# American Institute of Mathematical Sciences

May  2013, 18(3): 741-751. doi: 10.3934/dcdsb.2013.18.741

## Multidimensional stability of planar traveling waves for an integrodifference model

 1 Dept. of Mathematics and Statistics, Georgetown University, Washington DC 20057, United States 2 Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

Received  December 2011 Revised  September 2012 Published  December 2012

This paper studies the multidimensional stability of planar traveling waves for integrodifference equations. It is proved that for a Gaussian dispersal kernel, if the traveling wave is exponentially orbitally stable in one space dimension, then the corresponding planar wave is stable in $H^m(\mathbb{R}^N)$, $N\ge 4$, $m\ge [N/2]+1$, with the perturbation decaying at algebraic rate.
Citation: Judith R. Miller, Huihui Zeng. Multidimensional stability of planar traveling waves for an integrodifference model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 741-751. doi: 10.3934/dcdsb.2013.18.741
##### References:
 [1] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. [2] P. Fife and J. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. [3] R. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), 797-855. doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1. [4] M. Gil', "Difference Equations in Normed Spaces. Stability and Oscillations," North-Holland Mathematics Studies, 206, Elsevier Science B.V., Amsterdam, 2007. [5] J. Goodman, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc., 311 (1989), 683-695. doi: 10.2307/2001146. [6] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [7] S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. [8] T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1. [9] M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6. [10] M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295. [11] M. Kot, M. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. [12] C. Levermore and J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations, 17 (1992), 1901-1924. doi: 10.1080/03605309208820908. [13] B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338. doi: 10.1007/s00285-008-0175-1. [14] G. Lin and W. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations, J. Math. Anal. Appl., 361 (2010), 520-532. doi: 10.1016/j.jmaa.2009.07.035. [15] G. Lin, W. Li and S. Ruan, Asymptotic stability of monostable wavefronts in disctrete-time integral recursions, Sci. China Math., 53 (2010), 1185-1194. doi: 10.1007/s11425-009-0123-6. [16] R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937. doi: 10.1137/0513064. [17] R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953. doi: 10.1137/0513065. [18] R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator, J. Math. Biol., 16 (1982/83), 199-220. doi: 10.1007/BF00276502. [19] R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206. doi: 10.1137/0516087. [20] J. Miller and H. Zeng, Stability of travelling waves for systems of nonlinear integral recursions in spatial population biology, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 895-925. doi: 10.3934/dcdsb.2011.16.895. [21] M. Neubert, M. Kot and M. Lewis, Dispersal and pattern-formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43. [22] D. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355. [23] H. Weinberger, Asymptotic behavior of a model in population genetics, in "Nonlinear Partial Differential Equations and Applications" (Proc. Special Sem., Indiana Univ., Bloomington, Ind., 1976-1977), Lecture Notes in Math., Vol. 648, Springer, Berlin, (1978), 47-96. [24] H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [25] J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations, 17 (1992), 1889-1899. doi: 10.1080/03605309208820907. [26] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871. doi: 10.1512/iumj.1998.47.1604.

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##### References:
 [1] O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal., 2 (1978), 721-737. doi: 10.1016/0362-546X(78)90015-9. [2] P. Fife and J. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361. [3] R. Gardner and K. Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51 (1998), 797-855. doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1. [4] M. Gil', "Difference Equations in Normed Spaces. Stability and Oscillations," North-Holland Mathematics Studies, 206, Elsevier Science B.V., Amsterdam, 2007. [5] J. Goodman, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc., 311 (1989), 683-695. doi: 10.2307/2001146. [6] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [7] S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. [8] T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269. doi: 10.1090/S0002-9947-97-01668-1. [9] M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136. doi: 10.1016/0025-5564(86)90069-6. [10] M. Kot, Discrete-time travelling waves: Ecological examples, J. Math. Biol., 30 (1992), 413-436. doi: 10.1007/BF00173295. [11] M. Kot, M. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042. [12] C. Levermore and J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations, 17 (1992), 1901-1924. doi: 10.1080/03605309208820908. [13] B. Li, M. Lewis and H. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338. doi: 10.1007/s00285-008-0175-1. [14] G. Lin and W. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations, J. Math. Anal. Appl., 361 (2010), 520-532. doi: 10.1016/j.jmaa.2009.07.035. [15] G. Lin, W. Li and S. Ruan, Asymptotic stability of monostable wavefronts in disctrete-time integral recursions, Sci. China Math., 53 (2010), 1185-1194. doi: 10.1007/s11425-009-0123-6. [16] R. Lui, A nonlinear integral operator arising from a model in population genetics. I. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937. doi: 10.1137/0513064. [17] R. Lui, A nonlinear integral operator arising from a model in population genetics. II. Initial data with compact support, SIAM J. Math. Anal., 13 (1982), 938-953. doi: 10.1137/0513065. [18] R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator, J. Math. Biol., 16 (1982/83), 199-220. doi: 10.1007/BF00276502. [19] R. Lui, A nonlinear integral operator arising from a model in population genetics. III. Heterozygote inferior case, SIAM J. Math. Anal., 16 (1985), 1180-1206. doi: 10.1137/0516087. [20] J. Miller and H. Zeng, Stability of travelling waves for systems of nonlinear integral recursions in spatial population biology, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 895-925. doi: 10.3934/dcdsb.2011.16.895. [21] M. Neubert, M. Kot and M. Lewis, Dispersal and pattern-formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43. [22] D. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355. [23] H. Weinberger, Asymptotic behavior of a model in population genetics, in "Nonlinear Partial Differential Equations and Applications" (Proc. Special Sem., Indiana Univ., Bloomington, Ind., 1976-1977), Lecture Notes in Math., Vol. 648, Springer, Berlin, (1978), 47-96. [24] H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [25] J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations, 17 (1992), 1889-1899. doi: 10.1080/03605309208820907. [26] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47 (1998), 741-871. doi: 10.1512/iumj.1998.47.1604.
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