# American Institute of Mathematical Sciences

November  2013, 33(11&12): 4891-4921. doi: 10.3934/dcds.2013.33.4891

## Hopf bifurcation for a size-structured model with resting phase

 1 Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, China 2 Institut de Mathématiques de Bordeaux, UMR CNRS 5251 - Case 36, Université Bordeaux Segalen, 3ter place de la Victoire, 33076 Bordeaux, France

Received  December 2011 Revised  September 2012 Published  May 2013

This article investigates Hopf bifurcation for a size-structured population dynamic model that is designed to describe size dispersion among individuals in a given population. This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and study it in the frame work of integrated semigroup theory. We prove a Hopf bifurcation theorem and we present some numerical simulations to support our analysis.
Citation: Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
##### References:
 [1] H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, in "Delay Differential Equations and Dynamical Systems" (eds. S. N. Busenberg and M. Martelli), in: Lect. Notes Math., 1475, Springer-Verlag, Berlin, (1991), 53-63. doi: 10.1007/BFb0083479.  Google Scholar [2] O. Arino, A survey of structured cell population dynamics, Acta Biotheoret., 43 (1995), 3-25. doi: 10.1007/BF00709430.  Google Scholar [3] O. Arino and E. Sanchez, A survey of cell population dynamics, J. Theor. Med., 1 (1997), 35-51. doi: 10.1080/10273669708833005.  Google Scholar [4] O. Arino, E. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl., 215 (1997), 499-513. doi: 10.1006/jmaa.1997.5654.  Google Scholar [5] M. Bai and S. Cui, Well-posedness and asynchronous exponential growth of solutions of a two-phase cell division model, Electron. J. Differential Equations, 2010 (2010), 1-12.  Google Scholar [6] H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich and A. K. Dhar, Experimental design and estimation of growth rate distributions in size-structured shrimp populations, Inverse Problems, 25 (2009), 095003, (28pp). doi: 10.1088/0266-5611/25/9/095003.  Google Scholar [7] G. I. Bell and E. C. Anderson, Cell growth and division I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351. Google Scholar [8] S. Bertoni, Periodic solutions for non-linear equations of structured populations, J. Math. Anal. Appl., 220 (1998), 250-267. doi: 10.1006/jmaa.1997.5878.  Google Scholar [9] G. Buffoni and S. Pasquali, Structured population dynamics: Continuous size and discontinuous stage structures, J. Math. Biol., 54 (2007), 555-595. doi: 10.1007/s00285-006-0058-2.  Google Scholar [10] A. Calsina and J. Saldana, Global dynamics and optimal life history of a structured population model, SIAM J. Appl. Math., 59 (1999), 1667-1685. doi: 10.1137/S0036139997331239.  Google Scholar [11] A. Calsina and M. Sanchón, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl., 286 (2003), 435-452. doi: 10.1016/S0022-247X(03)00464-5.  Google Scholar [12] C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin and W. M. Liu, Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. Math. Biol., 27 (1989), 233-258. doi: 10.1007/BF00275810.  Google Scholar [13] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Differential Equations, 247 (2009), 956-1000. doi: 10.1016/j.jde.2009.04.003.  Google Scholar [14] J. Chu, P. Magal and R. Yuan, Hopf bifurcation for a maturity structured population dynamic model, J. Nonlinear Sci., 21 (2011), 521-562. doi: 10.1007/s00332-010-9091-9.  Google Scholar [15] M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72. doi: 10.1007/BF00280827.  Google Scholar [16] P. Guidotti and S. Merino, Hopf bifurcation in a scalar reaction diffusion equation, J. Differential Equations, 140 (1997), 209-222. doi: 10.1006/jdeq.1997.3307.  Google Scholar [17] J. M. Cushing, "An Introduction to Structured Population Dynamics," SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.  Google Scholar [18] J. M. Cushing, Model stability and instability in age structured populations, J. Theoret. Biol., 86 (1980), 709-730. doi: 10.1016/0022-5193(80)90307-0.  Google Scholar [19] J. M. Cushing, Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics, Comput. Math. Appl., 9 (1983), 459-478. doi: 10.1016/0898-1221(83)90060-3.  Google Scholar [20] G. Da Prato and A. Lunardi, Hopf bifurcation for fully nonlinear equations in Banach space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 315-329.  Google Scholar [21] A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518. doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar [22] A. Ducrot, P. Magal and O. Seydi, Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model, Journal of Applied Analysis and Computation, 1 (2011), 373-395.  Google Scholar [23] J. Dyson, R. Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Math. Biosci., 177&178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9.  Google Scholar [24] K.-J. Engel and R. Nagel, "One Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.  Google Scholar [25] J. Z. Farkas and P. Hinow, On a size-structured two-phase population model with infinite states-at-birth, Positivity, 14 (2010), 501-514. doi: 10.1007/s11117-009-0033-4.  Google Scholar [26] J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, 1985.  Google Scholar [27] M. Gyllenberg and G. F. Webb, Age-size structure in population with quiescence, Math. Bioscience, 86 (1987), 67-95. doi: 10.1016/0025-5564(87)90064-2.  Google Scholar [28] M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694. doi: 10.1007/BF00160231.  Google Scholar [29] H. J. A. M. Heijmans, On the stable size distribution of populations reproducing by fission into two unequal parts, Math. Bioscience, 72 (1984), 19-50. doi: 10.1016/0025-5564(84)90059-2.  Google Scholar [30] W. Huyer, A size structured population model with dispersion, J. Math. Anal. Appl., 181 (1994), 716-754. doi: 10.1006/jmaa.1994.1054.  Google Scholar [31] H. Inaba, Mathematical analysis for an evolutionary epidemic model, in "Mathematical Models in Medical and Health Sciences" (eds. M. A. Horn, G. Simonett and G. F. Webb), Vanderbilt Univ. Press, Nashville, TN, (1998), 213-236.  Google Scholar [32] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory" (eds. C. Castillo-Chavez et al), Springer-Verlag, New York, (2002), 337-359. doi: 10.1007/978-1-4613-0065-6_19.  Google Scholar [33] H. Koch and S. S. Antman, Stability and Hopf bifurcation for fully nonlinear parabolic-hyperbolic equations, SIAM J. Math. Anal., 32 (2000), 360-384. doi: 10.1137/S003614109833793X.  Google Scholar [34] S. A. L. M. Kooijman and J. A. J. Metz, On the dynamics of chemically stressed populations: The deduction of population consequences from effects on individuals, Ecotox. Env. Saf., 8 (1984), 254-274. Google Scholar [35] T. Kostova and J. Li, Oscillations and stability due to juvenile competitive effects on adult fertility, Comput. Math. Appl., 32 (1996), 57-70. doi: 10.1016/S0898-1221(96)00197-6.  Google Scholar [36] K. Y. Lee, R. O. Barr, S. H. Gage and A. N. Kharkar, Formulation of a mathematical model for insect pest ecosystem-the cereal leaf beetle problem, J. Theor. Biol., 59 (1976), 33-76. doi: 10.1016/S0022-5193(76)80023-9.  Google Scholar [37] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift Fur Angewandte Mathematik und Physik, 62 (2011), 191-222. doi: 10.1007/s00033-010-0088-x.  Google Scholar [38] P. Magal, Compact attractors for time-periodic age structured population models, Electronic Journal of Differential Equations, 2001 (2001), 1-35.  Google Scholar [39] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.  Google Scholar [40] P. Magal and S. Ruan, "Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications on Hopf Bifurcation in Age Structured Models," Mem. Amer. Math. Soc., 202, 2009. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar [41] P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. A, 466 (2010), 965-992. doi: 10.1098/rspa.2009.0435.  Google Scholar [42] J. A. Metz and E. O. Diekmann, "The Dynamics of Physiologically Structured Populations," Lecture Notes in Biomathematics, 68, Springer, Berlin Heidelberg New York, 1986.  Google Scholar [43] J. Prüss, On the qualitative behaviour of populations with age-specific interactions, Comput. Math. Appl., 9 (1983), 327-339. doi: 10.1016/0898-1221(83)90020-2.  Google Scholar [44] B. Sandstede and A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal., 39 (2008), 2033-2052. doi: 10.1137/060675587.  Google Scholar [45] G. Simonett, Hopf bifurcation and stability for a quasilinear reaction-diffusion system, in "Evolution Equations," (eds. G. Ferreyra, G. Goldstein and F. Neubrander), in Lect. Notes Pure and Appl. Math. 168, Dekker, New York, (1995), 407-418.  Google Scholar [46] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.  Google Scholar [47] J. H. Swart, Hopf bifurcation and the stability of non-linear age-depedent population models, Comput. Math. Appl., 15 (1988), 555-564. doi: 10.1016/0898-1221(88)90280-5.  Google Scholar [48] W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Canada, 11 (1954), 559-623. doi: 10.1139/f54-039.  Google Scholar [49] W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Bd. Canada, 191 (1975). Google Scholar [50] B. Rossa, "Asynchronous Exponential Growth of Linear $C_{0}$-Semigroups and a New Tumor Cell Population Model," Ph. D thesis, Vanderbilt University, 1991.  Google Scholar [51] B. Rossa, Asynchronous exponential growth in a size structured cell population with quiescent compartment, in "Proc. of the 3rd International Conf. on M. P. D." (eds. O. Arino, D. Axelrod and M. Kimmel), Pau, June (1992). Google Scholar [52] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.  Google Scholar [53] H. R. Thieme, Quasi-compact semigroups via bounded perturbation, in "Advances in Mathematical Population Dynamics-Molecules, Cells and Man" (eds. O. Arino, D. Axelrod and M. Kimmel), World Sci. Publ., River Edge, NJ, (1997), 691-711.  Google Scholar [54] G. F. Webb, "Theory of Nonlinear Age-Dependent population Dynamics," Marcel Dekker, New York, 1985.  Google Scholar [55] G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763. doi: 10.1090/S0002-9947-1987-0902796-7.  Google Scholar [56] G. F. Webb, Population models structured by age, size, and spatial position, in "Structured Population Models in Biology and Epidemiology" (eds. P. Magal and S. Ruan), in Lecture Notes in Math., 1936, Springer-Verlag, Berlin, (2008), 1-49. doi: 10.1007/978-3-540-78273-5_1.  Google Scholar [57] P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Math. Biosci., 205 (2007), 83-107. doi: 10.1016/j.mbs.2006.06.006.  Google Scholar

show all references

##### References:
 [1] H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, in "Delay Differential Equations and Dynamical Systems" (eds. S. N. Busenberg and M. Martelli), in: Lect. Notes Math., 1475, Springer-Verlag, Berlin, (1991), 53-63. doi: 10.1007/BFb0083479.  Google Scholar [2] O. Arino, A survey of structured cell population dynamics, Acta Biotheoret., 43 (1995), 3-25. doi: 10.1007/BF00709430.  Google Scholar [3] O. Arino and E. Sanchez, A survey of cell population dynamics, J. Theor. Med., 1 (1997), 35-51. doi: 10.1080/10273669708833005.  Google Scholar [4] O. Arino, E. Sánchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl., 215 (1997), 499-513. doi: 10.1006/jmaa.1997.5654.  Google Scholar [5] M. Bai and S. Cui, Well-posedness and asynchronous exponential growth of solutions of a two-phase cell division model, Electron. J. Differential Equations, 2010 (2010), 1-12.  Google Scholar [6] H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich and A. K. Dhar, Experimental design and estimation of growth rate distributions in size-structured shrimp populations, Inverse Problems, 25 (2009), 095003, (28pp). doi: 10.1088/0266-5611/25/9/095003.  Google Scholar [7] G. I. Bell and E. C. Anderson, Cell growth and division I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351. Google Scholar [8] S. Bertoni, Periodic solutions for non-linear equations of structured populations, J. Math. Anal. Appl., 220 (1998), 250-267. doi: 10.1006/jmaa.1997.5878.  Google Scholar [9] G. Buffoni and S. Pasquali, Structured population dynamics: Continuous size and discontinuous stage structures, J. Math. Biol., 54 (2007), 555-595. doi: 10.1007/s00285-006-0058-2.  Google Scholar [10] A. Calsina and J. Saldana, Global dynamics and optimal life history of a structured population model, SIAM J. Appl. Math., 59 (1999), 1667-1685. doi: 10.1137/S0036139997331239.  Google Scholar [11] A. Calsina and M. Sanchón, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl., 286 (2003), 435-452. doi: 10.1016/S0022-247X(03)00464-5.  Google Scholar [12] C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin and W. M. Liu, Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. Math. Biol., 27 (1989), 233-258. doi: 10.1007/BF00275810.  Google Scholar [13] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Differential Equations, 247 (2009), 956-1000. doi: 10.1016/j.jde.2009.04.003.  Google Scholar [14] J. Chu, P. Magal and R. Yuan, Hopf bifurcation for a maturity structured population dynamic model, J. Nonlinear Sci., 21 (2011), 521-562. doi: 10.1007/s00332-010-9091-9.  Google Scholar [15] M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72. doi: 10.1007/BF00280827.  Google Scholar [16] P. Guidotti and S. Merino, Hopf bifurcation in a scalar reaction diffusion equation, J. Differential Equations, 140 (1997), 209-222. doi: 10.1006/jdeq.1997.3307.  Google Scholar [17] J. M. Cushing, "An Introduction to Structured Population Dynamics," SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.  Google Scholar [18] J. M. Cushing, Model stability and instability in age structured populations, J. Theoret. Biol., 86 (1980), 709-730. doi: 10.1016/0022-5193(80)90307-0.  Google Scholar [19] J. M. Cushing, Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics, Comput. Math. Appl., 9 (1983), 459-478. doi: 10.1016/0898-1221(83)90060-3.  Google Scholar [20] G. Da Prato and A. Lunardi, Hopf bifurcation for fully nonlinear equations in Banach space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 315-329.  Google Scholar [21] A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518. doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar [22] A. Ducrot, P. Magal and O. Seydi, Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model, Journal of Applied Analysis and Computation, 1 (2011), 373-395.  Google Scholar [23] J. Dyson, R. Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Math. Biosci., 177&178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9.  Google Scholar [24] K.-J. Engel and R. Nagel, "One Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.  Google Scholar [25] J. Z. Farkas and P. Hinow, On a size-structured two-phase population model with infinite states-at-birth, Positivity, 14 (2010), 501-514. doi: 10.1007/s11117-009-0033-4.  Google Scholar [26] J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, 1985.  Google Scholar [27] M. Gyllenberg and G. F. Webb, Age-size structure in population with quiescence, Math. Bioscience, 86 (1987), 67-95. doi: 10.1016/0025-5564(87)90064-2.  Google Scholar [28] M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694. doi: 10.1007/BF00160231.  Google Scholar [29] H. J. A. M. Heijmans, On the stable size distribution of populations reproducing by fission into two unequal parts, Math. Bioscience, 72 (1984), 19-50. doi: 10.1016/0025-5564(84)90059-2.  Google Scholar [30] W. Huyer, A size structured population model with dispersion, J. Math. Anal. Appl., 181 (1994), 716-754. doi: 10.1006/jmaa.1994.1054.  Google Scholar [31] H. Inaba, Mathematical analysis for an evolutionary epidemic model, in "Mathematical Models in Medical and Health Sciences" (eds. M. A. Horn, G. Simonett and G. F. Webb), Vanderbilt Univ. Press, Nashville, TN, (1998), 213-236.  Google Scholar [32] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory" (eds. C. Castillo-Chavez et al), Springer-Verlag, New York, (2002), 337-359. doi: 10.1007/978-1-4613-0065-6_19.  Google Scholar [33] H. Koch and S. S. Antman, Stability and Hopf bifurcation for fully nonlinear parabolic-hyperbolic equations, SIAM J. Math. Anal., 32 (2000), 360-384. doi: 10.1137/S003614109833793X.  Google Scholar [34] S. A. L. M. Kooijman and J. A. J. Metz, On the dynamics of chemically stressed populations: The deduction of population consequences from effects on individuals, Ecotox. Env. Saf., 8 (1984), 254-274. Google Scholar [35] T. Kostova and J. Li, Oscillations and stability due to juvenile competitive effects on adult fertility, Comput. Math. Appl., 32 (1996), 57-70. doi: 10.1016/S0898-1221(96)00197-6.  Google Scholar [36] K. Y. Lee, R. O. Barr, S. H. Gage and A. N. Kharkar, Formulation of a mathematical model for insect pest ecosystem-the cereal leaf beetle problem, J. Theor. Biol., 59 (1976), 33-76. doi: 10.1016/S0022-5193(76)80023-9.  Google Scholar [37] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift Fur Angewandte Mathematik und Physik, 62 (2011), 191-222. doi: 10.1007/s00033-010-0088-x.  Google Scholar [38] P. Magal, Compact attractors for time-periodic age structured population models, Electronic Journal of Differential Equations, 2001 (2001), 1-35.  Google Scholar [39] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.  Google Scholar [40] P. Magal and S. Ruan, "Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications on Hopf Bifurcation in Age Structured Models," Mem. Amer. Math. Soc., 202, 2009. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar [41] P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. A, 466 (2010), 965-992. doi: 10.1098/rspa.2009.0435.  Google Scholar [42] J. A. Metz and E. O. Diekmann, "The Dynamics of Physiologically Structured Populations," Lecture Notes in Biomathematics, 68, Springer, Berlin Heidelberg New York, 1986.  Google Scholar [43] J. Prüss, On the qualitative behaviour of populations with age-specific interactions, Comput. Math. Appl., 9 (1983), 327-339. doi: 10.1016/0898-1221(83)90020-2.  Google Scholar [44] B. Sandstede and A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal., 39 (2008), 2033-2052. doi: 10.1137/060675587.  Google Scholar [45] G. Simonett, Hopf bifurcation and stability for a quasilinear reaction-diffusion system, in "Evolution Equations," (eds. G. Ferreyra, G. Goldstein and F. Neubrander), in Lect. Notes Pure and Appl. Math. 168, Dekker, New York, (1995), 407-418.  Google Scholar [46] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.  Google Scholar [47] J. H. Swart, Hopf bifurcation and the stability of non-linear age-depedent population models, Comput. Math. Appl., 15 (1988), 555-564. doi: 10.1016/0898-1221(88)90280-5.  Google Scholar [48] W. E. Ricker, Stock and recruitment, J. Fish. Res. 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