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
[Une inégalité pour les valeurs propres de Floquet et de Perron de systèmes différentiels monotones et d'équations structurées en âge]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2007-04-24
Accepté le : 2007-09-28
Publié le : 2007-11-07

DOI : 10.1016/j.crma.2007.10.001

Affiliations des auteurs :

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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     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},
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     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/

Bibliographie
Cité par

[1] S. Bernard; D. Gonze; B. Čajavec; H. Herzel; A. Kramer Synchronization-induced rhythmicity of circadian oscillators in the suprachiasmatic nucleus, PLoS Computational Biology, Volume 3 (2007) no. 4, p. e68 | DOI

[2] G. Chiorino; J.A.J. Metz; D. Tomasoni; P. Ubezio Desynchronization rate in cell populations: mathematical modeling and experimental data, J. Theor. Biol., Volume 208 (2001), pp. 185-199

[3] J. Clairambault; P. Michel; B. Perthame Circadian rhythm and tumour growth, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006) no. 1, pp. 17-22

[4] E. Filipski; P.F. Innominato; M.W. Wu; X.M. Li; S. Iacobelli; L.J. Xian; F. Lévi Effect of light and food schedules on liver and tumor molecular clocks in mice, J. Nat. Cancer Inst., Volume 97 (2005) no. 7, pp. 507-517

[5] A. Goldbeter A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase, Proc. Nat. Acad. Sci. USA, Volume 88 (1991), pp. 9107-9111

[6] J.-C. Leloup; A. Goldbeter Modeling the mammalian circadian clock: Sensitivity analysis and multiplicity of oscillatory mechanisms, J. Theor. Biol., Volume 230 (2004), pp. 541-562

[7] J.A.J. Metz; O. Diekmann The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68, Springer-Verlag, 1986

[8] P. Michel; S. Mischler; B. Perthame General relative entropy inequality: an illustration on growth models, J. Math. Pures Appl., Volume 84 (2005) no. 9, pp. 1235-1260

[9] E. Nagoshi; C. Saini; C. Bauer; T. Laroche; F. Naef; U. Schibler Circadian gene expression in individual fibroblasts: cell-autonomous and self-sustained oscillators pass time to daughter cells, Cell, Volume 119 (2004), pp. 693-705

[10] B. Perthame Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser, Basel, 2007

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