Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-29T08:06:51.275Z Has data issue: false hasContentIssue false
  • English
  • Français

On the large-time behaviour of the solution of a stochastic differential equation driven by a Poisson point process

Part of: Markov processes Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  26 June 2017

Elma Nassar  and
Etienne Pardoux
Elma Nassar*
Affiliation:
Aix-Marseille University
Etienne Pardoux*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
*
* Postal address: CMI, Aix-Marseille University, 39 Rue F. Joliot Curie, F-13453 Marseille, France.
** Postal address: CMI, Universite d'Aix Marseille, 39 rue F. Joliot Curie, F-13453 Marseille, France. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a stochastic differential equation driven by a Poisson point process, which models the continuous change in a population's environment, as well as the stochastic fixation of beneficial mutations that might compensate for this change. The fixation probability of a given mutation increases as the phenotypic lag Xt between the population and the optimum grows larger, and successful mutations are assumed to fix instantaneously (leading to an adaptive jump). Our main result is that the process is transient (i.e. converges to -∞, so that continued adaptation is impossible) if the rate of environmental change v exceeds a parameter m, which can be interpreted as the rate of adaptation in case every beneficial mutation becomes fixed with probability 1. If v < m, the process is Harris recurrent and possesses a unique invariant probability measure, while in the limiting case m = v, Harris recurrence with an infinite invariant measure or transience depends upon additional technical conditions. We show how our results can be extended to a class of time varying rates of environmental change.

Keywords

Stochastic differential equations with jumpslarge-time behaviourtransiencerecurrence

MSC classification

Primary: 60H10: Stochastic ordinary differential equations 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G57: Random measures 92D25: Population dynamics (general)
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 344 - 367
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA. Google Scholar
[2] Bürger, R. (1999). Evolution of genetic variability and the advantage of sex and recombination in changing environments. Genetics 153, 1055–1069. Google Scholar
[3] Bürger, R. and Lynch, M. (1995). Evolution and extinction in a changing environment: a quantitative-genetic analysis. Evolution 49, 151–163. Google Scholar
[4] Champagnat, N. (2006). A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stoch. Process. Appl. 116, 1127–1160. Google Scholar
[5] Dieckmann, U., Marrow, P. and Law, R. (1995). Evolutionary cycling in predator–prey interactions: population dynamics and the Red Queen. J. Theoret. Biol. 176, 91–102. Google Scholar
[6] Haldane, J. B. S. (1927). A mathematical theory of natural and artificial selection, part V: selection and mutation. Proc. Camb. Phil. Soc. 23, 838–844. Google Scholar
[7] Kauffman, S. and Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theoret. Biol. 128, 11–45. Google Scholar
[8] Kersting, G. (1986). On recurrence and transience of growth models. J. Appl. Prob. 23, 614–625. Google Scholar
[9] Kimura, M. (1962). On the probability of fixation of mutant genes in a population. Genetics 47, 713–719. CrossRefGoogle ScholarPubMed
[10] Kopp, M. and Hermisson, J. (2009). The genetic basis of phenotypic adaptation. I: fixation of beneficial mutations in the moving optimum model. Genetics 182, 233–249. CrossRefGoogle ScholarPubMed
[11] Kopp, M. and Hermisson, J. (2009). The genetic basis of phenotypic adaptation. II: the distribution of adaptive substitutions in the moving optimum model. Genetics 183, 1453–1476. Google Scholar
[12] Malécot, G. (1952). Les processus stochastiques et la méthode des fonctions génératrices ou caractéristiques. Publ. Inst. Statist. Univ. Paris 1, 1–16. Google Scholar
[13] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. II: Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517. Google Scholar
[14] Peischl, S. and Kirkpatrick, M. (2012). Establishment of new mutations in changing environments. Genetics 191, 895–906. Google Scholar
[15] Protter, P. E. (2005). Stochastic Integration and Differential Equations (Stoch. Modelling Appl. Prob. 21), 2nd edn. Springer, Berlin. Google Scholar
[16] Uecker, H. and Hermisson, J. (2011). On the fixation process of a beneficial mutation in a variable environment. Genetics 188, 915–930. Google Scholar