A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence
Abstract
We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth- and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [Méléard and Tran, EJP, 2012], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth- and death-events the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-13
Publication Date: October 18, 2014
DOI: 10.1214/EJP.v19-3506
References
- 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, 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. Marked metric measure spaces. Electron. Commun. Probab. 16 (2011), 174--188. MR2783338
- 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
- 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. Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 (2009), no. 1-2, 285--322. MR2520129
- Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields 155 (2013), no. 3-4, 789--838. MR3034793
- S. Kliem and A. Winter. Evolving phylogenies of trait-dependent branching with mutation and competition. In preparation (2014).
- Méléard, Sylvie; Tran, Viet Chi. Nonlinear historical superprocess approximations for population models with past dependence. Electron. J. Probab. 17 (2012), no. 47, 32 pp. MR2946154
- 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
- Perkins, Edwin. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125--324, Lecture Notes in Math., 1781, Springer, Berlin, 2002. MR1915445
This work is licensed under a Creative Commons Attribution 3.0 License.