On the Variance of Eigenvalues of the Community Matrix: Derivation and Appraisal

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Ecological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology.Abstract. Eigenvalues, the solutions to the characteristic polynomial, are important measures of community behavior. Their range and practical measurement present difficult challenges in ecology. We therefore present the derivation of variance of eigenvalues of the community matrix, var(X) = var (ai) + (n -1)aijaji, as well as a novel related formula, namely, the expectancy of pairwise eigenvalues (EPV), var(Xpairwise) = var(aii-pairwise) + aijaji.We propose that the two formulae may be useful in evaluating the relative contributions of inter-and intraspecific effects on the behavior of large systems. EPV allows estimating eigenvalue distribution of systems of unknown size.

Biomass and biodiversity in species-rich tritrophic communities

“…This is in line with previous mathematical results highlihting that, all other things being equal, an increase in the variance of the diagonal entries of the community matrix (the Jacobian matrix evaluated at a feasible equilibrium point) results in a greater variance of its eigenvalues along the real direction in the complex plane. Indeed, the variance V(λ) of the eigenvalues of any m × m matrix M reads (Jorgensen et al 2000, Barabás et al 2017. The diagonal entries of the community matrix are, in our case, proportional to the γ i (Barabás et al 2017).…”

Characterizing Species Interactions to Understand Press Perturbations: What Is the Community Matrix?

“…Numerous authors have also used the term "Jacobian matrix" as a synonym for "community matrix." Indeed, this has become the more typical use of the term among theoreticians (e.g., Berlow et al 2004, Dambacher et al 2002, de Ruiter et al 1995, Jorgensen et al 2000, Laska & Wootton 1998, Montoya et al 2009, Schmitz 1997, Stone & Roberts 1991, Vázquez et al 2015, Yodzis 1995. Although May (1972) used the term "interaction matrix" in reference to an approximation to a (mathematical) Jacobian matrix, his paper appears to have been the origin of this usage.…”

“…The assumption of negative self-effects for all species has become typical in qualitative modeling as well (e.g., Dambacher et al 2015). The biological and mathematical restrictions that such uniform assignments implicitly impose have been recognized for many years in the context of pulse perturbations (Haydon 1994, James et al 2015, Jorgensen et al 2000, Taylor 1988.…”

Mass-balanced trophic and loop models of complex benthic systems in northern Chile (SE Pacific) to improve sustainable interventions: a comparative analysis

“…For details about the model see Ortiz & Wolff (2002c). Recent extensions of the theoretical aspects of Loop Analysis are given by Jorgensen et al (2000) and Dambacher et al (2002Dambacher et al ( , 2003. Table 1 contains the resistance and/or sustainability estimations for quantitative and qualitative ecological subsystem models.…”