| Abstract: |
| Our understanding of the structure of ecological networks is inadequate in many areas. We need tools to explain why some species persist. Networks are often presented as graphs where each species is a node, and edges represent species interactions. Which links are crucial for the robustness of the network, links without which the system is fragile and species are lost? How many and which species must be removed from a fragile network for it to become robust?
We have found a fundamental criterion for the robustness of an ecological graph that is applicable by biologists. The problem is to determine which graphs {\it can} support a network with a ``robust`` steady state; i.e., a steady state that will exist despite any tiny changes in any constants appearing in the dynamical system equations.
We give a new, easy to apply, criterion, the ``Loop-Covering Principle``, that answers all these questions. It asserts that the graph of an ecological network {\it can} have a robust equilibrium if and only if it is ``loop coverable``. The well-known Competitive Exclusion Principle is a particular case, asserting that if there are more predator species than resource species, then there is no robust equilibrium. |
|