A New Proof for the Convergence of an Individual Based Model to the Trait Substitution Sequence
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A New Proof for the Convergence of an Individual Based Model to the Trait Substitution Sequence Ankit Gupta · J.A.J. Metz · Viet Chi Tran
Received: 13 March 2012 / Accepted: 18 September 2013 © Springer Science+Business Media Dordrecht 2013
Abstract 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. Keywords Birth and death process · Structured population · Adaptive dynamics · Individual based model · Averaging technique · Trait substitution sequence
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A. Gupta ( ) · V.C. Tran CMAP, Ecole Polytechnique, UMR CNRS 7641, Route de Saclay, 91128 Palaiseau Cédex, France e-mail: [email protected] J.A.J. Metz Mathematical Institute & Institute of Biology & NCB Naturalis, Leiden, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands e-mail: [email protected] J.A.J. Metz EEP, IIASA, Laxenburg, Austria V.C. Tran Equipe Probabilité Statistique, Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques, Université des Sciences et Technologies Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cédex, France e-mail: [email protected]
A. Gupta et al.
Mathematics Subject Classification (2000) 92D15 · 60J80 · 60K35 · 60F99
1 Introduction: The Logistic Birth and Death Model We consider a stochastic individual based model (IBM) with trait structure and evolving as a result of births and deaths, that was introduced by Dieckmann and Law [9] and Metz et al. [23] and in rigorous detail by Champagnat [2]. We study its limit in an evolutionary time scale when the population is large and the mutations are rare. Champagnat [2] established the first rigorous proof of the convergence of a sequence of such IBMs to the trait substitution sequence process (TSS) introduced by Metz et al. [22] (with Metz et al. [23] as a follow up). Following Dieckmann and Law [9], the TSS can be explained as follows. In the limit, the time scales of ecology and evolution are separated. Mutations are rare and before a mutant arises, the resident population stabilizes around an equilibrium. Under the “invasion implies substitution” Ansatz, there cannot be long term coexistence of two differe
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