Graphical representation of some duality relations in stochastic population models

Martin Hutzenthaler (Johann Wolfgang Goethe-Universität Frankfurt, Germany)
Roland Alkemper (Johannes-Gutenberg Universität Mainz, Germany)

Abstract


We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. MR2123250) and for the self-duality of Feller's branching diffusion with logistic growth (cf. MR2308333). The two dual processes are approximated by particle processes which are forward and backward processes in a graphical representation. We identify duality relations between the basic building blocks of the particle processes which lead to the two dualities mentioned above.

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Pages: 206-220

Publication Date: July 2, 2007

DOI: 10.1214/ECP.v12-1283

References

  1. Athreya, Siva R.; Swart, Jan M. Branching-coalescing particle systems. Probab. Theory Related Fields 131 (2005), no. 3, 376--414. MR2123250 (2006d:60149)
  2. Diaconis, P.; Freedman, D. Finite exchangeable sequences. Ann. Probab. 8 (1980), no. 4, 745--764. MR0577313 (81m:60032)
  3. Donnelly, Peter. The transient behavior of the Moran model in population genetics. Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 2, 349--358. MR0735377 (85k:92038)
  4. Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085 (88a:60130)
  5. 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 (80i:60132)
  6. Harris, T. E. Additive set-valued Markov processes and graphical methods. Ann. Probability 6 (1978), no. 3, 355--378. MR0488377 (58 #7925)
  7. Hutzenthaler, M.; Wakolbinger, A. Ergodic behavior of locally regulated branching populations. Ann. Appl. Probab. 17 (2007), no. 2, 474--501. MR2308333
  8. Krone, S.M. and Neuhauser, C. (1997). Ancestral processes with selection. Theor. Popul. Biol. 51,3, 210--237.
  9. 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 (86e:60089)
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