Entropy (ecology)

Ecological entropy is a measure of biodiversity in the study of biological ecology.

Definition

Assume that within a system of unlimited individuals, there exists Failed to parse (Missing texvc executable; please see math/README to configure.): n

species of organisms (Failed to parse (Missing texvc executable; please see math/README to configure.): A_1, A_2, ..., A_j, ..., A_n

), and the probability that an individual belongs to species Failed to parse (Missing texvc executable; please see math/README to configure.): A_j

is Failed to parse (Missing texvc executable; please see math/README to configure.): p_j

, where

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i=1}^n p_i=1

If there are Failed to parse (Missing texvc executable; please see math/README to configure.): q

individuals for each species, the population will be Failed to parse (Missing texvc executable; please see math/README to configure.): q_1 + q_2 + ... + q_j + ... + q_n

. The number of individuals is Failed to parse (Missing texvc executable; please see math/README to configure.): q^n = n \log q . Set Failed to parse (Missing texvc executable; please see math/README to configure.): N

as constant Failed to parse (Missing texvc executable; please see math/README to configure.): C

.

Diversity within each species, such as male, female, large, small, etc. can be expressed as

Failed to parse (Missing texvc executable; please see math/README to configure.): C \sum_{j} p_i \log q_j = C \sum_{j}p_i \log p_I + C \log q \quad (\mbox{since }q_j = qp_I)

Hence the specific diversity can be derived by subtracting the diversities within the species from the diversity of the whole population:

Failed to parse (Missing texvc executable; please see math/README to configure.): \ C \log q - (C \sum_{j} p_i \log p_i + C \log q) = -C \sum_{j}p_i \log p_i \

The expression is the same as that of information entropy, except that it is customary to use binary logarithm (base 2) for information entropy, the unit being bit, and to use common logarithm (base 10) for ecological entropy, the unit being bel. The thermodynamic entropy uses natural logarithm (base Failed to parse (Missing texvc executable; please see math/README to configure.): e ), so that the constant Failed to parse (Missing texvc executable; please see math/README to configure.): C

becomes the Boltzmann constant, the ratio of universal gas constant and Avogadro's number.

See also

• Entropy (thermodynamics)
• Entropy (information theory)