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Asymptotic analysis of a spatially and size-structured population model with delayed birth process
1. | Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China |
2. | Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241 |
References:
[1] |
D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. |
[2] |
A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20.
doi: 10.1006/jmaa.2001.6705. |
[3] |
M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Equ., 11 (2011), 531-552.
doi: 10.1007/s00028-011-0100-8. |
[4] |
G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291.
doi: 10.1016/0025-5564(79)90073-7. |
[5] |
O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1096.
doi: 10.1137/060659211. |
[6] |
O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187-200.
doi: 10.1007/978-3-7643-7794-6_12. |
[7] |
K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. |
[8] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. |
[9] |
M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123.
doi: 10.1016/S0096-3003(01)00131-X. |
[10] |
J. Z. Farkas, Stability conditions for a nonlinear size-structured model, Nonl. Anal. (RWA), 6 (2005), 962-969.
doi: 10.1016/j.nonrwa.2004.06.002. |
[11] |
J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.
doi: 10.1016/j.jmaa.2006.05.032. |
[12] |
X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 1 (2013), 109-131.
doi: 10.3934/dcdsb.2013.18.109. |
[13] |
G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation, Lect. Notes in Math., 1076 (1984), 86-100.
doi: 10.1007/BFb0072769. |
[14] |
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. |
[15] |
G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, Math. Appl. Sci. (New Orleans, La., 1986), Academic Press, 1988, 79-105. |
[16] |
B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832.
doi: 10.1080/03605308908820630. |
[17] |
M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467.
doi: 10.1016/0022-247X(92)90218-3. |
[18] |
T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407.
doi: 10.1006/jmaa.1999.6708. |
[19] |
T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443.
doi: 10.1006/jmaa.2000.7089. |
[20] |
T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Eq., 14 (2001), 19-36. |
[21] |
H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model, J. Math. Biol., 54 (2007), 101-146.
doi: 10.1007/s00285-006-0033-y. |
[22] |
Z. Liu, P. Magal and H. Tang, Hopf bifurcation for a spatially and age structured population dynamics model, Discr. Cont. Dyn. Syst. B, 20 (2015), 1735-1757.
doi: 10.3934/dcdsb.2015.20.1735. |
[23] |
Y. Liu and Z. He, Stability results for a size-structured population model with resources-dependence and inflow, J. Math. Anal. Appl., 360 (2009), 665-675.
doi: 10.1016/j.jmaa.2009.07.005. |
[24] |
R. Nagel ed., One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. vol. 1184, Springer-Verlag, 1986. |
[25] |
R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302.
doi: 10.1016/0022-1236(90)90096-4. |
[26] |
R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Math., 19 (1996), 83-100. |
[27] |
J. Pruss and W. Schappacher, Semigroup methods for age-structured population dynamics, Jahrbuch Uberblicke Mathematik, Vieweg, (1994), 74-90. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
S. Pizzera, An age dependent population equation with delayed birth process, Math. Meth. Appl. Sci., 27 (2004), 427-439.
doi: 10.1002/mma.462. |
[30] |
S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77.
doi: 10.1007/s00028-004-0159-6. |
[31] |
A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$, Discr. Cont. Dyn. Syst., 5 (1999), 663-683.
doi: 10.3934/dcds.1999.5.663. |
[32] |
W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). |
[33] |
K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. |
[34] |
K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910.
doi: 10.1137/0511080. |
[35] |
L. Weis, The stability of positive semigroups on $L_p$ spaces, Proceedings of the American Mathematical Society, 123 (1995), 3089-3094.
doi: 10.2307/2160665. |
[36] |
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcell Dekker, New York, 1985. |
show all references
References:
[1] |
D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. |
[2] |
A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20.
doi: 10.1006/jmaa.2001.6705. |
[3] |
M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Equ., 11 (2011), 531-552.
doi: 10.1007/s00028-011-0100-8. |
[4] |
G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291.
doi: 10.1016/0025-5564(79)90073-7. |
[5] |
O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1096.
doi: 10.1137/060659211. |
[6] |
O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187-200.
doi: 10.1007/978-3-7643-7794-6_12. |
[7] |
K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. |
[8] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. |
[9] |
M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123.
doi: 10.1016/S0096-3003(01)00131-X. |
[10] |
J. Z. Farkas, Stability conditions for a nonlinear size-structured model, Nonl. Anal. (RWA), 6 (2005), 962-969.
doi: 10.1016/j.nonrwa.2004.06.002. |
[11] |
J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.
doi: 10.1016/j.jmaa.2006.05.032. |
[12] |
X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 1 (2013), 109-131.
doi: 10.3934/dcdsb.2013.18.109. |
[13] |
G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation, Lect. Notes in Math., 1076 (1984), 86-100.
doi: 10.1007/BFb0072769. |
[14] |
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. |
[15] |
G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, Math. Appl. Sci. (New Orleans, La., 1986), Academic Press, 1988, 79-105. |
[16] |
B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832.
doi: 10.1080/03605308908820630. |
[17] |
M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467.
doi: 10.1016/0022-247X(92)90218-3. |
[18] |
T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407.
doi: 10.1006/jmaa.1999.6708. |
[19] |
T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443.
doi: 10.1006/jmaa.2000.7089. |
[20] |
T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Eq., 14 (2001), 19-36. |
[21] |
H. Inaba, Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model, J. Math. Biol., 54 (2007), 101-146.
doi: 10.1007/s00285-006-0033-y. |
[22] |
Z. Liu, P. Magal and H. Tang, Hopf bifurcation for a spatially and age structured population dynamics model, Discr. Cont. Dyn. Syst. B, 20 (2015), 1735-1757.
doi: 10.3934/dcdsb.2015.20.1735. |
[23] |
Y. Liu and Z. He, Stability results for a size-structured population model with resources-dependence and inflow, J. Math. Anal. Appl., 360 (2009), 665-675.
doi: 10.1016/j.jmaa.2009.07.005. |
[24] |
R. Nagel ed., One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. vol. 1184, Springer-Verlag, 1986. |
[25] |
R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302.
doi: 10.1016/0022-1236(90)90096-4. |
[26] |
R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Math., 19 (1996), 83-100. |
[27] |
J. Pruss and W. Schappacher, Semigroup methods for age-structured population dynamics, Jahrbuch Uberblicke Mathematik, Vieweg, (1994), 74-90. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
S. Pizzera, An age dependent population equation with delayed birth process, Math. Meth. Appl. Sci., 27 (2004), 427-439.
doi: 10.1002/mma.462. |
[30] |
S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77.
doi: 10.1007/s00028-004-0159-6. |
[31] |
A. Rhandi and R. Schnaubelt, Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$, Discr. Cont. Dyn. Syst., 5 (1999), 663-683.
doi: 10.3934/dcds.1999.5.663. |
[32] |
W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). |
[33] |
K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. |
[34] |
K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910.
doi: 10.1137/0511080. |
[35] |
L. Weis, The stability of positive semigroups on $L_p$ spaces, Proceedings of the American Mathematical Society, 123 (1995), 3089-3094.
doi: 10.2307/2160665. |
[36] |
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcell Dekker, New York, 1985. |
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