Tuesday, April 17, 2012

Community ecology is complicated: revisiting Robert May’s weak interactions

When it comes to explaining species diversity, Stefano Allesina differs from the traditional approach. Community ecology has long focused on the role of two species interactions in determining coexistence (Lotka-Volterra models, etc), particularly in theory. The question then is whether two species interactions are representative of the interactions that are maintaining the millions of species in the world, and Allesina strongly feels that they are not.

In the paper “Stability criteria for complex ecosystems”, Stephano Allesina and Si Tang revisit and expand on an idea proposed by Robert May in 1972. In his paper “Will a large complex system be stable?” Robert May showed analytically that the probability a large system of interacting species is stable – i.e. will return to equilibrium following perturbation – is a function of the number of species and their average interactions strength. Systems with many species are more likely to be stable when the interactions among species are weak.

May’s paper was necessarily limited by the available mathematics of the time. His approach examined a large community matrix, with a large number of interacting species. The sign and strength of the interactions among species were chosen at random. Stability then could be assessed based on the sign of the eigenvalues of the matrix – if the eigenvalues of the matrix are all negative the system is likely to be stable. Solving for the distribution of the eigenvalues of such a large system relied on the semi-circle law for random matrices, and looking at more realistic matrices, such as those representing predator-prey, mutualistic, or competitive interactions, was not possible in 1972. However, more modern theorems for the distribution of eigenvalues from large matrices allowed Allesina and Tang to reevaluate May’s conclusions and expand them to examine how specific types of interactions affect the stability of complex systems.

Allesina and Tang examined matrices where the interactions among species (sign and strength) were randomly selected, similar to those May analyzed. They also looked at more realistic community matrices, for example matrices in which pairs of species have opposite-signed interactions (+ & -) representing predator prey systems (since the effect of a prey species is positive on its predator, but that predator has a negative effect on its prey). A matrix could also contain pairs of species with interactions of the same sign, creating a system with both competition (- & -) and mutualism (+ & +). When these different types of matrices were analyzed for stability, Allesina and Tang found that there was a hierarchy in which mixed competition/mutualism matrices were the least likely to be stable, random matrices (similar to those May used) are intermediate, and predator–prey matrices were the most likely to be stable (figure below).

When the authors looked at more realistic situations where the mean interaction strength for the matrix wasn’t zero (e.g. so a system could have all competitive or all mutualistic interactions), they found such systems were much less likely to be stable. Similarly, realistic structures based on accepted food web models (cascade or niche type) also resulted in less stable systems.

The authors reexamined May’s results that showed that weak interactions made large systems more likely to be stable. In particular they examined how the distribution of interactions strengths, rather than the mean value alone, affected system stability. In contrast to accepted ideas, they found that when there were many weak interactions, predator-prey systems tended to become less stable, suggesting that weak interactions destabilize predator-prey systems. In contrast, weak interactions tended to stabilize competitive and mutualistic systems. The authors concluded, “Our analysis shows that, all other things being equal, weak interactions can be either stabilizing or destabilizing depending on the type of interactions between species.” 

Approaching diversity and coexistence from the idea of large systems and many weak interactions  flies in the face of how much community ecology is practiced today. For that reason, it wouldn't be surprising if this paper has little influence. Allesina suggests that focusing on two species interactions is ultimately misleading, since if species experience a wide range of interactions that vary in strength and direction, sampling only a single interaction will likely misrepresent the overall distribution of interactions. Even when researchers do carry out experiments with multiple species, finding a result of very weak interactions between species is often interpreted as a failure to elucidate the processes maintaining diversity in the system. That said, Allesina’s work (which is worth reading, few people explain complex ideas so clearly) doesn’t necessarily make itself amenable to being tested or applied to concrete questions. Still, there’s unexplored space between traditional, two-species interactions and systems of weak interactions among many species, and exploring this space could be very fruitful. 


  1. Very interesting, although completely counter intuitive. Would have thought that interaction between predators and prey was important to keep the eco-system in balance.

  2. I agree, and the paper does a fairly poor job of explaining what that particular result could mean. The ideas are interesting, but the authors could do a better job of reconciling the ideas (and the math) with our observations.

  3. Um, figure 2 does show that predator-prey interactions promote stability, compared to the other interaction types shown in the figure. Any values of species richness and connectance below and to the left of the green points (that's the predator-prey case) is stable, which encloses a larger area than for any of the other interaction types. Predator-prey interactions are more likely to lead to stable webs because a predator-prey interaction is a negative feedback loop. Competition and mutualism are positive feedback loops, and hence destabilizing.

  4. I assumed the first commenter was referring to the fact that increasing the number of *weak* interactions in predator-prey systems is destabilizing. Given measured food webs often show lots of these weak interactions, that particular finding is less intuitive. That said, of course figure 2 shows that in general, predator-prey systems are more likely to be stable than the other types.

  5. Re: the weak interactions result, it is a bit counterintuitive. But I suspect not so much that it can't be figured out. I'll have a bit of a think about it. Or you could just email Stefano and ask him. ;-)

  6. Re: weak interactions, one more thought. I'm sure it's important to distinguish between weak interactions in the sense of a low mean interaction strength, and weak interactions in the sense of the shape of the distribution around the mean (e.g., a skewed distribution with many weak and few very strong interactions). Changing the mean of the interaction strength distribution will have different consequences then changing the shape of the distribution while holding the mean constant. Can't remember which sense of 'weak interactions' applies to Stefano's paper, would need to go back and look.

  7. Out of curiosity, has any work like this been done with cases where interaction strengths are allowed to covary? I'm thinking of phenomena like mesopredator release, where the interaction strength on prey species X by a predatory species Y will be strong in the absence of a predatory species Z, and weak in the presence of predatory species Z.

    Or, alternatively, am I misunderstanding the definition of 'interaction strength'?

  8. Tor: The sorts of changes in "interaction strength" that you refer to reflect changes in species' abundances (here, removal of predator Z leads to increased abundance of mesopredator Y). That doesn't require any changes in model parameters, and so would not represent a change in interaction strength (on some common definitions of that term). Now, if you're imagining that mesopredator Y behaves differently in the absence of predator Z (say, spends more time hunting and less time hiding), that would be a change in model parameters and so would represent a change in interaction strength.

    Unfortunately, different people use the word "interaction strength" to mean different things. The definition used in this paper is a standard one in the theoretical literature. There's a review by Laska and Wootton from 1998 or so in Ecology that discusses and relates some common formal definitions of interaction strength.