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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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. Google Scholar

[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.  Google Scholar

[10]

M. Kot, Discrete-time travelling waves: ecological examples, J. Math. Biol., 30 (1992), 413-436. 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), 2027-2042. doi: 10.2307/2265698.  Google Scholar

[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.  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), 323-338. doi: 10.1007/s00285-008-0175-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems,, preprint, ().   Google Scholar

[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. Google Scholar

[31]

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

[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.  Google Scholar

[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.  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), 721-737. doi: 10.1016/0362-546X(78)90015-9.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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. Google Scholar

[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.  Google Scholar

[10]

M. Kot, Discrete-time travelling waves: ecological examples, J. Math. Biol., 30 (1992), 413-436. 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), 2027-2042. doi: 10.2307/2265698.  Google Scholar

[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.  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), 323-338. doi: 10.1007/s00285-008-0175-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems,, preprint, ().   Google Scholar

[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. Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

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