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?

We have been doing some work about qualitative stability of a network before and after an invasion by an alien species, the problem for us is to find sound information about the network before the invasion. I knew about the Quirk and Ruppert but not the other methods, so I’ll take a look at them, thanks!

No problem. Quasi sign-stability, in my opinion, is an incredibly useful tool for when we lack specific information about the strength of interactions in the system. It would definitely be interesting to see how that changes before and after invasion and it should be relatively easy to implement in R.

Sure, it’s worth looking into. Jeff Dambacher (http://scholar.google.ca/citations?user=F64PwC0AAAAJ&hl=en) has done some nice work on loop analysis, building on Levins’ classic work. Loop analysis is closely related to the topic you raise and worth looking into, I think. If one is clever, one can use loop analysis in combination with qualitative information on network structure to make testable predictions about how the network should respond to perturbations. This is useful both for validating hypothesized qualitative networks (one can test the predictions of alternative hypothesized qualitative networks in order to infer the correct network structure), and also as a way of generating testable predictions in the absence of quantitative network information. I think this is an under-utilized approach in community ecology. Florence Hulot and colleagues (Hulot et al. 2002 Nature, if memory serves) did some nice experimental work on this, as did Ed McCauley back in the late ’70s or early ’80s.

I am not familiar with loop analysis so I will have to look into that. It sounds promising though, thanks for the comment!

I am an engineering professor, inspired by ecological principles. I now have a result which I am very excited about and believe is useful to both engineers and ecologists. I consider a matrix with only signs and developed a sufficient condition for Qualitative Sign Instability. In other words, simply by looking at the signs of the matrix and with few simple arithmatic calculations, i can tell whether that sign matrix is Qualitative Unstable, i.e. unstable for any magnitudes in those entries of the matrix. The test is extremely easy, requiring simple arithmatic like counting the number and nature (positive, or negative or zero) of diagonal elements and the number and nature of l-cycle pairs (pp, mutualism, competition, ammensal, commensal and null links). Could anyone comment on the novelty and usefulness of the result and whether this type of result is of interest to you ecologists (it is definitely of interest to us engineers) and whether any one is aware of this kind of result in ecology community and if by chance there are any result like this in ecology literature. I appreciate your feedback very much.

To me that sounds a lot like the conditions outlined by Robert May’s 1974 paper on qualitative stability (here). I would also advise you to look into work by Dmitrii Logofet, who has done a significant amount of work in this field (he has a book titled

Matrices and Graphs Stability Problems in Mathematical Ecology, and also see recent work by Stefano Allesina.Thanks for your reply. Yes, of course, I am quite familiar with the work reported in all those references and actually used those concepts in my new result. Please note that I am giving conditions for Qualitative (Sign) Instability just like May gave those for Qualitative Stability. It is as if Qualitative (Sign) Stability is one extreme situation, and Qualitative (Sign) Instability (the problem I am addressing) is the other extreme situation and the matrices that don’t satisfy either May’s (and other extensions like Jeffries, Dambacher, Pimm, Logofet, others) QS conditions and now my (hopefully new) QI conditions are those matrices whose stability/instability depend on the magnitudes of the elements, which I label as MDSU (Magintude Dependent Stable/Unstable) matrices. So my conditions are sufficient conditions for Sign Instability.

Definitely these conditions were inspired by the ecological principles I learnt from May, Allesina, Logofet and other ecology literature. My results thus I believe will be of interest to engineers and ecologists alike. I will send the paper to my ecology colleagues (like you) and may be you can then give me your feedback comments which I appreciate very much. I don’t think I saw any of the conditions I developed in the ecology literature, definitely not in our engineering literature. That’s why I want to get the opinion of ecologists and make sure that I refer to any (and all) useful, relevant articles (and their authors) of the literature in ecology in my current and upcoming papers on this subject. I am a little torn as to where I should publish my results, to the ecology community or to the engineering community. I can tell that my engineering community will be fascinated but want to make sure whether this result can excite the ecology community. That’s why I started this dialogue. May be I can publish with both communities ! In fact, I have a desire to hold a workshop where engineers and ecologists could meet together and discuss these results. Again thanks a lot for your time and efforts in this matter.

Oh I see. So rather than telling us when the matrix is always stable, it is telling us that it will never be stable? That would be interesting. I would wonder how such a spectrum from QS to QI would relate to Allesina’s Quasi Sign-Stability work, as that is what I am currently focused on in my work. Perhaps quasi sign-stability tells us where a matrix with a given sign structure stands in relation to not only QS, but also QI?

Exactly, you got it. I now have to look at this Quasi sign stability work you and Allesina are doing. I have a feeling, yes, it could be related my QI conditions. Very fascinating and interesting. I love these ecological principles!! Could you send couple of papers you authored or links to them. Thanks a lot again.