vazelinahS

2021-08-19

To find the measure of diversity H for the following proportions of each:

hemlock, 0.521. beech, 0.324. birch, 0.081. maple, 0.074.

hemlock, 0.521. beech, 0.324. birch, 0.081. maple, 0.074.

Tasneem Almond

Skilled2021-08-20Added 91 answers

Modeling diversity of species:

One measure of the diversity of the species in an ecological community is modeled by the formula,

$H=[{P}_{1}{\mathrm{log}}_{2}{P}_{1}+{P}_{2}{\mathrm{log}}_{2}{P}_{2}+\cdots +{P}_{n}{\mathrm{log}}_{2}{P}_{n}]\dots \dots \dots ..\left(1\right)$

Where$P}_{1},{P}_{2},\cdots ,{P}_{n$ are the proportions of sample that belong to the each of n species found in the sample.

Given that${P}_{1}=0.521,{P}_{2}=0.324,{P}_{3}=0.081$ and ${P}_{4}=0.074$

Substitute these values in equation (1) to get the following,

$H=-[0.521{\mathrm{log}}_{2}0.521+0.324{\mathrm{log}}_{2}0.324+0.081{\mathrm{log}}_{2}0.081+0.074{\mathrm{log}}_{2}0.074]\dots \dots \dots \dots ..\left(2\right)$

Change-of -Base theorem:

For any positive real numbers x, a and b, where$a\ne q1$ and $b\ne q1$ , the following holds.

$\mathrm{log}}_{a}x=\frac{{\mathrm{log}}_{b}x}{{\mathrm{log}}_{b}a$

By using the change of base theorem the terms in the equation (2) becomes the following.

$\mathrm{log}}_{2}0.521=\frac{\mathrm{log}0.521}{{\mathrm{log}}_{2}$

$=\frac{-0.2832}{0.3010}$

$=-0.9406$

$\mathrm{log}}_{2}0.324=\frac{\mathrm{log}0.324}{{\mathrm{log}}_{2}$

$=\frac{-0.4895}{0.3010}$

$=-1.6259$

$\mathrm{log}}_{2}0.081=\frac{\mathrm{log}0.081}{\mathrm{log}2$

$=\frac{-1.0915}{0.3010}$

$=-3.6259$

$\mathrm{log}}_{2}0.074=\frac{\mathrm{log}0.074}{\mathrm{log}2$

$=\frac{-1.1308}{0.3010}$

$=-3.7563$

Substitute in equation (2) to get the following,

$H=-[0.521(-0.9406)+0.324(-1.6259)+0.081(-3.6259)+0.074(-3.7563)]$

$=-[-0.4900759-0.52680271-0.29370068-0.27796849]$

$=-[-1.5886]$

$=1.5886$

$\approx 1.589$

So, the measure of diversity H is 1.589.

Final statement:

The measure of diversity H is 1.589.

One measure of the diversity of the species in an ecological community is modeled by the formula,

Where

Given that

Substitute these values in equation (1) to get the following,

Change-of -Base theorem:

For any positive real numbers x, a and b, where

By using the change of base theorem the terms in the equation (2) becomes the following.

Substitute in equation (2) to get the following,

So, the measure of diversity H is 1.589.

Final statement:

The measure of diversity H is 1.589.

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