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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. AshwinM. V. BartuccelliT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. C. EvansH. 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.  Google Scholar

[12]

T. FariaW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. L. LiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. W.-H. SoJ. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081.  Google Scholar

[31]

J. W.-H. SoJ. 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.  Google Scholar

[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.  Google Scholar

[33]

Z.-C. WangW.-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.  Google Scholar

[34]

Z.-C. WangW.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. AshwinM. V. BartuccelliT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. C. EvansH. 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.  Google Scholar

[12]

T. FariaW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. L. LiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. W.-H. SoJ. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081.  Google Scholar

[31]

J. W.-H. SoJ. 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.  Google Scholar

[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.  Google Scholar

[33]

Z.-C. WangW.-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.  Google Scholar

[34]

Z.-C. WangW.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  the matured population $ w(t,x) $ for $ t = 0 $ and $ t = 0.25 $ with no control
Figure 2.  the matured population $ w(t,x) $ for $ t = 0.5 $ and $ t = 1 $ with no control
Figure 3.  the matured population $ w(t,x) $ for $ t = 0 $ and $ t = 0.25 $ with control $ u(a,w) $
Figure 4.  the matured population $ w(t,x) $ for $ t = 0.5 $ and $ t = 1 $ with control $ u(a,w) $
Figure 5.  the matured population $ w(t,x) $ for $ t = 0 $, $ 0.25 $
Figure 6.  the matured population $ w(t,x) $ for $ t=0.5 $, $ 1 $
Figure 7.  the matured population $ w(t,x) $ for $ 1.5 $, $ 2 $
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