April  2022, 42(4): 1599-1646. doi: 10.3934/dcds.2021166

Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author: Zhi-Cheng Wang

Received  April 2021 Revised  August 2021 Published  April 2022 Early access  November 2021

Fund Project: Both authors are supported by NNSF of China (12071193, 11731005) and NSF of Gansu Province of China (21JR7RA535)

This paper is concerned with a nonlocal time-space periodic reaction diffusion model with age structure. We first prove the existence and global attractivity of time-space periodic solution for the model. Next, by a family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Furthermore, we introduce a notion of transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and transition semi-waves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of transition semi-waves in the non-monotone case. In addition, we show that the obtained transition semi-waves are actually transition waves in the monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results.

Citation: Ning Wang, Zhi-Cheng Wang. Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1599-1646. doi: 10.3934/dcds.2021166
References:
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H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations. In: Perspectives in nonlinear partial differential equations. Providence: Amer. Math. Soc., Contemp. Math., 446 (2007), 101-123. doi: 10.1090/conm/446.

[2]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.

[3]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I-Species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[4]

X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.

[5]

F. Cao and W. Shen, Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media, Discrete Contin. Dyn. Syst., 37 (2017), 4697-4727.  doi: 10.3934/dcds.2017202.

[6]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, 279, Longman Scientific and Technical, Harlow, 1992.

[7]

D. Duehring and W. Huang, Periodic traveling waves for diffusion equations with time delayed and non-local responding reaction, J. Dynam. Differential Equations, 19 (2007), 457-477.  doi: 10.1007/s10884-006-9048-8.

[8]

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.

[9]

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.

[10]

T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.  doi: 10.1016/j.jde.2006.05.006.

[11]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[12]

B. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[13]

S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 459 (2003), 1563-1579.  doi: 10.1098/rspa.2002.1094.

[14]

Z. GuoF. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.

[15]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., 247, Longman Scientific and Technical, Harlow, 1991.

[16]

W. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.  doi: 10.1016/j.jde.2007.10.001.

[17]

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

[18]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496-2516.  doi: 10.1137/070709761.

[19]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Springer-Verlag, Berlin, 1995.

[20]

N. KinezakiK. KawasakiF. Takasu and N. Shigesada, Modeling biological invasions into periodically fragmented environments, Theor. Population Biol., 64 (2003), 291-302.  doi: 10.1016/S0040-5809(03)00091-1.

[21]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, Rhode Island, U.S.A., 1968.

[22]

J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z.

[23]

P. Li and S.-L. Wu, Monostable traveling waves for a time-periodic and delayed nonlocal reaction-diffusion equation, Z. Angew. Math. Phys., 69 (2018), 39, 16pp. doi: 10.1007/s00033-018-0936-7.

[24]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.

[25]

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.

[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]

M. Ma, J. Yue and C. Ou, Propagation direction of the bistable travelling wavefront for delayed non-local reaction diffusion equations, Proc. Royal Soc. A, 475 (2019), 20180898, 10 pp. doi: 10.1098/rspa.2018.0898.

[28]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277.  doi: 10.1016/j.jde.2007.03.014.

[29]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in non-local delayed diffusion equation, J. Dynam. Differential Equations, 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7.

[30]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[31]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[32]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.

[33]

G. Nadin and L. Rossi, Transition waves for Fisher-KPP equations with general time heterogeneous and space-periodic coefficients, Anal. PDE, 8 (2015), 1351-1377.  doi: 10.2140/apde.2015.8.1351.

[34]

G. Nadin and L. Rossi, Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations, Arch. Ration. Mech. Anal., 223 (2017), 1239-1267.  doi: 10.1007/s00205-016-1056-1.

[35]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatós, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.

[36]

Z. Ouyang and C. Ou, Global stability and convergence rate of traveling waves for a nonlocal model in periodic media, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 993-1007.  doi: 10.3934/dcdsb.2012.17.993.

[37]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[38]

W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.

[39] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, New York, 1997. 
[40]

H. L. Smith and H. R. Thieme, Strongly order preserving semiflows generated by functional- differential equations, J. Differential Equations, 93 (1991), 332-363.  doi: 10.1016/0022-0396(91)90016-3.

[41]

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, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[42]

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.

[43]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, J. Differential Equations, 247 (2009), 887-905.  doi: 10.1016/j.jde.2009.04.002.

[44]

N. WangZ.-C. Wang and X. Bao, Transition waves for lattice Fisher-KPP equations with time and space dependence, Proc. Roy. Soc. Edinburgh Sect. A, 151 (2021), 573-600.  doi: 10.1017/prm.2020.31.

[45]

Z.-C. Wang and W.-T. Li, Dynamics of a non-local delayed reaction-diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1081-1109.  doi: 10.1017/S0308210509000262.

[46]

Z.-C. WangW.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.

[47]

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.

[48]

Z.-C. WangL. Zhang and X.-Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dynam. Differential Equations, 30 (2018), 379-403.  doi: 10.1007/s10884-016-9546-2.

[49]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[50]

P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst., 29 (2011), 343-366.  doi: 10.3934/dcds.2011.29.343.

[51]

S.-L. Wu and C.-H. Hsu, Entire solutions of non-quasi-monotone delayed reaction-diffusion equations with applications, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1085-1112.  doi: 10.1017/S0308210512001412.

[52]

L. ZhangZ.-C. Wang and X.-Q. Zhao, Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity, Trans. Amer. Math. Soc., 372 (2019), 1751-1782.  doi: 10.1090/tran/7709.

[53]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

[54]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. 

show all references

References:
[1]

H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations. In: Perspectives in nonlinear partial differential equations. Providence: Amer. Math. Soc., Contemp. Math., 446 (2007), 101-123. doi: 10.1090/conm/446.

[2]

H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.  doi: 10.1002/cpa.21389.

[3]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model: I-Species persistence, J. Math. Biol., 51 (2005), 75-113.  doi: 10.1007/s00285-004-0313-3.

[4]

X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.

[5]

F. Cao and W. Shen, Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media, Discrete Contin. Dyn. Syst., 37 (2017), 4697-4727.  doi: 10.3934/dcds.2017202.

[6]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series, 279, Longman Scientific and Technical, Harlow, 1992.

[7]

D. Duehring and W. Huang, Periodic traveling waves for diffusion equations with time delayed and non-local responding reaction, J. Dynam. Differential Equations, 19 (2007), 457-477.  doi: 10.1007/s10884-006-9048-8.

[8]

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.

[9]

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.

[10]

T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.  doi: 10.1016/j.jde.2006.05.006.

[11]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[12]

B. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion-Convection Reaction, Progress in Nonlinear Differential Equations and their Applications, 60, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7964-4.

[13]

S. A. Gourley and Y. Kuang, Wavefront and global stability in a time-delayed population model with stage structure, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 459 (2003), 1563-1579.  doi: 10.1098/rspa.2002.1094.

[14]

Z. GuoF. Wang and X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.

[15]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., 247, Longman Scientific and Technical, Harlow, 1991.

[16]

W. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.  doi: 10.1016/j.jde.2007.10.001.

[17]

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

[18]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496-2516.  doi: 10.1137/070709761.

[19]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Springer-Verlag, Berlin, 1995.

[20]

N. KinezakiK. KawasakiF. Takasu and N. Shigesada, Modeling biological invasions into periodically fragmented environments, Theor. Population Biol., 64 (2003), 291-302.  doi: 10.1016/S0040-5809(03)00091-1.

[21]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, Rhode Island, U.S.A., 1968.

[22]

J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079.  doi: 10.1007/s11538-009-9457-z.

[23]

P. Li and S.-L. Wu, Monostable traveling waves for a time-periodic and delayed nonlocal reaction-diffusion equation, Z. Angew. Math. Phys., 69 (2018), 39, 16pp. doi: 10.1007/s00033-018-0936-7.

[24]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.

[25]

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.

[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]

M. Ma, J. Yue and C. Ou, Propagation direction of the bistable travelling wavefront for delayed non-local reaction diffusion equations, Proc. Royal Soc. A, 475 (2019), 20180898, 10 pp. doi: 10.1098/rspa.2018.0898.

[28]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237 (2007), 259-277.  doi: 10.1016/j.jde.2007.03.014.

[29]

S. Ma and J. Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in non-local delayed diffusion equation, J. Dynam. Differential Equations, 19 (2007), 391-436.  doi: 10.1007/s10884-006-9065-7.

[30]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[31]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[32]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.  doi: 10.1016/j.matpur.2012.05.005.

[33]

G. Nadin and L. Rossi, Transition waves for Fisher-KPP equations with general time heterogeneous and space-periodic coefficients, Anal. PDE, 8 (2015), 1351-1377.  doi: 10.2140/apde.2015.8.1351.

[34]

G. Nadin and L. Rossi, Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations, Arch. Ration. Mech. Anal., 223 (2017), 1239-1267.  doi: 10.1007/s00205-016-1056-1.

[35]

J. NolenJ.-M. RoquejoffreL. Ryzhik and A. Zlatós, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal., 203 (2012), 217-246.  doi: 10.1007/s00205-011-0449-4.

[36]

Z. Ouyang and C. Ou, Global stability and convergence rate of traveling waves for a nonlocal model in periodic media, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 993-1007.  doi: 10.3934/dcdsb.2012.17.993.

[37]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.

[38]

W. Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations, 16 (2004), 1011-1060.  doi: 10.1007/s10884-004-7832-x.

[39] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, New York, 1997. 
[40]

H. L. Smith and H. R. Thieme, Strongly order preserving semiflows generated by functional- differential equations, J. Differential Equations, 93 (1991), 332-363.  doi: 10.1016/0022-0396(91)90016-3.

[41]

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, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[42]

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.

[43]

H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations, J. Differential Equations, 247 (2009), 887-905.  doi: 10.1016/j.jde.2009.04.002.

[44]

N. WangZ.-C. Wang and X. Bao, Transition waves for lattice Fisher-KPP equations with time and space dependence, Proc. Roy. Soc. Edinburgh Sect. A, 151 (2021), 573-600.  doi: 10.1017/prm.2020.31.

[45]

Z.-C. Wang and W.-T. Li, Dynamics of a non-local delayed reaction-diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1081-1109.  doi: 10.1017/S0308210509000262.

[46]

Z.-C. WangW.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.  doi: 10.1016/j.jde.2007.03.025.

[47]

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.

[48]

Z.-C. WangL. Zhang and X.-Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dynam. Differential Equations, 30 (2018), 379-403.  doi: 10.1007/s10884-016-9546-2.

[49]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[50]

P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst., 29 (2011), 343-366.  doi: 10.3934/dcds.2011.29.343.

[51]

S.-L. Wu and C.-H. Hsu, Entire solutions of non-quasi-monotone delayed reaction-diffusion equations with applications, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1085-1112.  doi: 10.1017/S0308210512001412.

[52]

L. ZhangZ.-C. Wang and X.-Q. Zhao, Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity, Trans. Amer. Math. Soc., 372 (2019), 1751-1782.  doi: 10.1090/tran/7709.

[53]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

[54]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. 

Figure 1.  The long-time behaviour of the solution of (68). Changes in color represent changes in height. Baseline parameter values: $ D = 0.1, $ $ D_j = 0.1, $ $ \mu_j = 0.1, $ $ \tau = 1, $ $ p(x) = 0.5+0.2\cos(x), $ $ q = 0.3 $ and the periodicity of space $ L = 2\pi. $ The right panel is the two-dimensional projection of the left panel onto the $ xt $-plane
Figure 2.  The long-time behaviour of the solution of (69). Changes in color represent changes in height. Baseline parameter values: $ D = 0.1, $ $ D_j = 0.1, $ $ \mu_j = 0.1, $ $ \tau = 1, $ $ p_1(t) = 0.5+0.2\cos(\frac{t}{2}), $ $ q = 0.3 $ and the periodicity of time $ T = 4\pi. $ The right panel is the two-dimensional projection of the left panel onto the $ xt $-plane
Figure 3.  The long-time behaviour of the solution of (70). Changes in color represent changes in height. Baseline parameter values: $ D = 0.1, $ $ D_j = 0.1, $ $ \mu_j = 0.1, $ $ \tau = 1, $ $ p_2(t, x) = 0.7+0.2\cos(\frac{t}{2})+0.2\cos(x), $ $ q = 0.3, $ the periodicity of time $ T = 4\pi $ and the periodicity of space $ L = 2\pi. $ The right panel is the two-dimensional projection of the left panel onto the $ xt $-plane
Figure 4.  The shapes of the birth function $ b(x, u) = p(x)ue^{\bar{p}(x)u} $ and death function $ d(x, u) = qu $ for equation (71), where $ p(x) = 12+2\sin\left(x\right), $ $ \bar{p}(x) = -0.7-0.2\cos\left(x\right), $ $ q = 0.2. $ Changes in color represent changes in height. The right panel is the two-dimensional projection of the left panel onto the $ uoz $-plane
Figure 5.  The shapes of the birth function $ b(t, u) = p_1(t)ue^{\bar{p}_1(t)u} $ and death function $ d(x, u) = q_1u $ for equation (72), where $ p(x) = 12+2\cos\left(t\right), $ $ \bar{p}(x) = -0.7-0.2\sin\left(t\right), $ $ q = 0.4. $ Changes in color represent changes in height. The right panel is the two-dimensional projection of the left panel onto the $ uoz $-plane
Figure 6.  The long-time behaviour of the solution of (71). Changes in color represent changes in height. Baseline parameter values: $ D = 0.1, $ $ D_j = 0.1, $ $ \mu_j = 0.1, $ $ \tau = 10, $ $ p(x) = 12+2\sin\left(x\right), $ $ \bar{p}(x) = -0.7-0.2\cos\left(x\right), $ $ q = 0.2 $ and the periodicity of space $ L = 2\pi. $ The right panel is the two-dimensional projection of the left panel onto the $ xt $-plane
Figure 7.  The long-time behaviour of the solution of (72). Changes in color represent changes in height. Baseline parameter values: $ D = 0.1, $ $ D_j = 0.1, $ $ \mu_j = 0.1, $ $ \tau = 10, $ $ p_1(t) = 12+2\cos\left(t\right), $ $ \bar{p}_1(t) = -0.7-0.2\sin\left(t\right), $ $ q_1 = 0.4 $ and the periodicity of time $ T = 2\pi. $ The right panel is the two-dimensional projection of the left panel onto the $ xt $-plane
Figure 8.  The long-time behaviour of the solution of (73). Changes in color represent changes in height. Baseline parameter values: $ D = 0.1, $ $ D_j = 0.1, $ $ \mu_j = 0.1, $ $ \tau = 10, $ $ p_2(t, x) = 13+2\cos\left(\sqrt{2}t\right)+\sin\left(x\right), $ $ \bar{p}_2(t, x) = -1.1-0.2\cos\left(x\right), $ $ q = 0.3, $ the periodicity of time $ T = \sqrt{2}\pi $ and the periodicity of space $ L = 2\pi. $ The right panel is the two-dimensional projection of the left panel onto the $ xt $-plane
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