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.
Highlights
Introduction and main resultIn spatial population genetics, one often considers interacting particle systems where each site in the lattice can be occupied by one of two different types, respresenting different genetic types of the same species or even different species
It is natural to conjecture that if each type is able to invade an area that is so far occupied by the other type only, coexistence should be possible, i.e., there should exist invariant laws that are concentrated on configurations in which both types are present
We introduce a class of one-dimensional cancellative systems that will be our general framework and point out some interesting relations between their interface models and their dual models in the sense of cancellative systems duality
Summary
One often considers interacting particle systems where each site in the lattice can be occupied by one of two different types, respresenting different genetic types of the same species or even different species. The rebellious voter model, introduced in[SS08], with competition parameter 0 ≤ α ≤ 1 is the type-symmetric interacting particle system such that x(i) flips 0 ↔ 1 with rate. Following terminology first introduced in [CD95], we say that a type-symmetric interacting particle system X exhibits interface tightness if its corresponding interface model Yviewed from the left-most particle is positive recurrent on Sδ0 This implies that the process Ystarted from Y0 = δ0 spends a positive fraction of its time in δ0 and is ergodic with a unique invariant law on Sδ0. Theorem 1 (Strong interface tightness implies noncoexistence) Let X be either a neutral Neuhauser-Pacala model, or an affine voter model, or a rebellious voter model, with competition parameter 0 < α ≤ 1. As explained in [SS08], this model has special properties that give few clues on how to prove noncoexistence for any of the other models
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