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October  2011, 16(3): 895-925. doi: 10.3934/dcdsb.2011.16.895

Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology

Judith R. Miller 1, and  Huihui Zeng 1,

1. 

Dept. of Mathematics and Statistics, Georgetown University, Washington DC 20057, United States, United States

Received  June 2010 Revised  December 2010 Published  June 2011

We use spectral methods to prove a general stability theorem for traveling wave solutions to the systems of integrodifference equations arising in spatial population biology. We show that non-minimum-speed waves are exponentially asymptotically stable to small perturbations in appropriately weighted $L^\infty$ spaces, under assumptions which apply to examples including a Laplace or Gaussian dispersal kernel a monotone (or non-monotone) growth function behaving qualitatively like the Beverton-Holt function (or Ricker function with overcompensation), and a constant probability $p\in [0,1)$ (or $p=0$) of remaining sedentary for a single population; as well as to a system of two populations exhibiting non-cooperation (in particular, Hassell and Comins' model [6]) with $p=0$ and Laplace or Gaussian dispersal kernels which can be different for the two populations.
Keywords: Traveling wave, stability, integrodifference equation..
Mathematics Subject Classification: Primary: 92D25, 92D40; Secondary: 47G1.
Citation: Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895
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]

R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[3]

M. Gil', "Difference Equations in Normed Spaces. Stability and Oscillations," North-Holland Mathematics Studies 206, Elsevier Science B.V., Amsterdam, 2007.

[4]

D. Hardin, P. Takac and G. Webb, A comparison of dispersal strategies for survival of spatially heterogeneous populations, SIAM J. Appl. Math., 48 (1988), 1396-1423. doi: 10.1137/0148086.

[5]

D. Hardin, P. Takac and G. Webb, Dispersion population models discrete in time and continuous in space, J. Math. Biol., 28 (1990), 1-20. doi: 10.1007/BF00171515.

[6]

M. Hassell and H. Comins, Discrete time models for two-species competition, Theoretical Population Biology, 9 (1976), 202-221. doi: 10.1016/0040-5809(76)90045-9.

[7]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.

[8]

A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Éude de l'éuations de la diffusion avec crois-sance de la quantité de matièret son application a un problème biologique, Bull. Univ. Moscow, Ser. Internat., Sec. A, 1 (1937), 1-25.

[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. doi: 10.2307/2265698.

[12]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.

[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]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.

[15]

B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 389-399. doi: 10.3934/dcdsb.2009.12.389.

[16]

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.

[17]

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.

[18]

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.

[19]

R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator, J. Math. Biol., 16 (1982/83), 199-220.

[20]

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.

[21]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.

[22]

F. Lutscher, Density-dependent dispersal in integrodifference equations, J. Math. Biol., 56 (2008), 499-524. doi: 10.1007/s00285-007-0127-1.

[23]

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. doi: 10.1006/tpbi.1995.1020.

[24]

S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. RWA, 12 (2011), 535-544. doi: 10.1016/j.nonrwa.2010.06.038.

[25]

D. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0.

[26]

J. Travis, D. Murrell and C. Dytham, The evolution of density-dependent dispersal, Proc. R. Soc. Lond. B, 266 (1999), 1837-1842. doi: 10.1098/rspb.1999.0854.

[27]

R. Veit and M. Lewis, Dispersal, population growth and the Allee Effect: Dynamics of the House Finch invasion of eastern North America, American Naturalist, 148 (1996), 255-274. doi: 10.1086/285924.

[28]

D. Volkov and R. Lui, Spreading speed and travelling wave solutions of a partially sedentary population, IMA J. Appl. Math., 72 (2007), 801-816. doi: 10.1093/imamat/hxm025.

[29]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, preprint, arXiv:1003.1600.

[30]

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.

[31]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.

[32]

H. Weinberger, M. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[33]

H. Weinberger, M. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.

show all references

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]

R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[3]

M. Gil', "Difference Equations in Normed Spaces. Stability and Oscillations," North-Holland Mathematics Studies 206, Elsevier Science B.V., Amsterdam, 2007.

[4]

D. Hardin, P. Takac and G. Webb, A comparison of dispersal strategies for survival of spatially heterogeneous populations, SIAM J. Appl. Math., 48 (1988), 1396-1423. doi: 10.1137/0148086.

[5]

D. Hardin, P. Takac and G. Webb, Dispersion population models discrete in time and continuous in space, J. Math. Biol., 28 (1990), 1-20. doi: 10.1007/BF00171515.

[6]

M. Hassell and H. Comins, Discrete time models for two-species competition, Theoretical Population Biology, 9 (1976), 202-221. doi: 10.1016/0040-5809(76)90045-9.

[7]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.

[8]

A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Éude de l'éuations de la diffusion avec crois-sance de la quantité de matièret son application a un problème biologique, Bull. Univ. Moscow, Ser. Internat., Sec. A, 1 (1937), 1-25.

[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. doi: 10.2307/2265698.

[12]

M. Lewis, B. Li and H. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.

[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]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.

[15]

B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 389-399. doi: 10.3934/dcdsb.2009.12.389.

[16]

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.

[17]

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.

[18]

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.

[19]

R. Lui, Existence and stability of travelling wave solutions of a nonlinear integral operator, J. Math. Biol., 16 (1982/83), 199-220.

[20]

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.

[21]

R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6.

[22]

F. Lutscher, Density-dependent dispersal in integrodifference equations, J. Math. Biol., 56 (2008), 499-524. doi: 10.1007/s00285-007-0127-1.

[23]

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. doi: 10.1006/tpbi.1995.1020.

[24]

S. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Anal. RWA, 12 (2011), 535-544. doi: 10.1016/j.nonrwa.2010.06.038.

[25]

D. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0.

[26]

J. Travis, D. Murrell and C. Dytham, The evolution of density-dependent dispersal, Proc. R. Soc. Lond. B, 266 (1999), 1837-1842. doi: 10.1098/rspb.1999.0854.

[27]

R. Veit and M. Lewis, Dispersal, population growth and the Allee Effect: Dynamics of the House Finch invasion of eastern North America, American Naturalist, 148 (1996), 255-274. doi: 10.1086/285924.

[28]

D. Volkov and R. Lui, Spreading speed and travelling wave solutions of a partially sedentary population, IMA J. Appl. Math., 72 (2007), 801-816. doi: 10.1093/imamat/hxm025.

[29]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, preprint, arXiv:1003.1600.

[30]

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.

[31]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.

[32]

H. Weinberger, M. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[33]

H. Weinberger, M. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6.

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