I am interested in how the stability of ecological networks change as the diversity of the ecological community increases or the interactions between the species constituting the ecosystem get stronger. This was spurred by my work on excitatory-inhibitory neural network models - which share the mathematical framework of the predator-prey equations of mathematical ecology (loosely speaking, predators = inhibitory neurons and prey = excitatory neurons). I was interested in looking at whether the connectivity structure of such networks alters their stability (i.e., linear stability around a local attractor) and it was in this context that I came across the Wigner-May theorem. Very generally, what it says is that increasing the complexity of a system (namely, by increasing the number of components or by increasing the connectivity among them) makes a system more likely to be unstable.
May's original conclusions holds strictly only for randomly assembled networks. Clearly, natural ecosystems are anything but random. They exhibit various kinds of structures (e.g., trophic levels) which are absent in random networks. There have been some studies on the stability of networks with non-random features, such as hierarchical structures, etc. This was my motivation for looking at the stability of networks with ``small-world'' connectivity as well as with hierarchical connectivity structure.See the page on stability of small-world networks
However, I was wondering about where natural ecosystems
get their structure from. Obviously such systems acquire the structure
as they are built up - one species at a time - through a process of community
assembly. I was thinking of investigating how structured networks can emerge
from randomly connected networks - when I met Chris Wilmers at the Santa
Fe Institute Complex Systems Summer School in June 2000. Along with Marcus
Brede, we looked at a model of building a ecosystem by adding one species
at a time - provided the system does not become unstable at any point.
The details are given in the following link.
|Evolving Stable Ecological
SFI Complex Systems Summer School Project