# American Institute of Mathematical Sciences

January  2013, 18(1): 109-131. doi: 10.3934/dcdsb.2013.18.109

## Stability results for a size-structured population model with delayed birth process

 1 Department of Mathematics, East China Normal University, Shanghai, 200241, China, China

Received  October 2011 Revised  July 2012 Published  September 2012

In this paper, we discuss the qualitative behavior of an age/size-structured population equation with delay in the birth process. The linearization about stationary solutions is analyzed by semigroup and spectral methods. In particular, the spectrally determined growth property of the linearized semigroup is derived from its long-term regularity. These analytical results allow us to derive linearized stability and instability results under some conditions. The principal stability criterions are given in terms of a modified net reproduction rate. Finally, two examples are presented and simulated to illustrated the obtained conclusions.
Citation: Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109
##### References:
 [1] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. [2] G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291. [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), 1023-1096. [4] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187-200. [5] K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. [6] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York, 2000. [7] M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123. [8] J. Z. Farkas, Stability conditions for a nonlinear size structured model, Nonl. Anal. (RWA), 6 (2005), 962-969. [9] J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136. [10] 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. [11] 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. [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), 735-754. [13] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Lect. Notes in Math., 1076 (1984), 86-100. [14] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [15] B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832. [16] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407. [17] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443. [18] T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Equ., 14 2001, 19-36. [19] M. Iannelli, "Mathematical Theory of Age-structured Population Dynamics," Giardini Editori, Pisa, 1994. [20] Y. Liu and Z.-R. He, Stability results for a size-structured population model with resources-dependence and inflow, J. Math. Anal. Appl., 360 (2009), 665-675. [21] A. J. Metz and O. Diekmann, "The Dynamics of Psyiologically Structured Populations," Springer, Berlin, 1986. [22] R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302. [23] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983. [24] S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. [25] 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. [26] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. [27] K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. [28] K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910. [29] G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics," Marcell Dekker, New York, 1985.

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##### References:
 [1] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. [2] G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci., 46 (1979), 279-291. [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), 1023-1096. [4] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., 47 (2008), 187-200. [5] K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. [6] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York, 2000. [7] M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123. [8] J. Z. Farkas, Stability conditions for a nonlinear size structured model, Nonl. Anal. (RWA), 6 (2005), 962-969. [9] J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136. [10] 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. [11] 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. [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), 735-754. [13] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Lect. Notes in Math., 1076 (1984), 86-100. [14] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [15] B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832. [16] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407. [17] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443. [18] T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Equ., 14 2001, 19-36. [19] M. Iannelli, "Mathematical Theory of Age-structured Population Dynamics," Giardini Editori, Pisa, 1994. [20] Y. Liu and Z.-R. He, Stability results for a size-structured population model with resources-dependence and inflow, J. Math. Anal. Appl., 360 (2009), 665-675. [21] A. J. Metz and O. Diekmann, "The Dynamics of Psyiologically Structured Populations," Springer, Berlin, 1986. [22] R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302. [23] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983. [24] S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. [25] 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. [26] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. [27] K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. [28] K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910. [29] G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics," Marcell Dekker, New York, 1985.
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