doi: 10.3934/dcdsb.2022122
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Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

2. 

Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China

3. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada

* Corresponding author

Received  January 2022 Revised  April 2022 Early access June 2022

Fund Project: This work is supported by Natural Science Foundation of China (No. 11971013), the NSERC Individual Discovery Grant RGPIN-2020-03911 and NSERC Accelerator Grant RGPAS-2020-00090, the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20_0169) and Nanjing University of Aeronautics and Astronautics PhD short-term visiting scholar project (No. ZDGB2021026) at the University of Alberta

This paper is devoted to investigate the existence of traveling wave solutions for a partially degenerate Aedes aegypti model with nonlocal effects. By taking specific kernel forms and time scale transformation, we transform the nonlocal model into a singularly perturbed system with small parameters. A locally invariant manifold for wave profile system is obtained with the aid of the geometric singular perturbation theory, and then the existence of traveling wave solutions is proved provided that the basic reproduction number $ \mathcal{R}_0>1 $ through utilizing the Fredholm orthogonal method. Furthermore, we study the asymptotic behaviors of traveling wave solution with the help of asymptotic theory. The methods used in this work can help us overcome the difficulty that the solution map associated with the system is not compact. Numerically, we perform simulations to demonstrate the theoretical results.

Citation: Kai Wang, Hongyong Zhao, Hao Wang. Geometric singular perturbation of a nonlocal partially degenerate model for Aedes aegypti. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022122
References:
[1]

S. Ai, Travelling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differ. Equ., 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015.

[2]

J. F. M. Al-Omari and S. A. Gourley, Monotone wave-fronts in a structured population model with distributed maturation delay, IMA J. Appl. Math., 70 (2005), 858-879.  doi: 10.1093/imamat/hxh073.

[3]

K. Atkinson and W. Han, Theoretical Numerical Analysis. A Functional Analysis Framework, Springer, 2009. doi: 10.1007/978-1-4419-0458-4.

[4]

N. Becker, D. Petric, M. Zgomba, et al., Mosquitoes and their Control, Second Edition, Springer-Verlag, New York, 2010.

[5]

O. J. Brady, P. W. Gething, S. Bhatt, et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus, PLoS Negl. Trop. Dis., 6 (2012), e1760. doi: 10.1371/journal.pntd.0001760.

[6]

X. Chen and X. Zhang, Dynamics of the predator-prey model with the Sigmoid functional response, Stud. Appl. Math., 147 (2021), 300-318.  doi: 10.1111/sapm.12382.

[7]

P. N. DavisP. van HeijsterR. Marangell and M. R. Rodrigo, Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis, J. Dyn. Differ. Equ., 34 (2022), 1325-1347.  doi: 10.1007/s10884-021-10003-7.

[8]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.

[9]

Z. DuJ. Liu and Y. Ren, Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach, J. Differ. Equ., 270 (2021), 1019-1042.  doi: 10.1016/j.jde.2020.09.009.

[10]

Z. Du and Q. Qiao, The dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system, J. Differ. Equ., 269 (2020), 7214-7230.  doi: 10.1016/j.jde.2020.05.033.

[11]

A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh., 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[12]

S. R. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.  doi: 10.1137/0146063.

[13]

S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.

[14]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$, Trans. Am. Math. Soc., 286 (1984), 557-594.  doi: 10.2307/1999810.

[15]

J. FangX. Lai and F.-B. Wang, Spatial dynamics of a dengue transmission model in time-space periodic environment, J. Differ. Equ., 269 (2020), 149-175.  doi: 10.1016/j.jde.2020.04.034.

[16]

J. FangX. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[17]

J. Fang and X.-Q. Zhao, Monotone wave fronts for partially degenerate reaction-diffusion system, J. Dyn. Differ. Equ., 21 (2009), 663-680.  doi: 10.1007/s10884-009-9152-7.

[18]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[19]

R. A. Gardner, Existence of travelling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 44 (1984), 56-79.  doi: 10.1137/0144006.

[20]

R. Gardner and J. Smoller, The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index, J. Differ. Equ., 47 (1983), 133-161.  doi: 10.1016/0022-0396(83)90031-1.

[21]

N. G. Gratz, Critical review of the vector status of Aedes albopictus, Medical and Veterinary Entomology., 18 (2004), 215-227.  doi: 10.1111/j.0269-283X.2004.00513.x.

[22]

J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9 (2003), 925-936.  doi: 10.3934/dcds.2003.9.925.

[23]

W. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differ. Equ., 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060.

[24]

C. K. R. T. Jones, Geometrical Singular Perturbation Theory, Springer, Berlin, 1995. doi: 10.1007/BFb0095239.

[25]

W.-T. LiG. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.

[26]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[27]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differ. Equ., 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.

[28]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theoret. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.

[29]

M. S. MustafaV. RasotgiS. Jain and V. Gupta, Discovype of dengue virus (DENV-5): A new public health dilemma in dengue control, Med. J. Armed Forces India., 71 (2015), 67-70.  doi: 10.1016/j.mjafi.2014.09.011.

[30]

P. Reiter and D. Sprenger, The used tire trade: A mechanism for the worldwide dispersal of container breeding mosquitoes, J. Am. Mosq. Control. Assoc., 3 (1987), 494-501. 

[31]

S. Ruan and D. Xiao, Stability of steady states and existence of traveling wave in a vector disease model, Proc. Roy. Soc. Edinburgh., 134A (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[32]

Q. Shi, J. Shi and H. Wang, Spatial movement with distributed memory, J. Math. Biol., 82 (2021), Paper No. 33, 32 pp. doi: 10.1007/s00285-021-01588-0.

[33]

L. T. TakahashiN. A. MaidanaW. Castro Ferreira Jr.P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind, Bull. Math. Biol., 67 (2005), 509-528.  doi: 10.1016/j.bulm.2004.08.005.

[34]

K. Wang, H. Zhao and H. Wang, Traveling waves for a diffusive mosquito-borne epidemic model with general incidence, Z. Angew. Math. Phys., 73 (2022), Paper No. 31, 28 pp. doi: 10.1007/s00033-021-01666-9.

[35]

K. Wang, H. Zhao, H. Wang and R. Zhang, Traveling wave of a reaction-diffusion vector-borne disease model with nonlocal effects and distributed delay, J. Dyn. Differ. Equ., 2021. doi: 10.1007/s10884-021-10062-w.

[36]

Z.-C. WangW.-T. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatiotemporal delays, J. Differ. Equ., 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.

[37]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.

[38]

X. Wu and M. Ni, Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term, Discrete Contin. Dyn. Syst. Ser. S., 14 (2021), 3249-3266.  doi: 10.3934/dcdss.2020341.

[39]

R. Zhang, J. Wang and S. Liu, Traveling wave solutions for a class of discrete diffusive SIR epidemic model, J. Nonlinear Sci., 31 (2021), Paper No. 10, 33 pp. doi: 10.1007/s00332-020-09656-3.

[40]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differ. Equ., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.

[41]

R. Zhang and H. Zhao, Traveling wave solutions for Zika transmission model with nonlocal diffusion, J. Math. Anal. Appl., 513 (2022), 126201, 29 pp. doi: 10.1016/j.jmaa.2022.126201.

[42]

X. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via monotone iteration method, Proc. Am. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.

show all references

References:
[1]

S. Ai, Travelling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differ. Equ., 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015.

[2]

J. F. M. Al-Omari and S. A. Gourley, Monotone wave-fronts in a structured population model with distributed maturation delay, IMA J. Appl. Math., 70 (2005), 858-879.  doi: 10.1093/imamat/hxh073.

[3]

K. Atkinson and W. Han, Theoretical Numerical Analysis. A Functional Analysis Framework, Springer, 2009. doi: 10.1007/978-1-4419-0458-4.

[4]

N. Becker, D. Petric, M. Zgomba, et al., Mosquitoes and their Control, Second Edition, Springer-Verlag, New York, 2010.

[5]

O. J. Brady, P. W. Gething, S. Bhatt, et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus, PLoS Negl. Trop. Dis., 6 (2012), e1760. doi: 10.1371/journal.pntd.0001760.

[6]

X. Chen and X. Zhang, Dynamics of the predator-prey model with the Sigmoid functional response, Stud. Appl. Math., 147 (2021), 300-318.  doi: 10.1111/sapm.12382.

[7]

P. N. DavisP. van HeijsterR. Marangell and M. R. Rodrigo, Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis, J. Dyn. Differ. Equ., 34 (2022), 1325-1347.  doi: 10.1007/s10884-021-10003-7.

[8]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.

[9]

Z. DuJ. Liu and Y. Ren, Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach, J. Differ. Equ., 270 (2021), 1019-1042.  doi: 10.1016/j.jde.2020.09.009.

[10]

Z. Du and Q. Qiao, The dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system, J. Differ. Equ., 269 (2020), 7214-7230.  doi: 10.1016/j.jde.2020.05.033.

[11]

A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh., 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.

[12]

S. R. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.  doi: 10.1137/0146063.

[13]

S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.

[14]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$, Trans. Am. Math. Soc., 286 (1984), 557-594.  doi: 10.2307/1999810.

[15]

J. FangX. Lai and F.-B. Wang, Spatial dynamics of a dengue transmission model in time-space periodic environment, J. Differ. Equ., 269 (2020), 149-175.  doi: 10.1016/j.jde.2020.04.034.

[16]

J. FangX. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[17]

J. Fang and X.-Q. Zhao, Monotone wave fronts for partially degenerate reaction-diffusion system, J. Dyn. Differ. Equ., 21 (2009), 663-680.  doi: 10.1007/s10884-009-9152-7.

[18]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[19]

R. A. Gardner, Existence of travelling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 44 (1984), 56-79.  doi: 10.1137/0144006.

[20]

R. Gardner and J. Smoller, The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index, J. Differ. Equ., 47 (1983), 133-161.  doi: 10.1016/0022-0396(83)90031-1.

[21]

N. G. Gratz, Critical review of the vector status of Aedes albopictus, Medical and Veterinary Entomology., 18 (2004), 215-227.  doi: 10.1111/j.0269-283X.2004.00513.x.

[22]

J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9 (2003), 925-936.  doi: 10.3934/dcds.2003.9.925.

[23]

W. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differ. Equ., 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060.

[24]

C. K. R. T. Jones, Geometrical Singular Perturbation Theory, Springer, Berlin, 1995. doi: 10.1007/BFb0095239.

[25]

W.-T. LiG. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273.  doi: 10.1088/0951-7715/19/6/003.

[26]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[27]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differ. Equ., 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.

[28]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theoret. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.

[29]

M. S. MustafaV. RasotgiS. Jain and V. Gupta, Discovype of dengue virus (DENV-5): A new public health dilemma in dengue control, Med. J. Armed Forces India., 71 (2015), 67-70.  doi: 10.1016/j.mjafi.2014.09.011.

[30]

P. Reiter and D. Sprenger, The used tire trade: A mechanism for the worldwide dispersal of container breeding mosquitoes, J. Am. Mosq. Control. Assoc., 3 (1987), 494-501. 

[31]

S. Ruan and D. Xiao, Stability of steady states and existence of traveling wave in a vector disease model, Proc. Roy. Soc. Edinburgh., 134A (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[32]

Q. Shi, J. Shi and H. Wang, Spatial movement with distributed memory, J. Math. Biol., 82 (2021), Paper No. 33, 32 pp. doi: 10.1007/s00285-021-01588-0.

[33]

L. T. TakahashiN. A. MaidanaW. Castro Ferreira Jr.P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind, Bull. Math. Biol., 67 (2005), 509-528.  doi: 10.1016/j.bulm.2004.08.005.

[34]

K. Wang, H. Zhao and H. Wang, Traveling waves for a diffusive mosquito-borne epidemic model with general incidence, Z. Angew. Math. Phys., 73 (2022), Paper No. 31, 28 pp. doi: 10.1007/s00033-021-01666-9.

[35]

K. Wang, H. Zhao, H. Wang and R. Zhang, Traveling wave of a reaction-diffusion vector-borne disease model with nonlocal effects and distributed delay, J. Dyn. Differ. Equ., 2021. doi: 10.1007/s10884-021-10062-w.

[36]

Z.-C. WangW.-T. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatiotemporal delays, J. Differ. Equ., 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.

[37]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.

[38]

X. Wu and M. Ni, Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term, Discrete Contin. Dyn. Syst. Ser. S., 14 (2021), 3249-3266.  doi: 10.3934/dcdss.2020341.

[39]

R. Zhang, J. Wang and S. Liu, Traveling wave solutions for a class of discrete diffusive SIR epidemic model, J. Nonlinear Sci., 31 (2021), Paper No. 10, 33 pp. doi: 10.1007/s00332-020-09656-3.

[40]

T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differ. Equ., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.

[41]

R. Zhang and H. Zhao, Traveling wave solutions for Zika transmission model with nonlocal diffusion, J. Math. Anal. Appl., 513 (2022), 126201, 29 pp. doi: 10.1016/j.jmaa.2022.126201.

[42]

X. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via monotone iteration method, Proc. Am. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.

Figure 1.  The sketch of the distribution of the roots of $ \Sigma_1(\zeta) $
Figure 2.  Graphs of $ u_1(t, x) $ and $ u_2(t, x) $ with respect to the spatial variable $ x $ for fixed time: $ t = 3, 5, 20, 30, 35, 40, 50 $
Figure 3.  The dependence of the $ c_2^\ast $ on parameters
Figure 4.  Sensitivity of $ \mathcal{R}_0 $ to parameters of model (3)
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