So, we have a major question in ecology; “what is the relationship between diversity/complexity and stability.” And we have data on diversity and complexity, but how can we assess the stability of a system without numbers to fill in the Jacobian matrix? Typically to determine the stability of a system we take the real part of the largest eigenvalue, if it is negative then the system is stable, positive it is unstable.
In a previous post commenting on Robert May’s 1972 paper and the subsequent paper by Alan Roberts I in no way meant to diminish the role May has played in shaping the literature on stability and complexity. In fact I would probably credit May for bring matrix algebraic thinking to ecology. Along those lines I would like to focus on another, much less cited (~115 citations) paper written by May in 1974 on what is termed “qualitative stability.” With this paper May brings the thinking of economists Quirk and Ruppert to ecology. The idea is that in economics, as in ecology, we often lack the necessary quantitative data needed to analyze the stability of a matrix. It is difficult to empirically measure an interaction strength (in fact it is somewhat unclear as to what interaction strength even means and how it should be measured). However, May notes that Quirk and Ruppert in an earlier paper developed several rules which, if true, mean that the matrix will be relatively robust to small perturbations. The rules for qualitative stability are based primarily on the pattern of signs of the elements of the matrix rather than their magnitudes.
I think that this paper is greatly undervalued (as judged by its citation rate). The study of ecological networks is clearly a primarily qualitative field (although we appear to be quantitative, and are in other respects). I say this primarily because as I have begun gathering data for my research, mostly downloading food webs from the Interaction Web Database, Ecological Archives, and the PEaCE Lab I cannot help but notice that all of this data is on topology, and there are no values assigned to interaction strength.
I think that examining the qualitative stability of these networks could help us understand the stability of real systems. There have been to my knowledge about three different uses of some form of qualitative stability to test real systems in the literature. The first was done by Jeremy Fox in 2006 in a paper I talked about here. Fox took the adjacency matrix of a food web and performed a stability analysis on it, proposing that this represented the contribution of topology to qualitative stability. Another method developed by Allesina and Pascual in 2008 is something they termed “quasi sign-stability.” Here they take the interaction matrix and randomize interaction strengths, then determine the sign of the dominant eigenvalue. They iterate this process 10000 times to see how often the system is stable. A system that is more qualitatively stable should be stable proportionally more often than systems that are not. The third method, proposed by Samraat Pawar in 2009 is based on Allesina and Pascual’s quasi sign-stability and is called “interaction strength sensitivity” (ISS) which is a measure of the lack of quasi sign-stability. ISS is measured by the correlation between the maximum eigenvalue of the community matrix, and the maximum eigenvalue of the community matrix whose elements are all positive (absolute values).
I have recently been doing some work with quasi sign-stability in model and real food webs and I plan to write up a post about some of the issues I have been having soon. What do you think about the idea of qualitative stability as a measure for ecosystems, is this something we as ecologists should be pursuing?