An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations

Jean Clairambault; Stéphane Gaubert; Benoît Perthame

doi:10.1016/j.crma.2007.10.001

Partial Differential Equations

An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations

Presented by: Alain Bensoussan

Jean Clairambault ¹; Stéphane Gaubert ²; Benoît Perthame ^1,³

¹ INRIA, projet BANG, domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France
² INRIA, projet MAXPLUS, domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France
³ Département de mathématiques et applications, UMR 8553, École normale supérieure, 45, rue d'Ulm, 75230 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 549-554.

Abstract
Résumé

For monotone linear differential systems with periodic coefficients, the (first) Floquet eigenvalue measures the growth rate of the system. We define an appropriate arithmetico-geometric time average of the coefficients for which we can prove that the Perron eigenvalue is smaller than the Floquet eigenvalue. We apply this method to Partial Differential Equations, and we use it for an age-structured systems of equations for the cell cycle. This opposition between Floquet and Perron eigenvalues models the loss of circadian rhythms by cancer cells.

La (première) valeur propre de Floquet décrit le taux de croissance des systèmes différentiels linéaires monotones à coefficients périodiques. Nous définissons une moyenne arithmético-géométrique en temps des coefficients, qui nous permet de démontrer que la valeur propre de Perron pour le système ainsi moyenné est plus petite que celle de Floquet. La méthode s'applique aux Équations aux Dérivées Partielles et nous l'utilisons pour un système d'équations structurées en âge qui décrit le cycle cellulaire. Cette opposition entre valeurs propres de Floquet et de Perron modélise la perte de contrôle circadien pour le cycle cellulaire des cellules cancéreuses.

Received: 2007-04-24
Accepted: 2007-09-28
Published online: 2007-11-07

DOI: 10.1016/j.crma.2007.10.001

Author's affiliations:

Jean Clairambault ¹; Stéphane Gaubert ²; Benoît Perthame ^{1, 3}

¹ INRIA, projet BANG, domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France
² INRIA, projet MAXPLUS, domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France
³ Département de mathématiques et applications, UMR 8553, École normale supérieure, 45, rue d'Ulm, 75230 Paris cedex 05, France

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     author = {Jean Clairambault and St\'ephane Gaubert and Beno{\^\i}t Perthame},
     title = {An inequality for the {Perron} and {Floquet} eigenvalues of monotone differential systems and age structured equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {549--554},
     publisher = {Elsevier},
     volume = {345},
     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.001},
     language = {en},
}

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Jean Clairambault; Stéphane Gaubert; Benoît Perthame. An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 549-554. doi : 10.1016/j.crma.2007.10.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.001/

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