May  2012, 17(3): 993-1007. doi: 10.3934/dcdsb.2012.17.993

Global stability and convergence rate of traveling waves for a nonlocal model in periodic media

1. 

School of Mathematics and Physics, University of South China, Hengyang, 421001, China

2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St.Johns, Newfoundland, A1C 5S7, Canada

Received  May 2011 Revised  August 2011 Published  January 2012

In this paper, we study the stability and convergence rate of traveling wavefronts for a nonlocal population model in a periodic habitat \[ \left\{ \begin{array}{ll} \displaystyle\frac{\partial u(t,x)}{\partial t}=D(x)\frac{\partial ^2u(t,x)}{% \partial x^2}-d(x,u(t,x))+\int_R\Gamma (\tau ,x,y)b(y,u(t-\tau ,y))dy, & \\ u(\theta ,x)=\varphi (\theta ,x),\theta \in [-\tau ,0],& \end{array} \right. \] where $D(x), d(x,\cdot ), b(x,\cdot ), \Gamma (\tau ,x,y)$ are L-periodic functions with respect to space $x$ (and $y$) for some positive real constant $L $. Using the analysis of the principal eigenvalue of a non-local linear operator, we show that all noncritical wavefronts are globally exponentially stable, as long as the initial perturbation is uniformly bounded in a weighted space. This result can be generalized to n-dimensional case and three applications of our main results are also presented.
Citation: Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
References:
[1]

N. F. Britton, "Reaction-diffusion Equations and their Applications to Biology,'', Academic Press, (1986). Google Scholar

[2]

P. C. Fife, "Mathematical Aspect of Reacting and Diffusing Systems,'', Lecture Notes in Biomath., 28 (1979). Google Scholar

[3]

P. C. Fife and J. B. Mcleod, The approach solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335. doi: 10.1007/BF00250432. Google Scholar

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964). Google Scholar

[5]

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

[6]

A. N. Kolmogorov, I. G. Petrowsky and N. S. Piscounov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique,, Bull. Univ. d'État á Moscou, 1 (1937), 1. Google Scholar

[7]

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

[8]

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

[9]

M. Mei and J. W.-H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551. Google Scholar

[10]

M. Mei, C. 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. doi: 10.1137/090776342. Google Scholar

[11]

J. D. Murry, "Mathematical Biology,'', Vols. I and II, (2002). Google Scholar

[12]

A. N. Stokes, On two types of moving front in quasilinear diffusion,, Math. Biosci., 31 (1976), 307. doi: 10.1016/0025-5564(76)90087-0. Google Scholar

[13]

V. A. Vasiliev, Yu. M. Romanovskii, D. S. Chernavskki and G. Yakhno, "Autowave Processes in Kinetic Systems,'', Reidel, (1987). Google Scholar

[14]

P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat,, Discrete and Continuous Dynamical Systems Ser. A, 29 (2011), 343. doi: 10.3934/dcds.2011.29.343. Google Scholar

show all references

References:
[1]

N. F. Britton, "Reaction-diffusion Equations and their Applications to Biology,'', Academic Press, (1986). Google Scholar

[2]

P. C. Fife, "Mathematical Aspect of Reacting and Diffusing Systems,'', Lecture Notes in Biomath., 28 (1979). Google Scholar

[3]

P. C. Fife and J. B. Mcleod, The approach solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335. doi: 10.1007/BF00250432. Google Scholar

[4]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964). Google Scholar

[5]

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

[6]

A. N. Kolmogorov, I. G. Petrowsky and N. S. Piscounov, Étude de l'équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique,, Bull. Univ. d'État á Moscou, 1 (1937), 1. Google Scholar

[7]

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

[8]

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

[9]

M. Mei and J. W.-H. So, Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 551. Google Scholar

[10]

M. Mei, C. 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. doi: 10.1137/090776342. Google Scholar

[11]

J. D. Murry, "Mathematical Biology,'', Vols. I and II, (2002). Google Scholar

[12]

A. N. Stokes, On two types of moving front in quasilinear diffusion,, Math. Biosci., 31 (1976), 307. doi: 10.1016/0025-5564(76)90087-0. Google Scholar

[13]

V. A. Vasiliev, Yu. M. Romanovskii, D. S. Chernavskki and G. Yakhno, "Autowave Processes in Kinetic Systems,'', Reidel, (1987). Google Scholar

[14]

P. Weng and X.-Q. Zhao, Spatial dynamics of a nonlocal and delayed population model in a periodic habitat,, Discrete and Continuous Dynamical Systems Ser. A, 29 (2011), 343. doi: 10.3934/dcds.2011.29.343. Google Scholar

[1]

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

[2]

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

[3]

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

[4]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255

[5]

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

[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]

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

[8]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141

[9]

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

[10]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21

[11]

Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775

[12]

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

[13]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331

[14]

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

[15]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083

[16]

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

[17]

Wei Feng, Xin Lu. Global stability in a class of reaction-diffusion systems with time-varying delays. Conference Publications, 1998, 1998 (Special) : 253-261. doi: 10.3934/proc.1998.1998.253

[18]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

[19]

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

[20]

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

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]