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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[44]

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

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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[44]

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

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