"Complexity Measures for Ecological Assemblages"
Three approaches to an ecosystem measure of complexity are explored, all based on the general notion of connectivity. Two species or other taxa are said to be connected or linked ecologically if they occur together more often than expected by chance, whether they are linked trophically, auxotrophically, or simply have similar basic requirements. (1) The first measure P(k) is defined as the ratio of the number of vertices connected to k other vertices divided by the total number of vertices, where taxa are the vertices. (2) A second, dimC (S), is based on a box fractal dimensional approach, in which a sample S is partitioned into n equal subsamples, or equal aliquots. This complexity dimension also gives a way of counting the connections among taxa. (3) Finally, let A be the matrix with entries counting the number of connections between the taxa in X, and define B = exp(A), which essentially contains all connections of all lengths among the taxa. Another measure of complexity, F(X), is the logarithm of the sum of all non-diagonal entries in B divided by logn. Note that is an approximation of a box fractal dimension defined by the scale 1/n. Applications to assemblages of soil amoebas are used to illustrate these measures