Stability and complexity in model ecosystems

Back in 1972 Robert May wrote a paper that sparked a litany of research on the relationship between the complexity of a community and its associated stability. Previously the reigning paradigm had been developed by ecologists such as Elton and MacArthur who noted that the communities commonly observed in nature tended to be complex, with many pathways for energy to flow up trophic levels. Because of redundancy in energy pathways they believed that these systems should be stable, especially since obviously complex communities were still around, and not collapsing into instability.

May, however, generated randomly assembled communities (using SxS matrices) with random interaction strengths. He then systematically varied number of species, mean interaction strength, and the connectance (number of realized links) of these randomly generated webs. For each web that was generated May analyzed the stability based on the eigenvalues of the matrix (where in order to be stable the real part of the maximum eigenvalue must be negative). He found that as number of species, mean interaction strength, and/or connectance was increased, the stability of the system declined, results diametrically opposed to the reigning paradigm, and observation.

Obviously there are a number of issues involved with this type of analysis. Most significant is that real communities are not random, they have very specific topological structure that develops through community assembly and evolutionary processes. May noted that this was the case, however, and wrote that randomly assembled communities should serve as a null model and that future research should be devoted to finding those deviant strategies used by communities that allow them to be stable. To be fair, in the seventies in terms of network research, random networks were gaining a lot of steam in the world of physics and math as a result of the pioneering work of Erdos and Renyi.

So, this paper did what May had apparently intended, and it sparked a great deal of research over the past thirty years investigating why real communities are stable. Of particular interest is the research on topology of the network, showing how such architectures as nestedness and modularity can influence stability (mostly by altering the interaction strength combinations). Other work has focused on interaction strengths, demonstrating that most interactions are weak, as well as external influences (e.g. subsidies sensu Polis and Strong 1996) and other properties of networks.

What has surprised me most, however, as I read through the community ecology/food web literature is a surprising lack of knowledge about another paper that came out about a year after May’s by Alan Roberts. Roberts noted that May’s analysis allowed for what he called “ghost species,” those species whose equilibrium population size was negative (e.g. they would just go extinct). Clearly communities with such species are not likely to be stable, and communities with more species are more likely to have these ghosts. Critically, Roberts re-did May’s simulation with the added filter of feasibility. A feasible community was one whose equilibrium population densities are positive. When only feasible random communities were analyzed for their stability properties, May’s results were found to be exactly opposite. Roberts had clearly demonstrated that May was wrong (in 1974), yet his work is rarely cited despite being published in Nature. According to Google Scholar May 1972 is cited just over 1000 times (662 in Web of Science), while Robert’s paper has 116 (80 from Web of Science).

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15 Responses to Stability and complexity in model ecosystems

  1. Thanks for bringing that to my attention (and hopefully many others).

    • Jon Borrelli says:

      It is interesting that when feasibility is a criterion that is applied to these systems, they become much more likely to be stable. Of course, I do not think that this in any way diminishes the impact of May’s paper, I believe that the research his work has inspired has led to some fascinating work on communities.

  2. Pingback: Complexity and ecosystem stability « theoretical ecology

  3. Very interesting, I wonder if the recent work of Allesina & Tang (2012) Stability criteria for complex ecosystems. Nature 483: 205–208. Takes into accunt the Roberts work, I think not.

    Anyway I don’t have access to the Robert’s paper I apprecieate if somebody could send me the pdf:
    Roberts, A., 1974. The stability of a feasible random ecosystem. Nature, 251(5476), pp.607–608.

    • Jon Borrelli says:

      As far as I can tell Allesina and Tang do not focus on feasibility, however they do mention that they used “realistic structures” derived I believe from food web models. Such structure should, at least in theory, be feasible. Thus their findings are interesting because they support May’s original findings. I think it would be interesting to go back and try to replicate Robert’s work to see what is really going on, his letter to nature is not as informative as I would like (which perhaps is the reason for lack of citation). I should note that there are other papers out there on feasibility with (near as I can remember) conflicting results.

    • Jon Borrelli says:

      I can send you the paper if you let me know your email, if you haven’t gotten it yet.

      • Yes, I think It would be interesting to reproduce the Robert’s results, Allesina says that only predator-prey networks can potentially ellude May’s conclusion, but the work of Robert imply another thing.
        And I already got the Robert’s paper anyway, thanks!

  4. We are thinking on replicate the Robert’s results, are you already doing it?

  5. Jon Borrelli says:

    Yes I would definitely be interested in participating in a project like that.

  6. Pingback: Testing qualitative stability | Assembling my Network

  7. Mike Fowler says:

    Hi all – I’m a bit late to the discussion here, but it’s certainly an interesting topic.

    I think Roberts’ paper focuses on a rather specific question (local stability rather than a more general question of persistence), with a few arguably unrealistic assumption: that only feasible communities are formed in natural ecosystems (or of interest); that there is perfect biomass/energy conversion from resources to consumers (i.e., when there’s a +,- combination on the opposite elements of the interaction matrix).

    I don’t see why feasibility is guaranteed in natural/randomly assembled systems, and I’m pretty sure that energy conversion across resource-consumer pairs is imperfect, therefore I think May’s initial approach (that at least included asymmetry, if not appropriate conversion rates) might still be considered more general, even though it clearly has flaws when taken literally as some sort of realistic ‘ecosystem’ model, rather than as a ‘null/random’ model.

    I’ve tried to replicate Robert’s approach as best as I could – and I agree with Jon that he leaves out important detail from results presented in the paper. I find Roberts’ figure an unintuitive way to present the results. I also found that increasing (local) stability only occurs with increasing ecosystem size (T) when the interaction parameter ‘z’ scales with T. This does not match the underlying niche model assumptions that May bases his modelling approach on.

    When z is held constant for different T (e.g., z = 1/3), we recover May’s original result, that increasing T is associated with a decrease in the probability of feasibility and of a feasible system being locally stable.

    So, neither May nor Roberts are wrong, but the domains in which they are correct might differ 😉

    • Jon Borrelli says:

      Its never too late to hop into discussion, especially not with a great comment like that. I think you are right as well in saying that feasibility is not a necessary condition for real communities during assembly. However, I would argue then that if a community is not feasible then its likelihood of persistence becomes by necessity 0. I would think that during something like community assembly, if a new species invaded such that with the new configuration the system did not fulfill feasibility requirements we would see some sort of extinction event. Most likely the system would collapse into one that is in fact, feasible. This issue is one that greatly interests me, and I think will play a large part in my ongoing Ph. D. research.

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