The paper in Nature Ecology & Evolution is here: http://go.nature.com/2zPpmrG
This project grew out of a friendly banter with my former boss Stefano Allesina. He is known for the ease with which he employs the most varied methods to model species interactions in species-rich communities, from random interaction structures to cascade and niche models, and more. However, I always felt that his treatment of within-species interactions (self-effects; also known as self-regulation if they work to counteract population growth) might be a bit cavalier: he liked to simply assume that each species is self-regulated to the exact same degree. Now, it is easy to imagine self-regulation arising in, say, plants, because even if a plant species has no competitors, it cannot go on reproducing forever due to spatial constraints. The same might hold for certain territorial animals. However, for a herbivore eating the plant, this kind of limitation seems far less realistic. Instead, herbivore populations stop growing due to a shortage of food, or an overabundance of the herbivore's predators - neither of which are "self"-interactions. That is, common sense suggests that self-regulation is important in plants and maybe some territorial species, but probably not so important for others.
One day, I pointed out this perceived deficiency in my traditionally measured tone, shouting "Stefano, assigning the same self-regulation strength for all species makes absolutely no sense!" or something along those lines. He responded that he knows, but that his way was probably good enough. Being my obstinate self, I did not give up, and started looking at large interaction networks, both computer-generated and empirical, to prove him wrong. Instead, I quickly realized that he was right: these networks were practically impossible to stabilize unless the vast majority of species self-regulated to a substantial degree. Assuming real-world networks possess some degree of stability, this came as a rude surprise. Checking various scenarios and parameterizations, this was always the result unless some patently unrealistic assumption was added, such as all species feeding on one single plant, or the effects of predators on their prey being exactly equal to the effect of prey on predators, for every single predator-prey pair.
This was the point when we started looking for a way to theoretically understand when and why we see this pattern. With the help of Matt Michalska-Smith, we have tried and discarded a large pile of complicated and/or ill-suited methods that almost worked - until we hit upon this article by the mathematician Tim Rogers. His method was so well suited to our problem that we could hardly believe our eyes: with its use, we were now able to predict the stability properties of real-world networks analytically, with high accuracy. Unfortunately, one of the consequences of the theoretical development is that the vast majority of species in large networks must self-regulate if they are to be stable, except in certain small regions of parameter space.
I must confess that I am still trying to wrap my head around this result. It is now supported by studies on empirical and computer-generated networks as well as an analytical theory - yet the tug of the classical intuition (that self-regulation is by and large the privilege of plants and territorial animals) is strong. It is interesting to think about where those strong self-regulatory interactions, necessary for community stability, actually come from. We have done our share of hypothesizing in the paper. However, if someone could actually find an example of a large network which is both realistic and can be stabilized by a handful of self-regulating species, please email me (email@example.com) as soon as you can!