Noninvadability implies noncoexistence for a class of cancellative systems

摘要

There exist a number of results proving that for certain classes ofinteracting particle systems in population genetics, mutual invadability oftypes implies coexistence. In this paper we prove a sort of converse statementfor a class of one-dimensional cancellative systems that are used to modelbalancing selection. We say that a model exhibits strong interface tightness ifstarted from a configuration where to the left of the origin all sites are ofone type and to the right of the origin all sites are of the other type, theconfiguration as seen from the interface has an invariant law in which thenumber of sites where both types meet has finite expectation. We prove thatthis implies noncoexistence, i.e., all invariant laws of the process areconcentrated on the constant configurations. The proof is based on specialrelations between dual and interface models that hold for a large class ofone-dimensional cancellative systems and that are proved here for the firsttime.