# 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 - A, 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, 1475 (1991), 53. doi: 10.1007/BFb0083479. Google Scholar [2] O. Arino, A survey of structured cell population dynamics,, Acta Biotheoret., 43 (1995), 3. doi: 10.1007/BF00709430. Google Scholar [3] O. Arino and E. Sanchez, A survey of cell population dynamics,, J. Theor. Med., 1 (1997), 35. 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. 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. 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). 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. Google Scholar [8] S. Bertoni, Periodic solutions for non-linear equations of structured populations,, J. Math. Anal. Appl., 220 (1998), 250. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1006/jdeq.1997.3307. Google Scholar [17] J. M. Cushing, "An Introduction to Structured Population Dynamics,", SIAM, (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. 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. 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. 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. 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. 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 (2002), 73. 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, (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. 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. 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. 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. 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. doi: 10.1006/jmaa.1994.1054. Google Scholar [31] H. Inaba, Mathematical analysis for an evolutionary epidemic model,, in, (1998), 213. Google Scholar [32] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model,, in, (2002), 337. 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. 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. 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. 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. 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. 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. Google Scholar [39] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, Advances in Differential Equations, 14 (2009), 1041. 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. 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 (1986). Google Scholar [43] J. Prüss, On the qualitative behaviour of populations with age-specific interactions,, Comput. Math. Appl., 9 (1983), 327. 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. doi: 10.1137/060675587. Google Scholar [45] G. Simonett, Hopf bifurcation and stability for a quasilinear reaction-diffusion system,, in, 168 (1995), 407. Google Scholar [46] J. W. Sinko and W. Streifer, A new model for age-size structure of a population,, Ecology, 48 (1967), 910. 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. 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. 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, (1991). Google Scholar [51] B. Rossa, Asynchronous exponential growth in a size structured cell population with quiescent compartment,, in, (1992). Google Scholar [52] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035. Google Scholar [53] H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in, (1997), 691. Google Scholar [54] G. F. Webb, "Theory of Nonlinear Age-Dependent population Dynamics,", Marcel Dekker, (1985). Google Scholar [55] G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth,, Trans. Amer. Math. Soc., 303 (1987), 751. doi: 10.1090/S0002-9947-1987-0902796-7. Google Scholar [56] G. F. Webb, Population models structured by age, size, and spatial position,, in, 1936 (2008), 1. 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. 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, 1475 (1991), 53. doi: 10.1007/BFb0083479. Google Scholar [2] O. Arino, A survey of structured cell population dynamics,, Acta Biotheoret., 43 (1995), 3. doi: 10.1007/BF00709430. Google Scholar [3] O. Arino and E. Sanchez, A survey of cell population dynamics,, J. Theor. Med., 1 (1997), 35. 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. 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. 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). 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. Google Scholar [8] S. Bertoni, Periodic solutions for non-linear equations of structured populations,, J. Math. Anal. Appl., 220 (1998), 250. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1006/jdeq.1997.3307. Google Scholar [17] J. M. Cushing, "An Introduction to Structured Population Dynamics,", SIAM, (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. 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. 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. 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. 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. 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 (2002), 73. 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, (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. 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. 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. 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. 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. doi: 10.1006/jmaa.1994.1054. Google Scholar [31] H. Inaba, Mathematical analysis for an evolutionary epidemic model,, in, (1998), 213. Google Scholar [32] H. Inaba, Endemic threshold and stability in an evolutionary epidemic model,, in, (2002), 337. 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. 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. 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. 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. 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. 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. Google Scholar [39] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, Advances in Differential Equations, 14 (2009), 1041. 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. 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 (1986). Google Scholar [43] J. Prüss, On the qualitative behaviour of populations with age-specific interactions,, Comput. Math. Appl., 9 (1983), 327. 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. doi: 10.1137/060675587. Google Scholar [45] G. Simonett, Hopf bifurcation and stability for a quasilinear reaction-diffusion system,, in, 168 (1995), 407. Google Scholar [46] J. W. Sinko and W. Streifer, A new model for age-size structure of a population,, Ecology, 48 (1967), 910. 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. 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. 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, (1991). Google Scholar [51] B. Rossa, Asynchronous exponential growth in a size structured cell population with quiescent compartment,, in, (1992). Google Scholar [52] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035. Google Scholar [53] H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in, (1997), 691. Google Scholar [54] G. F. Webb, "Theory of Nonlinear Age-Dependent population Dynamics,", Marcel Dekker, (1985). Google Scholar [55] G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth,, Trans. Amer. Math. Soc., 303 (1987), 751. doi: 10.1090/S0002-9947-1987-0902796-7. Google Scholar [56] G. F. Webb, Population models structured by age, size, and spatial position,, in, 1936 (2008), 1. 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. doi: 10.1016/j.mbs.2006.06.006. Google Scholar
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