# American Institute of Mathematical Sciences

May  2021, 26(5): 2561-2580. doi: 10.3934/dcdsb.2020195

## Large-time behavior of matured population in an age-structured model

 1 School of Mathematics and Statistics, Xidian University, Xi'an, China 2 Institut de Mathématiques de Bordeaux, Université de Bordeaux, Bordeaux, France

* Corresponding author: Linlin Li

Received  October 2019 Revised  April 2020 Published  May 2021 Early access  June 2020

Fund Project: The first author is supported by the NSF of Shaanxi Province of China (2020JQ-289) and the Program for New Century Excellent Talents in University (XJS200701)

In this paper, we model a mosquito plasticity problem and investigate the large time behavior of matured population under different control strategies. We prove that when the control is small, then the matured population will become large for large time and when the control is large, then the matured population will become small for large time. In the intermediate case, we derive a time-delayed model for the matured population which can be governed by a sub-equation and a super-equation. We prove the existence of traveling fronts for the sub-equation and use it to prove that the matured population will finally be between the positive states of the sub-equation and super-equation. At last, we present numerical simulations.

Citation: Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195
##### References:
 [1] S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015. [2] B. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl., 248 (2000), 455-474.  doi: 10.1006/jmaa.2000.6921. [3] J. Al-Omari and S. A. Gourley, Monotone travelling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.  doi: 10.1007/s002850200159. [4] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Mathematical Modelling: Theory and Applications, 11, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. [5] P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8. [6] N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4. [7] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099. [8] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153. [9] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0. [10] D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, 279, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1992. [11] L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903. [12] T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.  doi: 10.1098/rspa.2005.1554. [13] C. P. Ferreira, S. P. Lyra, F. Azevedo, D. Greenhalgh and E. Massad, Modelling the impact of the long-term use of insecticide-treated bed nets on Anopheles mosquito biting time, Malaria J., 16 (2017). doi: 10.1186/s12936-017-2014-6. [14] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [15] S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333.  doi: 10.1007/BF00160498. [16] S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [17] S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991. [18] B. Huho, O. Briët, A. Seyoum, C. Sikaala and N. Bayoh, et al., Consistently high estimates for the proportion of human exposure to malaria vector populations occurring indoors in rural Africa, Internat. J. Epidemiology, 42 (2013), 235-247 doi: 10.1093/ije/dys214. [19] L. L. Li, C. P. Ferreira and B. Ainseba, Mathematical analysis of an age structured problem modeling phenotypic plasticity in mosquito behaviour, Nonlinear Anal. Real World Appl., 48 (2019), 410-423.  doi: 10.1016/j.nonrwa.2019.01.019. [20] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154. [21] S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004. [22] M. C. Memory, Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion, SIAM J. Math. Anal., 20 (1989), 533-546.  doi: 10.1137/0520037. [23] C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219-261.  doi: 10.1016/j.jde.2006.12.010. [24] M. R. Reddy, H. J. Overgaard, S. Abaga, V. P. Reddy, A. Caccone, A. E. Kiszewski and M. A. Slotman, Outdoor host seeking behaviour of Anopheles gambiae mosquitoes following initiation of malaria vector control on Bioko Island, Equatorial Guinea, Malaria J., 10 (2011). doi: 10.1186/1475-2875-10-184. [25] S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590. [26] T. L. Russell, N. J. Govella, S. Azizi, C. J. Drakeley, S. P. Kachur and G. F. Killeen, Increased proportions of outdoor feeding among residual malaria vector populations following increased use of insecticide-treated nets in rural Tanzania, Malaria J., 10 (2011). doi: 10.1186/1475-2875-10-80. [27] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. [28] H. L. Smith, A structured population model and a related functional-differential equation: Global attractors and uniform persistence, J. Dynam. Differential Equations, 6 (1994), 71-99.  doi: 10.1007/BF02219189. [29] H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785. [30] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081. [31] J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789. [32] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X. [33] Z.-C. Wang, W.-T. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010. [34] Z.-C. Wang, W.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.  doi: 10.1007/s10884-008-9103-8. [35] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [36] A. W. Yadouleton, G. Padonou, A. Asidi, N. Moiroux and S. Bio-Banganna, et al., Insecticide resistance status in Anopheles gambiae in southern Benin, Malaria J., 9 (2010). doi: 10.1186/1475-2875-9-83. [37] K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Hiroshima Math. J., 12 (1982), 321-348.  doi: 10.32917/hmj/1206133754. [38] X.-Q. Zhao and D. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differential Equations, 18 (2006), 1001-1019.  doi: 10.1007/s10884-006-9044-z.

show all references

##### References:
 [1] S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015. [2] B. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl., 248 (2000), 455-474.  doi: 10.1006/jmaa.2000.6921. [3] J. Al-Omari and S. A. Gourley, Monotone travelling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.  doi: 10.1007/s002850200159. [4] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Mathematical Modelling: Theory and Applications, 11, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3. [5] P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8. [6] N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4. [7] N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099. [8] X. Chen and J.-S. Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153. [9] X. Chen and J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326 (2003), 123-146.  doi: 10.1007/s00208-003-0414-0. [10] D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, 279, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1992. [11] L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903. [12] T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.  doi: 10.1098/rspa.2005.1554. [13] C. P. Ferreira, S. P. Lyra, F. Azevedo, D. Greenhalgh and E. Massad, Modelling the impact of the long-term use of insecticide-treated bed nets on Anopheles mosquito biting time, Malaria J., 16 (2017). doi: 10.1186/s12936-017-2014-6. [14] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [15] S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333.  doi: 10.1007/BF00160498. [16] S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [17] S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991. [18] B. Huho, O. Briët, A. Seyoum, C. Sikaala and N. Bayoh, et al., Consistently high estimates for the proportion of human exposure to malaria vector populations occurring indoors in rural Africa, Internat. J. Epidemiology, 42 (2013), 235-247 doi: 10.1093/ije/dys214. [19] L. L. Li, C. P. Ferreira and B. Ainseba, Mathematical analysis of an age structured problem modeling phenotypic plasticity in mosquito behaviour, Nonlinear Anal. Real World Appl., 48 (2019), 410-423.  doi: 10.1016/j.nonrwa.2019.01.019. [20] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154. [21] S. Ma and X. Zou, Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay, J. Differential Equations, 217 (2005), 54-87.  doi: 10.1016/j.jde.2005.05.004. [22] M. C. Memory, Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion, SIAM J. Math. Anal., 20 (1989), 533-546.  doi: 10.1137/0520037. [23] C. Ou and J. Wu, Persistence of wavefronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 235 (2007), 219-261.  doi: 10.1016/j.jde.2006.12.010. [24] M. R. Reddy, H. J. Overgaard, S. Abaga, V. P. Reddy, A. Caccone, A. E. Kiszewski and M. A. Slotman, Outdoor host seeking behaviour of Anopheles gambiae mosquitoes following initiation of malaria vector control on Bioko Island, Equatorial Guinea, Malaria J., 10 (2011). doi: 10.1186/1475-2875-10-184. [25] S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590. [26] T. L. Russell, N. J. Govella, S. Azizi, C. J. Drakeley, S. P. Kachur and G. F. Killeen, Increased proportions of outdoor feeding among residual malaria vector populations following increased use of insecticide-treated nets in rural Tanzania, Malaria J., 10 (2011). doi: 10.1186/1475-2875-10-80. [27] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. [28] H. L. Smith, A structured population model and a related functional-differential equation: Global attractors and uniform persistence, J. Dynam. Differential Equations, 6 (1994), 71-99.  doi: 10.1007/BF02219189. [29] H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785. [30] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081. [31] J. W.-H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789. [32] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X. [33] Z.-C. Wang, W.-T. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010. [34] Z.-C. Wang, W.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 573-607.  doi: 10.1007/s10884-008-9103-8. [35] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [36] A. W. Yadouleton, G. Padonou, A. Asidi, N. Moiroux and S. Bio-Banganna, et al., Insecticide resistance status in Anopheles gambiae in southern Benin, Malaria J., 9 (2010). doi: 10.1186/1475-2875-9-83. [37] K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Hiroshima Math. J., 12 (1982), 321-348.  doi: 10.32917/hmj/1206133754. [38] X.-Q. Zhao and D. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differential Equations, 18 (2006), 1001-1019.  doi: 10.1007/s10884-006-9044-z.
the matured population $w(t,x)$ for $t = 0$ and $t = 0.25$ with no control
the matured population $w(t,x)$ for $t = 0.5$ and $t = 1$ with no control
the matured population $w(t,x)$ for $t = 0$ and $t = 0.25$ with control $u(a,w)$
the matured population $w(t,x)$ for $t = 0.5$ and $t = 1$ with control $u(a,w)$
the matured population $w(t,x)$ for $t = 0$, $0.25$
the matured population $w(t,x)$ for $t=0.5$, $1$
the matured population $w(t,x)$ for $1.5$, $2$
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