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December  2021, 26(12): 6117-6130. doi: 10.3934/dcdsb.2021009

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

 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 2021 Early access  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.

Citation: Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6117-6130. 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
Numerical solutions of system (2.1) when $\tau = 10<\tau_{0}$
Numerical solutions of system (2.1) when $\tau = 20>\tau_{0}$
Numerical solutions of system (1.4) when $\tau = 10<\tau_{0}$ and $c = 1000$
Numerical solutions of system (1.4) when $\tau = 20>\tau_{0}$ and $c = 1000$
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