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Stability analysis for a size-structured juvenile-adult population model
1. | Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China, China |
References:
[1] |
T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407.
doi: 10.1006/jmaa.1999.6708. |
[2] |
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. |
[3] |
T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Equ., 14 (2001), 19-36. |
[4] |
J. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase, Discr. Contin. Dyn. Syst., 33 (2013), 4891-4921.
doi: 10.3934/dcds.2013.33.4891. |
[5] |
M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123.
doi: 10.1016/S0096-3003(01)00131-X. |
[6] |
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. |
[7] |
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. |
[8] |
J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized size-structured Daphnia model with inflow, Appl. Anal., 86 (2007), 1087-1103.
doi: 10.1080/00036810701545634. |
[9] |
J. Z. Farkas and T. Hagen, Asymptotic behavior of size-structured populations via juvenile-adult interaction, Discr. Cont. Dyn. Syst. B, 9 (2008), 249-266. |
[10] |
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. |
[11] |
R. Dilão, T. Domingos and E. M. Shahverdiev, Harvesting in a resource dependent age structured Leslie type population model, Math. Biosci., 189 (2004), 141-151.
doi: 10.1016/j.mbs.2004.01.008. |
[12] |
J. B. Shukla, K. Lata and A. K. Misra, Modeling the depletion of a renewable resource by population and industrialization: effect of technology on its conservation, Nat. Resour. Model., 24 (2011), 242-267.
doi: 10.1111/j.1939-7445.2011.00090.x. |
[13] |
J. B. Shukla, S. Sharma, B. Dubey and P. Sinha, Modeling the survival of a resource-dependent population: Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors, Nonl. Anal. (RWA), 10 (2009), 54-70.
doi: 10.1016/j.nonrwa.2007.08.014. |
[14] |
J. Xia, Z. Liu, R. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with holling type II functional response, SIAM J. Appl. Math., 70 (2009), 1178-1200.
doi: 10.1137/080728512. |
[15] |
E. M. C. D'Agata, P. Magal, S. Ruan and G. webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Diff. Int. Equ., 19 (2006), 573-600. |
[16] |
A. Ducrot and P. Magal, Traveling wave solution for infection age structured epidemic model with vital dynamics, Nonlinearity, 24 (2011), 2891-2911.
doi: 10.1088/0951-7715/24/10/012. |
[17] |
D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. |
[18] |
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 (2007/08), 1023-1069.
doi: 10.1137/060659211. |
[19] |
O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional analysis and evolution equations, 187-200, Birkhäuser, Basel, 2008.
doi: 10.1007/978-3-7643-7794-6_12. |
[20] |
K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498.
doi: 10.1137/0132040. |
[21] |
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. |
[22] |
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. |
[23] |
A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, in Infinite Dimensional Dynamical Systems, J. Mallet-Paret, J. Wu, Y. Yi, and H. Zhu (eds.), Fields Institute Communications, 64 (2013), 353-390. |
[24] |
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. |
[25] |
G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process, Discr. Cont. Dyn. Syst. B, 7 (2007), 735-754.
doi: 10.3934/dcdsb.2007.7.735. |
[26] |
S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439.
doi: 10.1002/mma.462. |
[27] |
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. |
[28] |
X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 18 (2013), 109-131.
doi: 10.3934/dcdsb.2013.18.109. |
[29] |
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. |
[30] |
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. |
[31] |
M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Giardini Editori, Pisa, 1994. |
[32] |
A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. |
[33] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985. |
[34] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. |
[35] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[36] |
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. |
[37] |
K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. |
show all references
References:
[1] |
T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407.
doi: 10.1006/jmaa.1999.6708. |
[2] |
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. |
[3] |
T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Equ., 14 (2001), 19-36. |
[4] |
J. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase, Discr. Contin. Dyn. Syst., 33 (2013), 4891-4921.
doi: 10.3934/dcds.2013.33.4891. |
[5] |
M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123.
doi: 10.1016/S0096-3003(01)00131-X. |
[6] |
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. |
[7] |
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. |
[8] |
J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized size-structured Daphnia model with inflow, Appl. Anal., 86 (2007), 1087-1103.
doi: 10.1080/00036810701545634. |
[9] |
J. Z. Farkas and T. Hagen, Asymptotic behavior of size-structured populations via juvenile-adult interaction, Discr. Cont. Dyn. Syst. B, 9 (2008), 249-266. |
[10] |
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. |
[11] |
R. Dilão, T. Domingos and E. M. Shahverdiev, Harvesting in a resource dependent age structured Leslie type population model, Math. Biosci., 189 (2004), 141-151.
doi: 10.1016/j.mbs.2004.01.008. |
[12] |
J. B. Shukla, K. Lata and A. K. Misra, Modeling the depletion of a renewable resource by population and industrialization: effect of technology on its conservation, Nat. Resour. Model., 24 (2011), 242-267.
doi: 10.1111/j.1939-7445.2011.00090.x. |
[13] |
J. B. Shukla, S. Sharma, B. Dubey and P. Sinha, Modeling the survival of a resource-dependent population: Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors, Nonl. Anal. (RWA), 10 (2009), 54-70.
doi: 10.1016/j.nonrwa.2007.08.014. |
[14] |
J. Xia, Z. Liu, R. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with holling type II functional response, SIAM J. Appl. Math., 70 (2009), 1178-1200.
doi: 10.1137/080728512. |
[15] |
E. M. C. D'Agata, P. Magal, S. Ruan and G. webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Diff. Int. Equ., 19 (2006), 573-600. |
[16] |
A. Ducrot and P. Magal, Traveling wave solution for infection age structured epidemic model with vital dynamics, Nonlinearity, 24 (2011), 2891-2911.
doi: 10.1088/0951-7715/24/10/012. |
[17] |
D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. |
[18] |
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 (2007/08), 1023-1069.
doi: 10.1137/060659211. |
[19] |
O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional analysis and evolution equations, 187-200, Birkhäuser, Basel, 2008.
doi: 10.1007/978-3-7643-7794-6_12. |
[20] |
K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498.
doi: 10.1137/0132040. |
[21] |
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. |
[22] |
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. |
[23] |
A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, in Infinite Dimensional Dynamical Systems, J. Mallet-Paret, J. Wu, Y. Yi, and H. Zhu (eds.), Fields Institute Communications, 64 (2013), 353-390. |
[24] |
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. |
[25] |
G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process, Discr. Cont. Dyn. Syst. B, 7 (2007), 735-754.
doi: 10.3934/dcdsb.2007.7.735. |
[26] |
S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439.
doi: 10.1002/mma.462. |
[27] |
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. |
[28] |
X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 18 (2013), 109-131.
doi: 10.3934/dcdsb.2013.18.109. |
[29] |
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. |
[30] |
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. |
[31] |
M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Giardini Editori, Pisa, 1994. |
[32] |
A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. |
[33] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985. |
[34] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. |
[35] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[36] |
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. |
[37] |
K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. |
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