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Slow passage through multiple bifurcation points
Stability results for a sizestructured population model with delayed birth process
1.  Department of Mathematics, East China Normal University, Shanghai, 200241, China, China 
References:
[1] 
D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345376. Google Scholar 
[2] 
G. Di Blasio, Nonlinear agedependent population growth with historydependent birth rate, Math. Biosci., 46 (1979), 279291. Google Scholar 
[3] 
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), 10231096. Google Scholar 
[4] 
O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187200. Google Scholar 
[5] 
K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 6180. Google Scholar 
[6] 
K. J. Engel and R. Nagel, "OneParameter Semigroups for Linear Evolution Equations," Springer, New York, 2000. Google Scholar 
[7] 
M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107123. Google Scholar 
[8] 
J. Z. Farkas, Stability conditions for a nonlinear size structured model, Nonl. Anal. (RWA), 6 (2005), 962969. Google Scholar 
[9] 
J. Z. Farkas and T. Hagen, Stability and regularity results for a sizestructured population model, J. Math. Anal. Appl., 328 (2007), 119136. Google Scholar 
[10] 
J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized sizestructured Daphnia model with inflow, Appl. Anal., 86 (2007), 10871103. Google Scholar 
[11] 
J. Z. Farkas and T. Hagen, Asymptotic behavior of sizestructured populations via juvenileadult interaction, Discr. Cont. Dyn. Syst. B, 9 (2008), 249266. Google Scholar 
[12] 
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), 735754. Google Scholar 
[13] 
G. Greiner, A typical PerronFrobenius theorem with applications to an agedependent population equation, Lect. Notes in Math., 1076 (1984), 86100. Google Scholar 
[14] 
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213229. Google Scholar 
[15] 
B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809832. Google Scholar 
[16] 
T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393407. Google Scholar 
[17] 
T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431443. Google Scholar 
[18] 
T. Hagen and M. Renardy, Studies on the linear equations of meltspinning of viscous fluids, Diff. Int. Equ., 14 2001, 1936. Google Scholar 
[19] 
M. Iannelli, "Mathematical Theory of Agestructured Population Dynamics," Giardini Editori, Pisa, 1994. Google Scholar 
[20] 
Y. Liu and Z.R. He, Stability results for a sizestructured population model with resourcesdependence and inflow, J. Math. Anal. Appl., 360 (2009), 665675. Google Scholar 
[21] 
A. J. Metz and O. Diekmann, "The Dynamics of Psyiologically Structured Populations," Springer, Berlin, 1986. Google Scholar 
[22] 
R. Nagel, The spectrum of unbounded operator matrices with nondiagonal domain, J. Funct. Anal., 89 (1990), 291302. Google Scholar 
[23] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983. Google Scholar 
[24] 
S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427439. Google Scholar 
[25] 
S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 6177. Google Scholar 
[26] 
J. W. Sinko and W. Streifer, A new model for agesize structure of a population, Ecology, 48 (1967), 910918. Google Scholar 
[27] 
K. E. Swick, A nonlinear agedependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484498. Google Scholar 
[28] 
K. E. Swick, Periodic solutions of a nonlinear agedependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901910. Google Scholar 
[29] 
G. F. Webb, "Theory of Nonlinear Agedependent Population Dynamics," Marcell Dekker, New York, 1985. Google Scholar 
show all references
References:
[1] 
D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345376. Google Scholar 
[2] 
G. Di Blasio, Nonlinear agedependent population growth with historydependent birth rate, Math. Biosci., 46 (1979), 279291. Google Scholar 
[3] 
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), 10231096. Google Scholar 
[4] 
O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187200. Google Scholar 
[5] 
K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 6180. Google Scholar 
[6] 
K. J. Engel and R. Nagel, "OneParameter Semigroups for Linear Evolution Equations," Springer, New York, 2000. Google Scholar 
[7] 
M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107123. Google Scholar 
[8] 
J. Z. Farkas, Stability conditions for a nonlinear size structured model, Nonl. Anal. (RWA), 6 (2005), 962969. Google Scholar 
[9] 
J. Z. Farkas and T. Hagen, Stability and regularity results for a sizestructured population model, J. Math. Anal. Appl., 328 (2007), 119136. Google Scholar 
[10] 
J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized sizestructured Daphnia model with inflow, Appl. Anal., 86 (2007), 10871103. Google Scholar 
[11] 
J. Z. Farkas and T. Hagen, Asymptotic behavior of sizestructured populations via juvenileadult interaction, Discr. Cont. Dyn. Syst. B, 9 (2008), 249266. Google Scholar 
[12] 
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), 735754. Google Scholar 
[13] 
G. Greiner, A typical PerronFrobenius theorem with applications to an agedependent population equation, Lect. Notes in Math., 1076 (1984), 86100. Google Scholar 
[14] 
G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213229. Google Scholar 
[15] 
B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809832. Google Scholar 
[16] 
T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393407. Google Scholar 
[17] 
T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431443. Google Scholar 
[18] 
T. Hagen and M. Renardy, Studies on the linear equations of meltspinning of viscous fluids, Diff. Int. Equ., 14 2001, 1936. Google Scholar 
[19] 
M. Iannelli, "Mathematical Theory of Agestructured Population Dynamics," Giardini Editori, Pisa, 1994. Google Scholar 
[20] 
Y. Liu and Z.R. He, Stability results for a sizestructured population model with resourcesdependence and inflow, J. Math. Anal. Appl., 360 (2009), 665675. Google Scholar 
[21] 
A. J. Metz and O. Diekmann, "The Dynamics of Psyiologically Structured Populations," Springer, Berlin, 1986. Google Scholar 
[22] 
R. Nagel, The spectrum of unbounded operator matrices with nondiagonal domain, J. Funct. Anal., 89 (1990), 291302. Google Scholar 
[23] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983. Google Scholar 
[24] 
S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427439. Google Scholar 
[25] 
S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 6177. Google Scholar 
[26] 
J. W. Sinko and W. Streifer, A new model for agesize structure of a population, Ecology, 48 (1967), 910918. Google Scholar 
[27] 
K. E. Swick, A nonlinear agedependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484498. Google Scholar 
[28] 
K. E. Swick, Periodic solutions of a nonlinear agedependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901910. Google Scholar 
[29] 
G. F. Webb, "Theory of Nonlinear Agedependent Population Dynamics," Marcell Dekker, New York, 1985. Google Scholar 
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