Population processes with unbounded extinction rate conditioned to non-extinction

Nicolas Champagnat 1, 2 Denis Villemonais 3, 2, 1
1 Probabilités et statistiques
IECL - Institut Élie Cartan de Lorraine
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : This article studies the quasi-stationary behaviour of population processes with unbounded absorption rate, including one-dimensional birth and death processes with catastrophes and multi-dimensional birth and death processes, modeling biological populations in interaction. To handle this situation, we develop original non-linear Lyapunov criteria. We obtain the exponential convergence in total variation of the conditional distributions to a unique quasi-stationary distribution, uniformly with respect to the initial distribution. Our results cover all one-dimensional birth and death processes which come down from infinity with catastrophe rate satisfying appropriate bounds, and multi- dimensional birth and death models with stronger intra-specific than inter-specific competition.
Type de document :
Pré-publication, Document de travail
2016
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  • HAL Id : hal-01395731, version 1
  • ARXIV : 1611.03010

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Nicolas Champagnat, Denis Villemonais. Population processes with unbounded extinction rate conditioned to non-extinction . 2016. 〈hal-01395731〉

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