant that is, $p_{1}=p_{2}=\cdots=p_{5}$. Compute $H$. (b) Assume that $S=10$ and that all species are eq

A community measure that takes both species abundance anc species richness into account is the Shannon diversity index $H$. To calculate $H$, the proportion $p_{i}$ of species $i$ in the community used. Assume that the community consists of $S$ species. Then $$ H=-\left(p_{1} \ln p_{1}+p_{2} \ln p_{2}+\cdots+p_{S} \ln p_{S}\right) $$ (a) Assume that $S=5$ and that all species are equally abundant that is, $p_{1}=p_{2}=\cdots=p_{5}$. Compute $H$. (b) Assume that $S=10$ and that all species are equally abundant that is, $p_{1}=p_{2}=\cdots=p_{10}$. Compute $H$. (c) A measure of equitability (or evenness) of the specie distribution can be measured by dividing the diversity index $H$ by $\ln S$. Compute $H / \ln S$ for $S=5$ and $S=10$. (d) Show that, in general, if there are $N$ species and all specie are equally abundant, then $$ \frac{H}{\ln S}=1 $$



Answers

The Shannon index (sometimes called the Shannon-Wiener index or Shannon-Weaver index) is a measure of diversity in an ecosystem. For the case of three species, it is defined as $$ H = -p_1 \ln p_1 - p_2 \ln p_2 - p_3 \ln p_3 $$ where $ p_i $ is the proportion of species $ i $ in the ecosystem.
(a) Express $ H $ as a function of two variables using the fact that $ p_1 + p_2 + p_3 = 1 $.
(b) What is the domain of $ H $?
(c) Find the maximum value of $ H $. For what values of $ p_1, p_2, p_3 $ does it occur?


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