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 and 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.

[2]

O. Arino, A survey of structured cell population dynamics, Acta Biotheoret., 43 (1995), 3-25. doi: 10.1007/BF00709430.

[3]

O. Arino and E. Sanchez, A survey of cell population dynamics, J. Theor. Med., 1 (1997), 35-51. doi: 10.1080/10273669708833005.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

J. M. Cushing, "An Introduction to Structured Population Dynamics," SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

K.-J. Engel and R. Nagel, "One Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.

[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.

[26]

J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, 1985.

[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.

[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.

[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.

[30]

W. Huyer, A size structured population model with dispersion, J. Math. Anal. Appl., 181 (1994), 716-754. doi: 10.1006/jmaa.1994.1054.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

P. Magal, Compact attractors for time-periodic age structured population models, Electronic Journal of Differential Equations, 2001 (2001), 1-35.

[39]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.

[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.

[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.

[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.

[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.

[44]

B. Sandstede and A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal., 39 (2008), 2033-2052. doi: 10.1137/060675587.

[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.

[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.

[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.

[48]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Canada, 11 (1954), 559-623. doi: 10.1139/f54-039.

[49]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Bd. Canada, 191 (1975).

[50]

B. Rossa, "Asynchronous Exponential Growth of Linear $C_{0} $-Semigroups and a New Tumor Cell Population Model," Ph. D thesis, Vanderbilt University, 1991.

[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).

[52]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.

[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.

[54]

G. F. Webb, "Theory of Nonlinear Age-Dependent population Dynamics," Marcel Dekker, New York, 1985.

[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.

[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.

[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.

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.

[2]

O. Arino, A survey of structured cell population dynamics, Acta Biotheoret., 43 (1995), 3-25. doi: 10.1007/BF00709430.

[3]

O. Arino and E. Sanchez, A survey of cell population dynamics, J. Theor. Med., 1 (1997), 35-51. doi: 10.1080/10273669708833005.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

J. M. Cushing, "An Introduction to Structured Population Dynamics," SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

K.-J. Engel and R. Nagel, "One Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.

[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.

[26]

J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford University Press, 1985.

[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.

[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.

[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.

[30]

W. Huyer, A size structured population model with dispersion, J. Math. Anal. Appl., 181 (1994), 716-754. doi: 10.1006/jmaa.1994.1054.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

P. Magal, Compact attractors for time-periodic age structured population models, Electronic Journal of Differential Equations, 2001 (2001), 1-35.

[39]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.

[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.

[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.

[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.

[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.

[44]

B. Sandstede and A. Scheel, Hopf bifurcation from viscous shock waves, SIAM J. Math. Anal., 39 (2008), 2033-2052. doi: 10.1137/060675587.

[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.

[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.

[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.

[48]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Canada, 11 (1954), 559-623. doi: 10.1139/f54-039.

[49]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Bd. Canada, 191 (1975).

[50]

B. Rossa, "Asynchronous Exponential Growth of Linear $C_{0} $-Semigroups and a New Tumor Cell Population Model," Ph. D thesis, Vanderbilt University, 1991.

[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).

[52]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.

[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.

[54]

G. F. Webb, "Theory of Nonlinear Age-Dependent population Dynamics," Marcel Dekker, New York, 1985.

[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.

[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.

[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.

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