# American Institute of Mathematical Sciences

March  2016, 15(2): 637-655. doi: 10.3934/cpaa.2016.15.637

## 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

Received  March 2015 Revised  November 2015 Published  January 2016

This paper is devoted to the study of a spatially and size-structured population dynamics model with delayed birth process. Our focus is on the asymptotic behavior of the system, in particular on the effect of the spatial location and the time lag on the long-term dynamics. To this end, within a semigroup framework, we derive the locally asymptotic stability and asynchrony results respectively for the considered population system under some conditions. For our discussion, we use the approaches concerning operator matrices, Hille-Yosida operators, spectral analysis as well as Perron-Frobenius theory. We also do two numerical simulations to illustrate the obtained stability and asynchrony results.
Citation: Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637
##### References:
 [1] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80.  Google Scholar [8] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407. doi: 10.1006/jmaa.1999.6708.  Google Scholar [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.  Google Scholar [20] T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Eq., 14 (2001), 19-36.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] R. Nagel ed., One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. vol. 1184, Springer-Verlag, 1986.  Google Scholar [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.  Google Scholar [26] R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Math., 19 (1996), 83-100.  Google Scholar [27] J. Pruss and W. Schappacher, Semigroup methods for age-structured population dynamics, Jahrbuch Uberblicke Mathematik, Vieweg, (1994), 74-90.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). Google Scholar [33] K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] G. F. Webb, Theory of Nonlinear Age-Dependent 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), 345-376. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80.  Google Scholar [8] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407. doi: 10.1006/jmaa.1999.6708.  Google Scholar [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.  Google Scholar [20] T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Eq., 14 (2001), 19-36.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] R. Nagel ed., One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. vol. 1184, Springer-Verlag, 1986.  Google Scholar [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.  Google Scholar [26] R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Math., 19 (1996), 83-100.  Google Scholar [27] J. Pruss and W. Schappacher, Semigroup methods for age-structured population dynamics, Jahrbuch Uberblicke Mathematik, Vieweg, (1994), 74-90.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975). Google Scholar [33] K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcell Dekker, New York, 1985.  Google Scholar
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