Nonlinear historical superprocess approximations for population models with past dependence

Sylvie Méléard (École Polytechnique)
Viet Chi Tran (Université des Sciences et Technologies de Lille)


We are interested in the evolving genealogy of a birth and death process with trait structure and ecological interactions. Traits are hereditarily transmitted from a parent to its offspring unless a mutation occurs. The dynamics may depend on the trait of the ancestors and on its past and allows interactions between individuals through their lineages. We define an interacting historical particle process  describing the  genealogies of the living individuals; it takes values in the space of point measures  on an infinite dimensional càdlàg path space. This individual-based process can be approximated by  a nonlinear historical superprocess, under the assumptions of large populations, small individuals and allometric demographies. Because of the interactions, the branching property fails and we use martingale problems and fine couplings between our population and independent branching particles. Our convergence theorem is illustrated by two examples of current interest in biology. The first one relates the biodiversity history of a population and its phylogeny, while the second treats a spatial model where individuals compete through their past trajectories.

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Pages: 1-32

Publication Date: June 18, 2012

DOI: 10.1214/EJP.v17-2093


  • Adler, Robert; Tribe, Roger. Uniqueness for a historical SDE with a singular interaction. J. Theoret. Probab. 11 (1998), no. 2, 515--533. MR1622584
  • Bansaye, Vincent; Delmas, Jean-François; Marsalle, Laurence; Tran, Viet Chi. Limit theorems for Markov processes indexed by continuous time Galton-Watson trees. Ann. Appl. Probab. 21 (2011), no. 6, 2263--2314. MR2895416
  • Barton, Nick H.; Etheridge, Alison M. The effect of selection on genealogies, Genetics 166 (2004), 1115--1131.
  • Barton, Nick H.; Etheridge, Alison M.; Véber, Amandine. A new model for evolution in a spatial continuum. Electron. J. Probab. 15 (2010), no. 7, 162--216. MR2594876
  • Berestycki, Nathanaël. Recent progress in coalescent theory. Ensaios Matemáticos [Mathematical Surveys], 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. 193 pp. ISBN: 978-85-85818-40-1 MR2574323
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396
  • Billingsley, Patrick. Probability and measure. Third edition. Wiley Series in Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. xiv+593 pp. ISBN: 0-471-00710-2 MR1324786
  • Bolker, Ben; Pacala, Stephen. Using moment equations to understand stochastically driven spatial pattern formation in ecological systems, Theoretical Population Biology 52 (1997), 179--197.
  • Bolker, Ben; Pacala, Stephen. Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal, The American Naturalist 153 (1999), 575--602.
  • Champagnat, Nicolas; Ferrière, Régis; Méléard, Sylvie. Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation, Theoretical Population Biology 69 (2006), 297--321.
  • Champagnat, Nicolas; Ferrière, Régis; Méléard, Sylvie. From individual stochastic processes to macroscopic models in adaptive evolution. Stoch. Models 24 (2008), suppl. 1, 2--44. MR2466448
  • Champagnat, Nicolas; Méléard, Sylvie. Polymorphic evolution sequence and evolutionary branching. Probab. Theory Related Fields 151 (2011), no. 1-2, 45--94. MR2834712
  • Dawson, Donald A. Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI—1991, 1--260, Lecture Notes in Math., 1541, Springer, Berlin, 1993. MR1242575
  • Dawson, D. A. Geostochastic calculus. Canad. J. Statist. 6 (1978), no. 2, 143--168. MR0532855
  • Dawson, Donald A.; Perkins, Edwin A. Historical processes. Mem. Amer. Math. Soc. 93 (1991), no. 454, iv+179 pp. MR1079034
  • Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Tree-valued Fleming-Viot dynamics with mutation and selection, Annals of Applied Probability (2011), in press.
  • Dieckmann, Ulf; Doebeli, Michael. On the origin of species by sympatric speciation, Nature 400 (1999), 354--357.
  • Dieckmann, Ulf; Law, Richard. Relaxation projections and the method of moments, The Geometry of Ecological Interactions, Cambridge University Press, 2000, pp. 412--455.
  • Etheridge, Alison M. An introduction to superprocesses. University Lecture Series, 20. American Mathematical Society, Providence, RI, 2000. xii+187 pp. ISBN: 0-8218-2706-5 MR1779100
  • Etheridge, Alison; Pfaffelhuber, Peter; Wakolbinger, Anton. An approximate sampling formula under genetic hitchhiking. Ann. Appl. Probab. 16 (2006), no. 2, 685--729. MR2244430
  • Evans, Steven N.; Perkins, Edwin A. Measure-valued branching diffusions with singular interactions. Canad. J. Math. 46 (1994), no. 1, 120--168. MR1260341
  • Evans, Steven N.; Steinsaltz, David. Damage segregation at fissioning may increase growth rates: A superprocess model, Theoretical Population Biology 71 (2007), 473--490.
  • Fournier, Nicolas; Méléard, Sylvie. A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 (2004), no. 4, 1880--1919. MR2099656
  • Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Tree-valued resampling dynamics. martingale problems and applications, Probability Theory and Related Fields (2011), in press.
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1 MR0959133
  • Jagers, Peter. A general stochastic model for population development. Skand. Aktuarietidskr. 1969, 84--103. MR0431406
  • Jagers, Peter. Branching processes with biological applications. Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistics. Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. xiii+268 pp. ISBN: 0-471-43652-6 MR0488341
  • Jagers, Peter; Nerman, Olle. The asymptotic composition of supercritical multi-type branching populations. Séminaire de Probabilités, XXX, 40--54, Lecture Notes in Math., 1626, Springer, Berlin, 1996. MR1459475
  • Jakubowski, Adam. On the Skorokhod topology. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 3, 263--285. MR0871083
  • Joffe, A.; Métivier, M. Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 (1986), no. 1, 20--65. MR0827331
  • Kisdi, Eva. Evolutionary branching under asymmetric competition, J. Theor. Biol. 197 (1999), no. 2, 149--162.
  • Krone, Steve; Neuhauser, Claudia. Ancestral processes with selection, Theoretical Population Biology 51 (1997), 210--237.
  • Méléard, Sylvie; Tran, Viet Chi. Trait substitution sequence process and canonical equation for age-structured populations. J. Math. Biol. 58 (2009), no. 6, 881--921. MR2495555
  • Méléard, Sylvie; Tran, Viet Chi. Slow and fast scales for superprocess limits of age-structured populations. Stochastic Process. Appl. 122 (2012), no. 1, 250--276. MR2860449
  • Morlon, Hélène; Parsons, Todd L.; Plotkin, Joshua. Reconciling molecular phylogenies with the fossil record, Proceedings of the National Academy of Sciences 108 (2001), 16327--16332.
  • Perkins, Edwin. Conditional Dawson-Watanabe processes and Fleming-Viot processes, Seminar on Stochastic Processes 1991 (Birkhaüser, ed.), Progr. Probab., vol. 29, 1992, pp. 142--155.
  • Perkins, Edwin. On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc. 115 (1995), no. 549, vi+89 pp. MR1249422
  • Roughgarden, J.A. Theory of population genetics and evolutionary ecology: an introduction, Macmillan, New York, 1979.
  • Rudin, Walter. Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1 MR0924157
  • Tran, Viet Chi. Large population limit and time behaviour of a stochastic particle model describing an age-structured population. ESAIM Probab. Stat. 12 (2008), 345--386. MR2404035

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