American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media
  • DCDS-B Home
  • This Issue
  • Next Article
    Dynamic analysis of an $ SEIR $ epidemic model with a time lag in awareness allocated funds
doi: 10.3934/dcdsb.2021009

Existence of periodic wave trains for an age-structured model with diffusion

Zhihua Liu 1, , Yayun Wu 1, and  Xiangming Zhang 2,,

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People's Republic of China
2. 

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, People's Republic of China

* Corresponding author: Xiangming Zhang

Received  May 2020 Revised  October 2020 Published  December 2020

Fund Project: Research was partially supported by NSFC (Grant Nos. 11871007 and 11811530272), the Fundamental Research Funds for the Central Universities and the Young Scholars Science Foundation of Lanzhou Jiaotong University (2020027)

In this paper, we make a mathematical analysis of an age-structured model with diffusion including a generalized Beverton-Holt fertility function. The existence of periodic wave train solutions of the age structure model with diffusion are investigated by using the theory of integrated semigroup and a Hopf bifurcation theorem for second order semi-linear equations. We also carry out numerical simulations to illustrate these results.

Keywords: Age structure, Diffusion, Non-densely defined Cauchy problem, Hopf bifurcation, Periodic wave trains.
Mathematics Subject Classification: 34C20; 37L10; 92D25.
Citation: Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021009
References:
[1]

T. S. Bellows, Jr., The descriptive properties of some models for density dependence, J. Animal Ecology, 50 (1981), 139–156. doi: 10.2307/4037.  Google Scholar

[2]

R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Springer Netherlands, HMSO, 1957. Google Scholar

[3]

M. J. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.  doi: 10.1080/00036810701474140.  Google Scholar

[4]

J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.  Google Scholar

[5]

J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250.  doi: 10.1007/BF01832847.  Google Scholar

[6]

A. Ducrot, Z. H. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.  doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar

[7]

A. Ducrot and P. Magal, A center manifold for second order semi-linear differential equations on the real line and applications to the existence of wave trains for the Gurtin-McCamy equation, Trans. Amer. Math. Soc., 372 (2019), 3487-3537.  doi: 10.1090/tran/7780.  Google Scholar

[8]

W. M. Getz, A hypothesis regarding the abruptness of density dependence and the growth rate of populations, Ecology, 77 (1996), 2014-2026.  doi: 10.2307/2265697.  Google Scholar

[9]

C.-H. Hsu, C.-R. Yang, T.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.  doi: 10.1016/j.jde.2011.11.008.  Google Scholar

[10]

J. P. Keener, Waves in excitable media, SIAM J. Appl. Math., 39 (1980), 528-548.  doi: 10.1137/0139043.  Google Scholar

[11]

Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonlinear Sci., 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x.  Google Scholar

[12]

Z. Liu, P. 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.  Google Scholar

[13]

P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 1-35.   Google Scholar

[14]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084.   Google Scholar

[15]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar

[16]

K. Maginu, Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion systems, SIAM J. Appl. Math., 45 (1985), 750-774.  doi: 10.1137/0145044.  Google Scholar

[17]

S. M. Merchant and W. Nagata, Selection and stability of wave trains behind predator invasions in a model with non-local prey competition, IMA J. Appl. Math., 80 (2015), 1155-1177.  doi: 10.1093/imamat/hxu048.  Google Scholar

[18]

S. M. Merchant and W. Nagata, Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.  doi: 10.1016/j.physd.2010.04.014.  Google Scholar

[19]

M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995. Google Scholar

[20]

J. D. Murray, Mathematical Biology. I. An Introduction, Springer-Verlag, New York, 2002.  Google Scholar

[21]

J. D. M. Rademacher and A. Scheel, Instabilities of wave trains and Turing patterns in large domains, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2679-2691.  doi: 10.1142/S0218127407018683.  Google Scholar

[22]

J. D. M. Rademacher and A. Scheel, The saddle-node of nearly homogeneous wave trains in reaction-diffusion systems, J. Dynam. Differential Equations, 19 (2007), 479-496.  doi: 10.1007/s10884-006-9059-5.  Google Scholar

[23]

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

[24]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.  doi: 10.1007/s002850000070.  Google Scholar

[25] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.   Google Scholar
[26]

J.-M. Vanden-Broeck and E. I. P$\breve{a}$r$\breve{a}$u, Two-dimensional generalized solitary waves and periodic waves under an ice sheet, Philos. Trans. Roy. Soc. A, 369 (2011), 2957-2972.  doi: 10.1098/rsta.2011.0108.  Google Scholar

[27]

Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150.  doi: 10.1016/j.jmaa.2011.07.038.  Google Scholar

[28]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, 1985.  Google Scholar

[29]

G. B. Whitham, Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.  doi: 10.1017/S0022112067000424.  Google Scholar

[30]

X. Zhang and Z. Liu, Bifurcation analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850109, 20 pp. doi: 10.1142/S0218127418501092.  Google Scholar

[31]

X. Zhang and Z. Liu, Hopf bifurcation for a susceptible-infective model with infection-age structure, J. Nonlinear Sci., 30 (2020), 317-367.  doi: 10.1007/s00332-019-09575-y.  Google Scholar

[32]

X. Zhang and Z. Liu, Periodic oscillations in age-structured ratio-dependent predator-prey model with Michaelis-Menten type functional response, Phys. D, 389 (2019), 51-63.  doi: 10.1016/j.physd.2018.10.002.  Google Scholar

show all references

References:
[1]

T. S. Bellows, Jr., The descriptive properties of some models for density dependence, J. Animal Ecology, 50 (1981), 139–156. doi: 10.2307/4037.  Google Scholar

[2]

R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Springer Netherlands, HMSO, 1957. Google Scholar

[3]

M. J. Bohner and H. Warth, The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.  doi: 10.1080/00036810701474140.  Google Scholar

[4]

J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.  Google Scholar

[5]

J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250.  doi: 10.1007/BF01832847.  Google Scholar

[6]

A. Ducrot, Z. H. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.  doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar

[7]

A. Ducrot and P. Magal, A center manifold for second order semi-linear differential equations on the real line and applications to the existence of wave trains for the Gurtin-McCamy equation, Trans. Amer. Math. Soc., 372 (2019), 3487-3537.  doi: 10.1090/tran/7780.  Google Scholar

[8]

W. M. Getz, A hypothesis regarding the abruptness of density dependence and the growth rate of populations, Ecology, 77 (1996), 2014-2026.  doi: 10.2307/2265697.  Google Scholar

[9]

C.-H. Hsu, C.-R. Yang, T.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.  doi: 10.1016/j.jde.2011.11.008.  Google Scholar

[10]

J. P. Keener, Waves in excitable media, SIAM J. Appl. Math., 39 (1980), 528-548.  doi: 10.1137/0139043.  Google Scholar

[11]

Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonlinear Sci., 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x.  Google Scholar

[12]

Z. Liu, P. 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.  Google Scholar

[13]

P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 1-35.   Google Scholar

[14]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084.   Google Scholar

[15]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78273-5.  Google Scholar

[16]

K. Maginu, Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion systems, SIAM J. Appl. Math., 45 (1985), 750-774.  doi: 10.1137/0145044.  Google Scholar

[17]

S. M. Merchant and W. Nagata, Selection and stability of wave trains behind predator invasions in a model with non-local prey competition, IMA J. Appl. Math., 80 (2015), 1155-1177.  doi: 10.1093/imamat/hxu048.  Google Scholar

[18]

S. M. Merchant and W. Nagata, Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.  doi: 10.1016/j.physd.2010.04.014.  Google Scholar

[19]

M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995. Google Scholar

[20]

J. D. Murray, Mathematical Biology. I. An Introduction, Springer-Verlag, New York, 2002.  Google Scholar

[21]

J. D. M. Rademacher and A. Scheel, Instabilities of wave trains and Turing patterns in large domains, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2679-2691.  doi: 10.1142/S0218127407018683.  Google Scholar

[22]

J. D. M. Rademacher and A. Scheel, The saddle-node of nearly homogeneous wave trains in reaction-diffusion systems, J. Dynam. Differential Equations, 19 (2007), 479-496.  doi: 10.1007/s10884-006-9059-5.  Google Scholar

[23]

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

[24]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.  doi: 10.1007/s002850000070.  Google Scholar

[25] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.   Google Scholar
[26]

J.-M. Vanden-Broeck and E. I. P$\breve{a}$r$\breve{a}$u, Two-dimensional generalized solitary waves and periodic waves under an ice sheet, Philos. Trans. Roy. Soc. A, 369 (2011), 2957-2972.  doi: 10.1098/rsta.2011.0108.  Google Scholar

[27]

Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150.  doi: 10.1016/j.jmaa.2011.07.038.  Google Scholar

[28]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, 1985.  Google Scholar

[29]

G. B. Whitham, Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.  doi: 10.1017/S0022112067000424.  Google Scholar

[30]

X. Zhang and Z. Liu, Bifurcation analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850109, 20 pp. doi: 10.1142/S0218127418501092.  Google Scholar

[31]

X. Zhang and Z. Liu, Hopf bifurcation for a susceptible-infective model with infection-age structure, J. Nonlinear Sci., 30 (2020), 317-367.  doi: 10.1007/s00332-019-09575-y.  Google Scholar

[32]

X. Zhang and Z. Liu, Periodic oscillations in age-structured ratio-dependent predator-prey model with Michaelis-Menten type functional response, Phys. D, 389 (2019), 51-63.  doi: 10.1016/j.physd.2018.10.002.  Google Scholar

Figure 1.  Numerical solutions of system (2.1) when $ \tau = 10<\tau_{0} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Numerical solutions of system (2.1) when $ \tau = 20>\tau_{0} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Numerical solutions of system (1.4) when $ \tau = 10<\tau_{0} $ and $ c = 1000 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Numerical solutions of system (1.4) when $ \tau = 20>\tau_{0} $ and $ c = 1000 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171
[2]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
[3]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342
[4]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[5]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013
[6]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[7]

Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020054
[8]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053
[9]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[10]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227
[11]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018
[12]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[13]

Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020281
[14]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004
[15]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028
[16]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169
[17]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[18]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398
[19]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004
[20]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

2019 Impact Factor: 1.27

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences