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 & 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.  doi: 10.1016/0362-546X(78)90015-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1.  Google Scholar

[4]

M. Gil', "Difference Equations in Normed Spaces. Stability and Oscillations,", North-Holland Mathematics Studies, 206 (2007).   Google Scholar

[5]

J. Goodman, Stability of viscous scalar shock fronts in several dimensions,, Trans. Amer. Math. Soc., 311 (1989), 683.  doi: 10.2307/2001146.  Google Scholar

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar

[7]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

[8]

T. Kapitula, Multidimensional stability of planar travelling waves,, Trans. Amer. Math. Soc., 349 (1997), 257.  doi: 10.1090/S0002-9947-97-01668-1.  Google Scholar

[9]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models,, Math. Biosci., 80 (1986), 109.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

[10]

M. Kot, Discrete-time travelling waves: Ecological examples,, J. Math. Biol., 30 (1992), 413.  doi: 10.1007/BF00173295.  Google Scholar

[11]

M. Kot, M. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.   Google Scholar

[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.  doi: 10.1080/03605309208820908.  Google Scholar

[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.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

[14]

G. Lin and W. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations,, J. Math. Anal. Appl., 361 (2010), 520.  doi: 10.1016/j.jmaa.2009.07.035.  Google Scholar

[15]

G. Lin, W. Li and S. Ruan, Asymptotic stability of monostable wavefronts in disctrete-time integral recursions,, Sci. China Math., 53 (2010), 1185.  doi: 10.1007/s11425-009-0123-6.  Google Scholar

[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.  doi: 10.1137/0513064.  Google Scholar

[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.  doi: 10.1137/0513065.  Google Scholar

[18]

R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.  doi: 10.1007/BF00276502.  Google Scholar

[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.  doi: 10.1137/0516087.  Google Scholar

[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.  doi: 10.3934/dcdsb.2011.16.895.  Google Scholar

[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.   Google Scholar

[22]

D. Sattinger, On the stability of waves of nonlinear parabolic systems,, Advances in Math., 22 (1976), 312.   Google Scholar

[23]

H. Weinberger, Asymptotic behavior of a model in population genetics,, in, (1978), 1976.   Google Scholar

[24]

H. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353.  doi: 10.1137/0513028.  Google Scholar

[25]

J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I,, Comm. Partial Differential Equations, 17 (1992), 1889.  doi: 10.1080/03605309208820907.  Google Scholar

[26]

K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves,, Indiana Univ. Math. J., 47 (1998), 741.  doi: 10.1512/iumj.1998.47.1604.  Google Scholar

show all references

References:
[1]

O. Diekmann and H. Kaper, On the bounded solutions of a nonlinear convolution equation,, Nonlinear Anal., 2 (1978), 721.  doi: 10.1016/0362-546X(78)90015-9.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1.  Google Scholar

[4]

M. Gil', "Difference Equations in Normed Spaces. Stability and Oscillations,", North-Holland Mathematics Studies, 206 (2007).   Google Scholar

[5]

J. Goodman, Stability of viscous scalar shock fronts in several dimensions,, Trans. Amer. Math. Soc., 311 (1989), 683.  doi: 10.2307/2001146.  Google Scholar

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar

[7]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776.  doi: 10.1137/070703016.  Google Scholar

[8]

T. Kapitula, Multidimensional stability of planar travelling waves,, Trans. Amer. Math. Soc., 349 (1997), 257.  doi: 10.1090/S0002-9947-97-01668-1.  Google Scholar

[9]

M. Kot and W. Schaffer, Discrete-time growth-dispersal models,, Math. Biosci., 80 (1986), 109.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

[10]

M. Kot, Discrete-time travelling waves: Ecological examples,, J. Math. Biol., 30 (1992), 413.  doi: 10.1007/BF00173295.  Google Scholar

[11]

M. Kot, M. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms,, Ecology, 77 (1996), 2027.   Google Scholar

[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.  doi: 10.1080/03605309208820908.  Google Scholar

[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.  doi: 10.1007/s00285-008-0175-1.  Google Scholar

[14]

G. Lin and W. Li, Spreading speeds and traveling wavefronts for second order integrodifference equations,, J. Math. Anal. Appl., 361 (2010), 520.  doi: 10.1016/j.jmaa.2009.07.035.  Google Scholar

[15]

G. Lin, W. Li and S. Ruan, Asymptotic stability of monostable wavefronts in disctrete-time integral recursions,, Sci. China Math., 53 (2010), 1185.  doi: 10.1007/s11425-009-0123-6.  Google Scholar

[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.  doi: 10.1137/0513064.  Google Scholar

[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.  doi: 10.1137/0513065.  Google Scholar

[18]

R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator,, J. Math. Biol., 16 (): 199.  doi: 10.1007/BF00276502.  Google Scholar

[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.  doi: 10.1137/0516087.  Google Scholar

[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.  doi: 10.3934/dcdsb.2011.16.895.  Google Scholar

[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.   Google Scholar

[22]

D. Sattinger, On the stability of waves of nonlinear parabolic systems,, Advances in Math., 22 (1976), 312.   Google Scholar

[23]

H. Weinberger, Asymptotic behavior of a model in population genetics,, in, (1978), 1976.   Google Scholar

[24]

H. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353.  doi: 10.1137/0513028.  Google Scholar

[25]

J. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I,, Comm. Partial Differential Equations, 17 (1992), 1889.  doi: 10.1080/03605309208820907.  Google Scholar

[26]

K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves,, Indiana Univ. Math. J., 47 (1998), 741.  doi: 10.1512/iumj.1998.47.1604.  Google Scholar

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