# American Institute of Mathematical Sciences

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

 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.
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.

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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] 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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