### Noninvadability implies noncoexistence for a class of cancellative systems

**Jan M. Swart**

*(Institute of Information Theory and Automation of the ASCR (UTIA))*

#### 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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-12

Publication Date: May 23, 2013

DOI: 10.1214/ECP.v18-2471

#### References

- Andjel, Enrique D.; Liggett, Thomas M.; Mountford, Thomas. Clustering in one-dimensional threshold voter models.
*Stochastic Process. Appl.*42 (1992), no. 1, 73--90. MR1172508 - Belhaouari, S.; Mountford, T.; Sun, Rongfeng; Valle, G. Convergence results and sharp estimates for the voter model
interfaces.
*Electron. J. Probab.*11 (2006), no. 30, 768--801 (electronic). MR2242663 - Cox, J. T.; Durrett, R. Nonlinear voter models.
*Random walks, Brownian motion, and interacting particle systems,*189--201, Progr. Probab., 28,*Birkhäuser Boston, Boston, MA,*1991. MR1146446 - Cox, J. T.; Durrett, R. Hybrid zones and voter model interfaces.
*Bernoulli*1 (1995), no. 4, 343--370. MR1369166 - Cox, J. Theodore; Perkins, Edwin A. Survival and coexistence in stochastic spatial Lotka-Volterra
models.
*Probab. Theory Related Fields*139 (2007), no. 1-2, 89--142. MR2322693 - Durrett, Rick; Neuhauser, Claudia. Coexistence results for some competition models.
*Ann. Appl. Probab.*7 (1997), no. 1, 10--45. MR1428748 - Durrett, Rick. Mutual invadability implies coexistence in spatial models.
*Mem. Amer. Math. Soc.*156 (2002), no. 740, viii+118 pp. MR1879853 - Griffeath, David. Additive and cancellative interacting particle systems.
Lecture Notes in Mathematics, 724.
*Springer, Berlin,*1979. iv+108 pp. ISBN: 3-540-09508-X MR0538077 - Handjani, Shirin J. The complete convergence theorem for coexistent threshold voter
models.
*Ann. Probab.*27 (1999), no. 1, 226--245. MR1681118 - Jansen, Sabine and Kurt, Noemi: On the notion(s) of duality for Markov processes. Preprint, 50 pages, arXiv:1210.7193v1
- Liggett, Thomas M. Interacting particle systems.
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles
of Mathematical Sciences], 276.
*Springer-Verlag, New York,*1985. xv+488 pp. ISBN: 0-387-96069-4 MR0776231 - Liggett, Thomas M. Coexistence in threshold voter models.
*Ann. Probab.*22 (1994), no. 2, 764--802. MR1288131 - Neuhauser, Claudia; Pacala, Stephen W. An explicitly spatial version of the Lotka-Volterra model with interspecific competition.
*Ann. Appl. Probab.*9 (1999), no. 4, 1226--1259. MR1728561 - Sturm, Anja; Swart, Jan. Voter models with heterozygosity selection.
*Ann. Appl. Probab.*18 (2008), no. 1, 59--99. MR2380891 - Sturm, Anja; Swart, Jan M. Subcritical contact processes seen from a typical infected site. Preprint, 41 pages, arXiv:1110.4777v2
- Swart, Jan M.; Vrbenský, Karel. Numerical analysis of the rebellious voter model.
*J. Stat. Phys.*140 (2010), no. 5, 873--899. MR2673338

This work is licensed under a Creative Commons Attribution 3.0 License.