Optimal nesting of species for exact cover: Many against many

Abstract

Experiments with resource-defined fitness sharing (RFS) applied to shape nesting problems indicate a remarkable ability to discover exact covers of resources [1, 2]. These exact covers are represented by a maximally sized set of cooperating (non-competing) species. Recent papers by Horn [3, 4] introduce the first formal analyses of this empirical phenomenon. In [3], a minimal case of two species, a and b, against a third, c, is considered: the two-against-one scenario. It is shown that if the team of a and b form an exact cover, then c will be extinct at niching equilibrium. In [4], this result is generalized to the case of two-against-many: if a and b form an exact cover against an arbitrary number of competing species, under very general assumptions, a and b will be the only survivors at niching equilibirum. In the current paper, we extend these results to the most general scenario: many-against-many. We prove that, under certain very general assumptions, any size team of species forming an exact cover will dominate a population with any number of competing species: at niching equilibirum, all such competitors will be extinct. The results are more general than shape-nesting problems, applying as well to the NP-complete problem exact cover by k-sets. © 2008 Springer-Verlag Berlin Heidelberg.