Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

On the variations of the principal eigenvalue with respect to a parameter in growth-fragmentation models

Pages: 1801 – 1819

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n7.a1

Authors

Fabien Campillo (Inria, MATHNEURO, Montpellier, France; and Institut Montpelliérain Alexander Grothendieck, Montpellier, France)

Nicolas Champagnat (Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, Vandoeuvre-lès-Nancy, France; and Inria, TOSCA, Villers-lès-Nancy, France)

Coralie Fritsch (Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, Vandoeuvre-lès-Nancy, France; Inria, TOSCA, Villers-lès-Nancy, France; and CMAP, UMR CNRS 7641, École Polytechnique, Palaiseau, France)

Abstract

We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we deduce the variations of the eigenvalue with respect to the parameter of the model.

Keywords

growth-fragmentation model, eigenproblem, integro-differential equation, invasion fitness, individual-based model, infinite dimensional branching process, piecewise-deterministic Markov process, bacterial population

2010 Mathematics Subject Classification

35Q92, 45C05, 60J25, 60J80, 60J85, 92D25

Full Text (PDF format)

Received 1 February 2016

Accepted 25 February 2017

Published 16 October 2017