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.