On the variance of eigenvalues of the community matrix: Derivation and appraisal

Eigenvalues, the solutions to the characteristic polynomial, are important measures of community behavior. Their range and practical measurement prese nt difficult challenges in ecology, We therefore present the derivation of variance of eigenvalues of the community matrix, var(lambda) = var (a(ii)) + (n - 1)a(ij)a(ji), as well as a novel related formula, namely, the expect ancy of pairwise eigenvalues (EPV), var(lambda (pairwise)) = var(a(ii-pairw ise)) + a(ij)a(ji). We propose that the two formulae may be useful in evalu ating the relative contributions of inter- and intraspecific effects on the behavior of large systems. EPV allows estimating eigenvalue distribution o f systems of unknown size.