The Annals of Applied Probability

Voter models with heterozygosity selection

Anja Sturm and Jan Swart

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This paper studies variations of the usual voter model that favor types that are locally less common. Such models are dual to certain systems of branching annihilating random walks that are parity preserving. For both the voter models and their dual branching annihilating systems we determine all homogeneous invariant laws, and we study convergence to these laws started from other initial laws.

Article information

Ann. Appl. Probab., Volume 18, Number 1 (2008), 59-99.

First available in Project Euclid: 9 January 2008

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Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35]
Secondary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Heterozygosity selection negative frequency dependent selection rebellious voter model branching annihilation parity preservation cancellative systems survival coexistence


Sturm, Anja; Swart, Jan. Voter models with heterozygosity selection. Ann. Appl. Probab. 18 (2008), no. 1, 59--99. doi:10.1214/07-AAP444.

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  • Blath, J., Etheridge, A. M. and Meredith, M. E. (2007). Coexistence in locally regulated competing populations and survival of BARW. Ann. Appl. Probab. 17 1474--1507.
  • Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann. Probab. 18 1462--1482.
  • Bramson, M. and Durrett, R. (1988). A simple proof of the stability theorem of Gray and Griffeath. Probab. Theory Related Fields 80 293--298.
  • Bramson, M., Ding, W. and Durrett, R. (1991). Annihilating branching processes. Stoch. Process. Appl. 37 1--17.
  • Cardy, J. L. and Täuber, U. C. (1996). Theory of branching and annihilating walks. Phys. Rev. Lett. 77 4780--4783.
  • Cardy, J. L. and Täuber, U. C. (1998). Field theory of branching and annihilating random walks. J. Stat. Phys. 90 1--56.
  • Cox, J. T. and Durrett, R. (1991). Nonlinear voter models. In Random Walks, Brownian Motion and Interacting Particle Systems. A Festschrift in Honor of Frank Spitzer 189--201. Birkäuser, Boston.
  • Cox, J. T. and Perkins, E. A. (2005). Rescaled Lotka--Volterra models converge to super-Brownian motion. Ann. Probab. 33 904--947.
  • Cox, J. T. and Perkins, E. A. (2007). Survival and coexistence in stochastic spatial Lotka--Volterra models. Probab. Theory Related Fields 139 89--142.
  • Durrett, R. and Griffeath, D. (1982). Contact processes in several dimensions. Z. Wahrsch. Verw. Gebiete 59 535--552.
  • Durrett, R. (1991). A new method for proving the existence of phase transitions. In Spatial Stochastic Processes, Festschr. in Honor of Ted Harris 70th Birthday (K. S. Alexander and J. C. Watkins, eds.) Prog. Probab. 19 141--169. Birkhäuser, Boston.
  • Durrett, R. (1995). Ten lectures on particle systems. Lecture Notes in Math. 1608 97--201. Springer, Berlin.
  • Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math. 724. Springer, Berlin.
  • Handjani, S. J. (1999). The complete convergence theorem for coexistent threshold voter models. Ann. Probab. 27 226--245.
  • Harris, T. E. (1976). On a class of set-valued Markov processes. Ann. Probab. 4 175--194.
  • Kallenberg, O. (2002). Foundations of Modern Probability. Springer, New York.
  • Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
  • Liggett, T. M. (1994). Coexistence in threshold voter models. Ann. Probab. 22 764--802.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Process. Springer, Berlin.
  • Mountford, T. S. (1995). A coupling of infinite particle systems. J. Math. Kyoto Univ. 35 43--52.
  • Neuhauser, C. and Pacala, S. W. (1999). An explicitly spatial version of the Lotka--Volterra model with interspecific competition. Ann. Appl. Probab. 9 1226--1259.
  • Simonelli, I. (1995). A limit theorem for a class of interacting particle systems. Ann. Probab. 23 141--156.
  • Sturm, A. and Swart, J. M. (2007). Tightness of voter model interfaces. Unpublished manuscript. Available at arxiv:0706.4405v2.
  • Sudbury, A. (1990). The branching annihilating process: An interacting particle system. Ann. Probab. 18 581--601.
  • Täuber, U. C. (2003). Scale invariance and dynamic phase transitions in diffusion-limited reactions. Adv. in Solid State Phys. 43 659--676.