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Phenotypic diversity and population growth in a fluctuating environment

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Clément Dombry ,
Christian Mazza  and
Vincent Bansaye
Clément Dombry*
Affiliation:
Université de Poitiers
Christian Mazza*
Affiliation:
Université de Fribourg
Vincent Bansaye*
Affiliation:
École Polytechnique
*
∗ Postal address: Laboratoire de Mathématiques et Applications, Téléport 2-BP30179, Boulevard Pierre et Marie Curie, 86962 Futuroscope Chasseneuil Cedex, France. Email address: clement.dombry@math.univ-poitiers.fr
∗∗ Postal address: Département de Mathématique, Université de Fribourg, Chemin du Musée 23, CH-1700 Fribourg, Switzerland. Email address: christian.mazza@unifr.ch
∗∗∗ Postal address: CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France. Email address: bansaye@polytechnique.edu
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Abstract

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Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait t ∈ in environment e ∈ ε is given by some (fixed) distribution ϒ t,e on ℕ. Then, the phenotypes are attributed using a distribution (strategy) πt,e on the trait space . We look for the optimal strategy πt,e , t ∈ , e ∈ ε, maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus nonhereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of nonhereditary strategies: thanks to a reduction to simple branching processes in a random environment, we derive an exact expression for the net growth rate and a characterization of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

Keywords

Branching process in a random environmentphenotypic diversityLyapounov exponentoptimal strategyextinctiontypical genealogy

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K37: Processes in random environments

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 375 - 398
Copyright
Copyright © Applied Probability Trust 2011 

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