June  2018, 38(6): 2987-3022. doi: 10.3934/dcds.2018128

Propagation of monostable traveling fronts in discrete periodic media with delay

1. 

School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China

2. 

Department of Mathematics, National Central University, Chungli 32001, Taiwan

E-mail address: slwu@xidian.edu.cn(S.-L. Wu)(Corresponding author)

Received  September 2017 Published  April 2018

This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.

Citation: Shi-Liang Wu, Cheng-Hsiung Hsu. Propagation of monostable traveling fronts in discrete periodic media with delay. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2987-3022. doi: 10.3934/dcds.2018128
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: J. A. Goldstein (Ed. ), Partial Differential Equations and Related Topics, in: Lecture Notes in Math. , Springer, New York, 446 (1975), 5-49.  Google Scholar

[2]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅱ -Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[3]

X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.  Google Scholar

[4]

X. ChenJ.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Rational Mech. Anal., 189 (2008), 189-236.  doi: 10.1007/s00205-007-0103-3.  Google Scholar

[5]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.  Google Scholar

[6]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.  doi: 10.1007/BF02450783.  Google Scholar

[7]

J. FangJ. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Differential Equations, 245 (2008), 2749-2770.  doi: 10.1016/j.jde.2008.09.001.  Google Scholar

[8]

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., 459 (2003), 1563-1579.  doi: 10.1098/rspa.2002.1094.  Google Scholar

[9]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[10]

J.-S. Guo and C.-H. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833.  doi: 10.1016/j.jde.2009.03.010.  Google Scholar

[11]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK, 1991.  Google Scholar

[12]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar

[13]

T. Kawahara and M. Tanaka, Interactions of traveling fronts: An exact solution of a nonlinear diffusion equation, Phys. Lett. A, 97 (1983), 311-314.  doi: 10.1016/0375-9601(83)90648-5.  Google Scholar

[14]

W. Kerscher and R. Nagel, Asymptotic behavior of one-parameter semigroups of positive operators, Acta Applicandae Math., 2 (1984), 297-309.  doi: 10.1007/BF02280856.  Google Scholar

[15]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[16]

X. Liang and M. Hiroshi, Maximizing the spreading speed of KPP fronts in two-dimensional stratified media, Proc. Lond. Math. Soc., 109 (2014), 1137-1174.  doi: 10.1112/plms/pdu031.  Google Scholar

[17]

X. LiangX. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Trans. Amer. Math. Soc., 362 (2010), 5605-5633.  doi: 10.1090/S0002-9947-2010-04931-1.  Google Scholar

[18]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[19]

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.  Google Scholar

[20]

R. Lowen, Kuratowski's measure of noncompactness revisited, Quart. J. Math. Oxford Ser. (2), 39 (1988), 235-254.  doi: 10.1093/qmath/39.2.235.  Google Scholar

[21]

R. Lui, Biological growth and spread modeled by systems of recursions. Ⅰ. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[22]

R. Lui, Biological growth and spread modeled by systems of recursions. Ⅱ. Biological theory, Math. Biosci., 93 (1989), 297-331.  doi: 10.1016/0025-5564(89)90027-8.  Google Scholar

[23]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.  Google Scholar

[24]

S. Ma and X. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations, Discrete Contin. Dyn. Syst., 21 (2008), 259-275.  doi: 10.3934/dcds.2008.21.259.  Google Scholar

[25]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.   Google Scholar

[26]

M. MeiC.-K. LinC.-T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: Ⅱ nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020.  Google Scholar

[27]

M. MeiC. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790.  doi: 10.1137/090776342.  Google Scholar

[28]

H. Smith and H. 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.  Google Scholar

[29]

J. W.-H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. Ⅰ. Travelling wavefronts on the unbounded domains, Proc. R. Soc. Lond. Ser. A., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[30]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonl. Analy.: RWA, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[31]

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

[32]

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.  Google Scholar

[33]

X.-S. Wang and X.-Q. Zhao, Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differential Equations, 259 (2015), 7238-7259.  doi: 10.1016/j.jde.2015.08.019.  Google Scholar

[34]

Z.-C. WangW.-T. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.  doi: 10.1007/s10884-008-9103-8.  Google Scholar

[35]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[36]

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.  Google Scholar

[37]

P. WengH. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.  doi: 10.1093/imamat/68.4.409.  Google Scholar

[38]

P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[39]

P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 343-366.   Google Scholar

[40]

S.-L. Wu and C.-H. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays, Nonlinearity, 25 (2012), 1-17.  doi: 10.1088/0951-7715/25/9/2785.  Google Scholar

[41]

S.-L. WuZ. Shi and F. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049.  Google Scholar

[42]

Y. Wu and X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123-1139.  doi: 10.3934/dcds.2008.20.1123.  Google Scholar

[43]

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

[44]

X. -Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in: J. A. Goldstein (Ed. ), Partial Differential Equations and Related Topics, in: Lecture Notes in Math. , Springer, New York, 446 (1975), 5-49.  Google Scholar

[2]

H. BerestyckiF. Hamel and L. Roques, Analysis of the periodically fragmented environment model: Ⅱ -Biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.  doi: 10.1016/j.matpur.2004.10.006.  Google Scholar

[3]

X. Chen and J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.  doi: 10.1006/jdeq.2001.4153.  Google Scholar

[4]

X. ChenJ.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Rational Mech. Anal., 189 (2008), 189-236.  doi: 10.1007/s00205-007-0103-3.  Google Scholar

[5]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.  Google Scholar

[6]

O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.  doi: 10.1007/BF02450783.  Google Scholar

[7]

J. FangJ. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Differential Equations, 245 (2008), 2749-2770.  doi: 10.1016/j.jde.2008.09.001.  Google Scholar

[8]

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., 459 (2003), 1563-1579.  doi: 10.1098/rspa.2002.1094.  Google Scholar

[9]

J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0.  Google Scholar

[10]

J.-S. Guo and C.-H. Wu, Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system, J. Differential Equations, 246 (2009), 3818-3833.  doi: 10.1016/j.jde.2009.03.010.  Google Scholar

[11]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK, 1991.  Google Scholar

[12]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar

[13]

T. Kawahara and M. Tanaka, Interactions of traveling fronts: An exact solution of a nonlinear diffusion equation, Phys. Lett. A, 97 (1983), 311-314.  doi: 10.1016/0375-9601(83)90648-5.  Google Scholar

[14]

W. Kerscher and R. Nagel, Asymptotic behavior of one-parameter semigroups of positive operators, Acta Applicandae Math., 2 (1984), 297-309.  doi: 10.1007/BF02280856.  Google Scholar

[15]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[16]

X. Liang and M. Hiroshi, Maximizing the spreading speed of KPP fronts in two-dimensional stratified media, Proc. Lond. Math. Soc., 109 (2014), 1137-1174.  doi: 10.1112/plms/pdu031.  Google Scholar

[17]

X. LiangX. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Trans. Amer. Math. Soc., 362 (2010), 5605-5633.  doi: 10.1090/S0002-9947-2010-04931-1.  Google Scholar

[18]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[19]

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.  Google Scholar

[20]

R. Lowen, Kuratowski's measure of noncompactness revisited, Quart. J. Math. Oxford Ser. (2), 39 (1988), 235-254.  doi: 10.1093/qmath/39.2.235.  Google Scholar

[21]

R. Lui, Biological growth and spread modeled by systems of recursions. Ⅰ. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[22]

R. Lui, Biological growth and spread modeled by systems of recursions. Ⅱ. Biological theory, Math. Biosci., 93 (1989), 297-331.  doi: 10.1016/0025-5564(89)90027-8.  Google Scholar

[23]

S. Ma and X. Zou, Propagation and its failure in a lattice delayed differential equation with global interaction, J. Differential Equations, 212 (2005), 129-190.  doi: 10.1016/j.jde.2004.07.014.  Google Scholar

[24]

S. Ma and X. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations, Discrete Contin. Dyn. Syst., 21 (2008), 259-275.  doi: 10.3934/dcds.2008.21.259.  Google Scholar

[25]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.   Google Scholar

[26]

M. MeiC.-K. LinC.-T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation: Ⅱ nonlocal nonlinearity, J. Differential Equations, 247 (2009), 511-529.  doi: 10.1016/j.jde.2008.12.020.  Google Scholar

[27]

M. MeiC. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762-2790.  doi: 10.1137/090776342.  Google Scholar

[28]

H. Smith and H. 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.  Google Scholar

[29]

J. W.-H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. Ⅰ. Travelling wavefronts on the unbounded domains, Proc. R. Soc. Lond. Ser. A., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.  Google Scholar

[30]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonl. Analy.: RWA, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.  Google Scholar

[31]

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

[32]

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.  Google Scholar

[33]

X.-S. Wang and X.-Q. Zhao, Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differential Equations, 259 (2015), 7238-7259.  doi: 10.1016/j.jde.2015.08.019.  Google Scholar

[34]

Z.-C. WangW.-T. Li and S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations, 20 (2008), 563-607.  doi: 10.1007/s10884-008-9103-8.  Google Scholar

[35]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[36]

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.  Google Scholar

[37]

P. WengH. Huang and J. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Math., 68 (2003), 409-439.  doi: 10.1093/imamat/68.4.409.  Google Scholar

[38]

P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[39]

P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst. Ser. A, 29 (2011), 343-366.   Google Scholar

[40]

S.-L. Wu and C.-H. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays, Nonlinearity, 25 (2012), 1-17.  doi: 10.1088/0951-7715/25/9/2785.  Google Scholar

[41]

S.-L. WuZ. Shi and F. Yang, Entire solutions in periodic lattice dynamical systems, J. Differential Equations, 255 (2013), 3505-3535.  doi: 10.1016/j.jde.2013.07.049.  Google Scholar

[42]

Y. Wu and X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst., 20 (2008), 1123-1139.  doi: 10.3934/dcds.2008.20.1123.  Google Scholar

[43]

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

[44]

X. -Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.  Google Scholar

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