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Dynamic analysis of an $ SEIR $ epidemic model with a time lag in awareness allocated funds
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J. P. Keener,
Waves in excitable media, SIAM J. Appl. Math., 39 (1980), 528-548.
doi: 10.1137/0139043. |
[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. |
[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. |
[13] |
P. Magal,
Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 1-35.
|
[14] |
P. Magal and S. Ruan,
On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
W. E. Ricker,
Stock and recruitment, J. Fish. Res. Bd. Canada, 11 (1954), 559-623.
doi: 10.1139/f54-039. |
[24] |
S. J. Schreiber,
Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.
doi: 10.1007/s002850000070. |
[25] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.
![]() |
[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. |
[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. |
[28] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, 1985. |
[29] |
G. B. Whitham,
Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.
doi: 10.1017/S0022112067000424. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J. P. Keener,
Waves in excitable media, SIAM J. Appl. Math., 39 (1980), 528-548.
doi: 10.1137/0139043. |
[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. |
[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. |
[13] |
P. Magal,
Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 1-35.
|
[14] |
P. Magal and S. Ruan,
On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
W. E. Ricker,
Stock and recruitment, J. Fish. Res. Bd. Canada, 11 (1954), 559-623.
doi: 10.1139/f54-039. |
[24] |
S. J. Schreiber,
Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.
doi: 10.1007/s002850000070. |
[25] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.
![]() |
[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. |
[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. |
[28] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, 1985. |
[29] |
G. B. Whitham,
Non-linear dispersion of water waves, J. Fluid Mech., 27 (1967), 399-412.
doi: 10.1017/S0022112067000424. |
[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. |
[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. |
[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. |


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