June  2022, 11(3): 895-923. doi: 10.3934/eect.2021030

Long-time behavior of a size-structured population model with diffusion and delayed birth process

1. 

School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

2. 

School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China

* Corresponding author: Xianlong Fu

Received  November 2020 Revised  March 2021 Published  June 2022 Early access  July 2021

Fund Project: This work is supported by NNSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 18dz2271000), Natural Science Foundation of Jiangsu Province of China (grant No. BK20200749), Nanjing University of Posts and Telecommunications Science Foundation (rant No. NY220093)

This work focuses on the long time behavior for a size-dependent population system with diffusion and Riker type birth function. Some dynamical properties of the considered system is investigated by using $ C_0 $-semigroup theory and spectral analysis arguments. Some sufficient conditions are obtained respectively for asymptotical stability, asynchronous exponential growth at the null equilibrium as well as Hopf bifurcation occurring at the positive steady state of the system. In the end several examples and their simulations are also provided to illustrate the achieved results.

Citation: Dongxue Yan, Xianlong Fu. Long-time behavior of a size-structured population model with diffusion and delayed birth process. Evolution Equations and Control Theory, 2022, 11 (3) : 895-923. doi: 10.3934/eect.2021030
References:
[1]

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.

[2]

P. Bi and X. Fu, Hopf bifurcation in an age-dependent population model with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250146, 16 pp. doi: 10.1142/S0218127412501465.

[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]

J. ChuA. DucrotP. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Differ. Equ., 247 (2009), 956-1000.  doi: 10.1016/j.jde.2009.04.003.

[5]

K. CookeP. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.  doi: 10.1007/s002850050194.

[6]

O. DiekmannPh. 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), 1023-1069.  doi: 10.1137/060659211.

[7]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations, Birkhäuser, Basel, 2008,187–200. doi: 10.1007/978-3-7643-7794-6_12.

[8]

A. DucrotP. Magal and and O. Seydi, Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model, J. Appl. Anal. Comput., 1 (2011), 373-395.  doi: 10.11948/2011026.

[9]

K.-J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. 

[10]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[11]

X. FuZ. Liu and P. Magal, Hopf bifurcation in an age-structured population model with two delays, Commun. Pure Appl. Anal., 14 (2015), 657-676.  doi: 10.3934/cpaa.2015.14.657.

[12]

X. Fu and Q. Wu, Asymptotic behaviors of a size-structured population model, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 1025-1042.  doi: 10.1007/s10255-017-0717-7.

[13]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Infinite-Dimensional Systems, 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]

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.

[16]

P. GettoM. GyllenbergY. Nakata and F. Scarabel, Stability analysis of a state dependent delay differential equation for cell maturation: Analytical and numerical methods, J. Math. Biol., 79 (2019), 281-328.  doi: 10.1007/s00285-019-01357-0.

[17]

Z. HeD. Ni and S. Wang, Existence and stability of steady states for hierarchical age-structured population models, Electron. J. Differ. Equ., 124 (2019), 1-14. 

[18] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Math. Soc. Lect. Note Ser., Vol. 41, Cambridge Univ. Press, Cambridge-New York, 1981. 
[19]

Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.

[20]

P. Magal and S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.

[21]

P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 965-992.  doi: 10.1098/rspa.2009.0435.

[22]

R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math., Vol. 1184, Springer-Verlag, 1986. doi: 10.1007/BFb0074922.

[23]

R. Nagel, The spectrum of unbounded operator matrices with nondiagonal domain, J. Funct. Anal., 89 (1990), 291-302.  doi: 10.1016/0022-1236(90)90096-4.

[24]

R. NagelG. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Math., 19 (1996), 83-100.  doi: 10.1080/16073606.1996.9631827.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

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.

[27]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in $L^1$, Discrete Contin. Dynam. Systems, 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663.

[28]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975).

[29]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623.  doi: 10.1139/f54-039.

[30]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574.  doi: 10.1007/s00285-010-0381-5.

[31]

H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015.

[32]

J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math., 90 (1980), 153-161.  doi: 10.1007/BF01303264.

[33]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcell Dekker, New York, 1985.

[34]

X. Wang, H. Wang and M. Y. Li, $R_0$ and sensitivity analysis of a predator-prey model with seasonality and maturation delay, Math. Biosci., 315 (2019), 108225, 11 pp. doi: 10.1016/j.mbs.2019.108225.

[35]

D. Yan and X. Fu, Asymptotic analysis of a spatially and size-structured population model with delayed birth process, Commun. Pure Appl. Anal., 15 (2016), 637-655.  doi: 10.3934/cpaa.2016.15.637.

[36]

D. Yan and X. Fu, Long-time behavior of spatially and size-structured population dynamics with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750032, 23 pp. doi: 10.1142/S0218127417500328.

[37]

C. Zheng, F. Zhang and J. Li, Stability analysis of a population model with maturation delay and Ricker birth function, Abstr. Appl. Anal., (2014), Art. ID 136707, 1–8. doi: 10.1155/2014/136707.

show all references

References:
[1]

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.

[2]

P. Bi and X. Fu, Hopf bifurcation in an age-dependent population model with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250146, 16 pp. doi: 10.1142/S0218127412501465.

[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]

J. ChuA. DucrotP. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Differ. Equ., 247 (2009), 956-1000.  doi: 10.1016/j.jde.2009.04.003.

[5]

K. CookeP. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.  doi: 10.1007/s002850050194.

[6]

O. DiekmannPh. 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), 1023-1069.  doi: 10.1137/060659211.

[7]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations, Birkhäuser, Basel, 2008,187–200. doi: 10.1007/978-3-7643-7794-6_12.

[8]

A. DucrotP. Magal and and O. Seydi, Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model, J. Appl. Anal. Comput., 1 (2011), 373-395.  doi: 10.11948/2011026.

[9]

K.-J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. 

[10]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[11]

X. FuZ. Liu and P. Magal, Hopf bifurcation in an age-structured population model with two delays, Commun. Pure Appl. Anal., 14 (2015), 657-676.  doi: 10.3934/cpaa.2015.14.657.

[12]

X. Fu and Q. Wu, Asymptotic behaviors of a size-structured population model, Acta Math. Appl. Sin. Engl. Ser., 33 (2017), 1025-1042.  doi: 10.1007/s10255-017-0717-7.

[13]

G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Infinite-Dimensional Systems, 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]

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.

[16]

P. GettoM. GyllenbergY. Nakata and F. Scarabel, Stability analysis of a state dependent delay differential equation for cell maturation: Analytical and numerical methods, J. Math. Biol., 79 (2019), 281-328.  doi: 10.1007/s00285-019-01357-0.

[17]

Z. HeD. Ni and S. Wang, Existence and stability of steady states for hierarchical age-structured population models, Electron. J. Differ. Equ., 124 (2019), 1-14. 

[18] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Math. Soc. Lect. Note Ser., Vol. 41, Cambridge Univ. Press, Cambridge-New York, 1981. 
[19]

Z. LiuP. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.

[20]

P. Magal and S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.

[21]

P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 965-992.  doi: 10.1098/rspa.2009.0435.

[22]

R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math., Vol. 1184, Springer-Verlag, 1986. doi: 10.1007/BFb0074922.

[23]

R. Nagel, The spectrum of unbounded operator matrices with nondiagonal domain, J. Funct. Anal., 89 (1990), 291-302.  doi: 10.1016/0022-1236(90)90096-4.

[24]

R. NagelG. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaestiones Math., 19 (1996), 83-100.  doi: 10.1080/16073606.1996.9631827.

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[26]

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.

[27]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in $L^1$, Discrete Contin. Dynam. Systems, 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663.

[28]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191 (1975).

[29]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623.  doi: 10.1139/f54-039.

[30]

Y. SuS. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574.  doi: 10.1007/s00285-010-0381-5.

[31]

H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015.

[32]

J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math., 90 (1980), 153-161.  doi: 10.1007/BF01303264.

[33]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcell Dekker, New York, 1985.

[34]

X. Wang, H. Wang and M. Y. Li, $R_0$ and sensitivity analysis of a predator-prey model with seasonality and maturation delay, Math. Biosci., 315 (2019), 108225, 11 pp. doi: 10.1016/j.mbs.2019.108225.

[35]

D. Yan and X. Fu, Asymptotic analysis of a spatially and size-structured population model with delayed birth process, Commun. Pure Appl. Anal., 15 (2016), 637-655.  doi: 10.3934/cpaa.2016.15.637.

[36]

D. Yan and X. Fu, Long-time behavior of spatially and size-structured population dynamics with delayed birth process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750032, 23 pp. doi: 10.1142/S0218127417500328.

[37]

C. Zheng, F. Zhang and J. Li, Stability analysis of a population model with maturation delay and Ricker birth function, Abstr. Appl. Anal., (2014), Art. ID 136707, 1–8. doi: 10.1155/2014/136707.

Figure 1.  "a" represents the stationary solution, the initial conditions corresponding to curves $ b $ and $ c $ are: (a) $ u_0(x) = \frac{1}{1+x^3}-0.01 $; (b) $ u_0(x) = \frac{1}{1+x^3}+0.05 $;
Figure 2.  The case of asynchronous exponential growth
Figure 3.  The case of Hopf bifurcation
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