American Institute of Mathematical Sciences

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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

Zigen Ouyang 1, and  Chunhua Ou 2,

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.
Keywords: Green's function., time delay, Nonlocal reaction-diffusion equations, global stability, traveling waves.
Mathematics Subject Classification: Primary: 35K57, 34K20; Secondary: 92D2.
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

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