• Previous Article
    A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations
  • KRM Home
  • This Issue
  • Next Article
    Second-order mixed-moment model with differentiable ansatz function in slab geometry
October 2018, 11(5): 1235-1253. doi: 10.3934/krm.2018048

Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

* Corresponding author

Received  June 2017 Revised  July 2017 Published  May 2018

Fund Project: The first author is supported by NSFC grant (No.11571066) and the second author is supported by NSFC grant (No.11771071)

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed $c>c_*$, where $c_*>0$ is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves ($c = c_*$), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.

Citation: Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048
References:
[1]

M. AguerreaC. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8.

[2]

I. L. ChernM. MeiX. F. Yang and Q. F. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations, 259 (2015), 1503-1541. doi: 10.1016/j.jde.2015.03.003.

[3]

J. Fang and X. Q. Zhao, Esistence and uniqueness of traveling waves for non-monotone integral equations with in applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009.

[4]

T. FariaW. Huang and J. Wu, Traveling 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.

[5]

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

[6]

T. Faria and S. Trofimchuk, Positive heteroclinics and traveling waves for scalar population models with a single delay, Appl. Math. Comput., 185 (2007), 594-603. doi: 10.1016/j.amc.2006.07.059.

[7]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68. doi: 10.1112/jlms/jdt050.

[8]

S. A. Gourley and J. Wu, Delayed nonlocal diffusive system in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.

[9]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[10]

R. HuangM. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621.

[11]

R. HuangM. MeiK. J. Zhang and Q. F. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersal equations, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1331-1353. doi: 10.3934/dcds.2016.36.1331.

[12]

Y. C. Jiang and K. J. Zhang, Time-delayed reaction-diffusion equation with boundary effect: (Ⅰ) converegence to non-critical traveling waves, Applicable Analysis, 97 (2018), 230-254. doi: 10.1080/00036811.2016.1258696.

[13]

W. T. LiS. G. Ruan and Z. C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delays, J. Nonlinear Sci., 17 (2007), 505-525. doi: 10.1007/s00332-007-9003-9.

[14]

C. K. LinC. T. LinY. P. Lin and M. Mei, Exponential stability of non-monotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391.

[15]

C. K. Lin and M. Mei, On traveling wavefronts of the Nicholson's blowflies equations with diffusion, Proc. Roy. Soc. Edinburgh Set. A, 140 (2010), 135-152. doi: 10.1017/S0308210508000784.

[16]

S. W. 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.

[17]

A. Matsumura and M. Mei, Convergence to traveling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22. doi: 10.1007/s002050050134.

[18]

M. MeiC. K. LinC. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation, (Ⅰ) Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026.

[19]

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.

[20]

M. Mei, C. H. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math., 42 (2010), 2762–2790; erratum, SIAM J. Appl. Math., 44 (2012), 538–540. doi: 10.1137/110850633.

[21]

M. Mei and J. W. H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333.

[22]

M. MeiJ. W. H. SoM. Y. Li and S. S. P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358.

[23]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Num. Anal. Model Ser. B, 2 (2011), 379-401.

[24]

A. J. Nicholson, Competition for food amongst Lucilia Cuprina larvae, Proceedings of the 8th International Congress of Entomology, Stockhom, (1984), 227–281.

[25]

A. J. Nicholson, An outline of dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65. doi: 10.1071/ZO9540009.

[26]

J. W. H. So and Y. Yang, Dirichlet problem for the diffusion Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348. doi: 10.1006/jdeq.1998.3489.

[27]

J. So and X. Zou, Traveling waves for the diffusion Nicholson's blowflies equation, Appl. Math. Comput., 122 (2001), 385-392. doi: 10.1016/S0096-3003(00)00055-2.

[28]

E. TrofimchukV. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332. doi: 10.1016/j.jde.2008.06.023.

[29]

E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion with delay, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 407-423. doi: 10.3934/dcds.2008.20.407.

[30]

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

show all references

References:
[1]

M. AguerreaC. Gomez and S. Trofimchuk, On uniqueness of semi-wavefronts, Math. Ann., 354 (2012), 73-109. doi: 10.1007/s00208-011-0722-8.

[2]

I. L. ChernM. MeiX. F. Yang and Q. F. Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations, 259 (2015), 1503-1541. doi: 10.1016/j.jde.2015.03.003.

[3]

J. Fang and X. Q. Zhao, Esistence and uniqueness of traveling waves for non-monotone integral equations with in applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009.

[4]

T. FariaW. Huang and J. Wu, Traveling 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.

[5]

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

[6]

T. Faria and S. Trofimchuk, Positive heteroclinics and traveling waves for scalar population models with a single delay, Appl. Math. Comput., 185 (2007), 594-603. doi: 10.1016/j.amc.2006.07.059.

[7]

A. Gomez and S. Trofimchuk, Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89 (2014), 47-68. doi: 10.1112/jlms/jdt050.

[8]

S. A. Gourley and J. Wu, Delayed nonlocal diffusive system in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137-200.

[9]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[10]

R. HuangM. Mei and Y. Wang, Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3621-3649. doi: 10.3934/dcds.2012.32.3621.

[11]

R. HuangM. MeiK. J. Zhang and Q. F. Zhang, Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersal equations, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1331-1353. doi: 10.3934/dcds.2016.36.1331.

[12]

Y. C. Jiang and K. J. Zhang, Time-delayed reaction-diffusion equation with boundary effect: (Ⅰ) converegence to non-critical traveling waves, Applicable Analysis, 97 (2018), 230-254. doi: 10.1080/00036811.2016.1258696.

[13]

W. T. LiS. G. Ruan and Z. C. Wang, On the diffusive Nicholson's blowflies equation with nonlocal delays, J. Nonlinear Sci., 17 (2007), 505-525. doi: 10.1007/s00332-007-9003-9.

[14]

C. K. LinC. T. LinY. P. Lin and M. Mei, Exponential stability of non-monotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46 (2014), 1053-1084. doi: 10.1137/120904391.

[15]

C. K. Lin and M. Mei, On traveling wavefronts of the Nicholson's blowflies equations with diffusion, Proc. Roy. Soc. Edinburgh Set. A, 140 (2010), 135-152. doi: 10.1017/S0308210508000784.

[16]

S. W. 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.

[17]

A. Matsumura and M. Mei, Convergence to traveling fronts of solutions of the $p$-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22. doi: 10.1007/s002050050134.

[18]

M. MeiC. K. LinC. T. Lin and J. W. H. So, Traveling wavefronts for time-delayed reaction-diffusion equation, (Ⅰ) Local nonlinearity, J. Differential Equations, 247 (2009), 495-510. doi: 10.1016/j.jde.2008.12.026.

[19]

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.

[20]

M. Mei, C. H. Ou and X. Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations, SIAM J. Appl. Math., 42 (2010), 2762–2790; erratum, SIAM J. Appl. Math., 44 (2012), 538–540. doi: 10.1137/110850633.

[21]

M. Mei and J. W. H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551-568. doi: 10.1017/S0308210506000333.

[22]

M. MeiJ. W. H. SoM. Y. Li and S. S. P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 579-594. doi: 10.1017/S0308210500003358.

[23]

M. Mei and Y. Wang, Remark on stability of traveling waves for nonlocal Fisher-KPP equations, Int. J. Num. Anal. Model Ser. B, 2 (2011), 379-401.

[24]

A. J. Nicholson, Competition for food amongst Lucilia Cuprina larvae, Proceedings of the 8th International Congress of Entomology, Stockhom, (1984), 227–281.

[25]

A. J. Nicholson, An outline of dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65. doi: 10.1071/ZO9540009.

[26]

J. W. H. So and Y. Yang, Dirichlet problem for the diffusion Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348. doi: 10.1006/jdeq.1998.3489.

[27]

J. So and X. Zou, Traveling waves for the diffusion Nicholson's blowflies equation, Appl. Math. Comput., 122 (2001), 385-392. doi: 10.1016/S0096-3003(00)00055-2.

[28]

E. TrofimchukV. Tkachenko and S. Trofimchuk, Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay, J. Differential Equations, 245 (2008), 2307-2332. doi: 10.1016/j.jde.2008.06.023.

[29]

E. Trofimchuk and S. Trofimchuk, Admissible wavefront speeds for a single species reaction-diffusion with delay, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 407-423. doi: 10.3934/dcds.2008.20.407.

[30]

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

[1]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029

[2]

Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526

[3]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

[4]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681

[5]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[6]

Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115

[7]

Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157

[8]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319

[9]

Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709

[10]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

[11]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[12]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347

[13]

Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011

[14]

Masaharu Taniguchi. Traveling fronts in perturbed multistable reaction-diffusion equations. Conference Publications, 2011, 2011 (Special) : 1368-1377. doi: 10.3934/proc.2011.2011.1368

[15]

Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks & Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79

[16]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

[17]

Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275

[18]

Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471

[19]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[20]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (32)
  • HTML views (87)
  • Cited by (0)

Other articles
by authors

[Back to Top]