Optimal Nesting of Species for Exact Cover of Resources: Two Against One

Experiments for resource-defined fitness sharing (RFS) show a remarkable ability to find tilings in shape nesting problems (Horn, 2002, 2005). These tilings are essentially exact covers for a set of resources, and represent a maximally sized set of cooperating (non-competing) species. This paper initiates a formal analysis of this empirical phenomenon by examining a minimal case: two species a and b "cooperate" to exactly cover the resources, while a third species c "competes" with a and b by overlapping both in terms of covered resources. The analysis reveals that in cases in which a and b maximally compete with c for resources, species c will become extinct, while the optimal set of species, a and b, will survive. This result is clearly proven using algebra on the niching equilibrium equations for RFS, a purely static analysis