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Voter Model and Biased Voter Model in Heterogeneous Environments

Part of: Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  14 July 2016

N. Lanchier  and
C. Neuhauser
N. Lanchier
Affiliation:
Université de Rouen
C. Neuhauser*
Affiliation:
University of Minnesota
*
∗∗Postal address: Ecology, Evolution and Behavior, University of Minnesota, 1987 Upper Buford Circle, St Paul, MN 55108, USA.
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Abstract

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With the rapid adoption of transgenic crops, gene flow from transgenic crops to wild relatives through pollen dispersal is of significant concern and warrants both empirical and theoretical studies to assess the risk of introduction of transgenes into wild populations. We propose to use the (biased) voter model in a heterogeneous environment to investigate the effects of recurrent gene flow from transgenic crop to wild relatives. The model is defined on the d-dimensional integer lattice that is divided into two parts, Δ and Zd \ Δ. Individuals carrying the transgene and individuals carrying the wild type gene compete according to the evolution rules of a (biased) voter model on Zd \ Δ, while the process is conditioned to have only individuals carrying the transgene on Δ. Our main findings suggest that unless transgenes confer increased fitness in wild relatives, introgression of transgenes into populations of wild plants is slow and may even be reversible without intervention. Our study also addresses the effects of different spatial planting patterns of transgenic crops on the rate of introgression.

Keywords

Interacting particle systemsvoter model interfacesrandom walktransgenic crops

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C22: Interacting particle systems
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 770 - 787
Copyright
Copyright © Applied Probability Trust 2007 

Footnotes

∗

Current address: Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA. Email address: lanchier@math.asu.edu

Partially supported by NSF Grants DMS-00-72262 and DMS-00-83468

References

[1] Bramson, M. and Griffeath, D. (1980). On the Williams–Bjerknes tumor growth model. II. Math. Proc. Camb. Philos. Soc. 88, 339–357.CrossRefGoogle Scholar
[2] Bramson, M. and Griffeath, D. (1981). On the Williams–Bjerknes tumor growth model. I. Ann. Prob. 9, 173–185.CrossRefGoogle Scholar
[3] Cox, J. T. and Durrett, R. (1995). Hybrid zones and voter model interfaces. Bernoulli 1, 343–370.Google Scholar
[4] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth, Pacific Grove, CA.Google Scholar
[5] Ellstrand, N. C., Prentice, H. C. and Hancock, J. F. (1999). Gene flow and introgression from domesticated plants into their wild relatives. Ann. Rev. Ecol. Syst. 30, 539–563.Google Scholar
[6] Harris, T. E. (1972). Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89.Google Scholar
[7] Haygood, R., Ives, A. R. and Andow, D. A. (2003). Consequences of recurrent gene flow from crops to wild relatives. Proc. R. Soc. London B 270, 1879–1886.Google Scholar
[8] Holley, R. A. and Liggett, T.M. (1975). Ergodic theorems for weakly interacting systems and the voter model. Ann. Prob. 3, 643–663.Google Scholar
[9] Huxel, G. R. (1999). Rapid displacement of native species by invasive species: effects of hybridization. Biol. Conserv. 89, 143–152.CrossRefGoogle Scholar
[10] Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
[11] Spitzer, F. (1976). Principles of Random Walks, 2nd edn. Springer, New York.CrossRefGoogle Scholar
[12] Watrud, L. S. et al. (2004). Evidence of landscape-level, pollen-mediated gene flow from genetically modified creeping bentgrass with CP4 EPSPS as a marker. Proc. Nat. Acad. Sci. USA 101, 14533–14538.CrossRefGoogle ScholarPubMed
[13] Wolf, D. E., Takebayashi, N. and Rieseberg, L. H. (2001). Predicting the risk of extinction through hybridization. Conserv. Biol. 15, 1039–1053.Google Scholar