Noninvadability implies noncoexistence for a class of cancellative systems

TL;DR

This paper demonstrates that in certain one-dimensional cancellative systems modeling balancing selection, strong interface tightness leads to noncoexistence, meaning only uniform states are invariant, contrasting with previous results linking mutual invadability to coexistence.

Contribution

It establishes a converse relationship between interface tightness and noncoexistence in a class of cancellative systems, introducing new duality relations for these models.

Findings

Strong interface tightness implies noncoexistence.

Invariant laws are concentrated on constant configurations.

New dual and interface model relations are proved.

Abstract

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