Acta Applicandae Mathematicae

, Volume 131, Issue 1, pp 1–27 | Cite as

A New Proof for the Convergence of an Individual Based Model to the Trait Substitution Sequence



We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and the process is viewed in the timescale of mutations. Using averaging techniques due to Kurtz (in Lecture Notes in Control and Inform. Sci., vol. 177, pp. 186–209, 1992), we give a new proof of the convergence of the individual based model to the trait substitution sequence of Metz et al. (in Trends in Ecology and Evolution 7(6), 198–202, 1992), first worked out by Dieckman and Law (in Journal of Mathematical Biology 34(5–6), 579–612, 1996) and rigorously proved by Champagnat (in Theoretical Population Biology 69, 297–321, 2006): rigging the model such that “invasion implies substitution”, we obtain in the limit a process that jumps from one population equilibrium to another when mutations occur and invade the population.


Birth and death process Structured population Adaptive dynamics Individual based model Averaging technique Trait substitution sequence 

Mathematics Subject Classification (2000)

92D15 60J80 60K35 60F99 



The authors thank Sylvie Méléard for invaluable discussions. We also wish to give special thanks to a reviewer whose suggestions were very helpful in improving the presentation of our paper. This work benefited from the support of the ANR MANEGE (ANR-09-BLAN-0215), from the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement-Ecole Polytechnique-Museum National d’Histoire Naturelle-Fondation X. V.C.T. was also supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Ankit Gupta
    • 1
  • J. A. J. Metz
    • 2
    • 3
  • Viet Chi Tran
    • 1
    • 4
  1. 1.CMAP, Ecole PolytechniqueUMR CNRS 7641Palaiseau CédexFrance
  2. 2.Mathematical Institute & Institute of Biology & NCB Naturalis, LeidenLeidenThe Netherlands
  3. 3.EEPIIASALaxenburgAustria
  4. 4.Equipe Probabilité Statistique, Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de MathématiquesUniversité des Sciences et Technologies Lille 1Villeneuve d’Ascq CédexFrance

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