Asymptotic analysis of a spatially and size-structured population model with delayed birth process

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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

Dongxue Yan ^1, and Xianlong Fu ^2,

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

Abstract
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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.

Keywords: stability, asynchronous exponential growth., delayed boundary condition, $C_0$-semigroup, Population dynamics.

Mathematics Subject Classification: 34G20, 37G15, 35B32, 35K55, 92D2.

Citation: Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure and 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.
[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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