Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T06:32:39.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Modeling of Density-Dependent Diploid Populations and the Extinction Vortex

Part of: Markov processes

Published online by Cambridge University Press:  22 February 2016

Camille Coron
Camille Coron*
Affiliation:
École Polytechnique
*
∗ Current address: Université d'Orsay, Laboratoire de Mathématiques, 91405 Orsay Cedex, France. Email address: camille.coron@math.u-psud.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We model and study the genetic evolution and conservation of a population of diploid hermaphroditic organisms, evolving continuously in time and subject to resource competition. In the absence of mutations, the population follows a three-type, nonlinear birth-and-death process, in which birth rates are designed to integrate Mendelian reproduction. We are interested in the long-term genetic behavior of the population (adaptive dynamics), and in particular we compute the fixation probability of a slightly nonneutral allele in the absence of mutations, which involves finding the unique subpolynomial solution of a nonlinear three-dimensional recurrence relationship. This equation is simplified to a one-dimensional relationship which is proved to admit exactly one bounded solution. Adding rare mutations and rescaling time, we study the successive mutation fixations in the population, which are given by the jumps of a limiting Markov process on the genotypes space. At this time scale, we prove that the fixation rate of deleterious mutations increases with the number of already fixed mutations, which creates a vicious circle called the extinction vortex.

Keywords

Population geneticsdiploid populationnonlinear birth-and-death processfixation probabilityDirichlet problemmultidimensional nonlinear recurrence equationextinction vortex

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general) 92D15: Problems related to evolution
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 446 - 477
Copyright
© Applied Probability Trust 

References

Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch. Process. Appl. 116, 1127–1160.CrossRefGoogle Scholar
Champagnat, N. and Lambert, A. (2007). Evolution of discrete populations and the canonical diffusion of adaptive dynamics. Ann. Appl. Prob. 17, 102–155.CrossRefGoogle Scholar
Champagnat, N. and Méléard, S. (2011). Polymorphic evolution sequence and evolutionary branching. Prob. Theory Relat. Fields 151, 45–94.CrossRefGoogle Scholar
Champagnat, N., Ferrière, R. and Méléard, S. (2006). Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theoret. Pop. Biol. 69, 297–321.CrossRefGoogle ScholarPubMed
Collet, P., Martı´nez, S. and San Martı´n, J. (2013). Birth and death chains. In Quasi-Stationary Distributions. Springer, Berlin, pp. 66–111.CrossRefGoogle Scholar
Collet, P., Méléard, S. and Metz, J. A. J. (2013). A rigorous model study of the adaptive dynamics of Mendelian diploids. J. Math. Biol. 67, 569–607.CrossRefGoogle ScholarPubMed
Coron, C., Méléard, S., Porcher, E. and Robert, A. (2013). Quantifying the mutational meltdown in diploid populations. Amer. Naturalist 181, 623–636.CrossRefGoogle ScholarPubMed
Crow, J. and Kimura, M. (1970). An Introduction to Population Genetics Theory, 2nd edn. Harper and Row, New York.Google Scholar
Dieudonné, J. (1969). Éléments d'Analyse, Vol. 1. Gauthier-Villars, Paris.Google Scholar
Durrett, R. (2010). Probability Theory and Examples, 4th edn. Cambridge University Press.CrossRefGoogle Scholar
Lande, R. (1994). Risk of population extinction from fixation of new deleterious mutations. Evolution 48, 1460–1469.CrossRefGoogle ScholarPubMed
Lynch, M., Conery, J. and Burger, R. (1995). Mutation accumulation and the extinction of small populations. Amer. Naturalist 146, 489–518.CrossRefGoogle Scholar
Lynch, M. and Gabriel, W. (1990). Mutation load and the survival of small populations. Evolution 44, 1725–1737.CrossRefGoogle ScholarPubMed
Metz, J. et al. (1996). Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. In Stochastic and Spatial Structures of Dynamical Systems, eds van Strien, S. J. and Verduyn-Lunel, S. M., North Holland, Amsterdam, pp. 183–231.Google Scholar
Norris, J. (1997). Markov Chains. Cambridge University Press.CrossRefGoogle Scholar
Orr, H. A. (1998). The population genetics of adaptation: the distribution of factors fixed during adaptive evolution. Evolution 52, 935–949.CrossRefGoogle ScholarPubMed
Orr, H. A. (1999). The evolutionary genetics of adaptation: a simulation study. Genet. Res. 74, 207–214.Google ScholarPubMed
Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.CrossRefGoogle Scholar
Soulé, M. E. (ed.) (1986). Conservation Biology: The Science of Scarcity and Diversity. Sinauer Associates, Sunderland, MA.Google Scholar