American Institute of Mathematical Sciences

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 and 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, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. [2] P. C. Fife, "Mathematical Aspect of Reacting and Diffusing Systems,'' Lecture Notes in Biomath., 28, Springer-Verlag, Berlin-New York, 1979. [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), 335-361. doi: 10.1007/BF00250432. [4] A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [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), 1563-1579. doi: 10.1098/rspa.2002.1094. [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, Ser. Internat. A, 1 (1937), 1-26. [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), 495-510. doi: 10.1016/j.jde.2008.12.026. [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), 511-529. doi: 10.1016/j.jde.2008.12.020. [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), 551-568. [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), 2762-2790. doi: 10.1137/090776342. [11] J. D. Murry, "Mathematical Biology,'' Vols. I and II, Springer-Verlag, New York, 2002. [12] A. N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosci., 31 (1976), 307-315. doi: 10.1016/0025-5564(76)90087-0. [13] V. A. Vasiliev, Yu. M. Romanovskii, D. S. Chernavskki and G. Yakhno, "Autowave Processes in Kinetic Systems,'' Reidel, Dorgrecht, 1987. [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), 343-366. doi: 10.3934/dcds.2011.29.343.

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References:
 [1] N. F. Britton, "Reaction-diffusion Equations and their Applications to Biology,'' Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. [2] P. C. Fife, "Mathematical Aspect of Reacting and Diffusing Systems,'' Lecture Notes in Biomath., 28, Springer-Verlag, Berlin-New York, 1979. [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), 335-361. doi: 10.1007/BF00250432. [4] A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. [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), 1563-1579. doi: 10.1098/rspa.2002.1094. [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, Ser. Internat. A, 1 (1937), 1-26. [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), 495-510. doi: 10.1016/j.jde.2008.12.026. [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), 511-529. doi: 10.1016/j.jde.2008.12.020. [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), 551-568. [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), 2762-2790. doi: 10.1137/090776342. [11] J. D. Murry, "Mathematical Biology,'' Vols. I and II, Springer-Verlag, New York, 2002. [12] A. N. Stokes, On two types of moving front in quasilinear diffusion, Math. Biosci., 31 (1976), 307-315. doi: 10.1016/0025-5564(76)90087-0. [13] V. A. Vasiliev, Yu. M. Romanovskii, D. S. Chernavskki and G. Yakhno, "Autowave Processes in Kinetic Systems,'' Reidel, Dorgrecht, 1987. [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), 343-366. doi: 10.3934/dcds.2011.29.343.
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