Traveling wave solutions in an integro-differential competition model

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January 2012, 17(1): 417-428. doi: 10.3934/dcdsb.2012.17.417

Traveling wave solutions in an integro-differential competition model

Liang Zhang ^1, and Bingtuan Li ^2,

1.	Institute of Applied Mathematics, College of Science, Northwest A & F University, Yangling, Shaanxi 712100, China
2.	Department of Mathematics, University of Louisville, Louisville, KY 40292

Received June 2010 Revised April 2011 Published October 2011

Abstract
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We study the existence of traveling wave solutions for the two-species Lotka-Volterra competition model in the form of integro-differential equations. The model incorporates asymmetric dispersal kernels that describe long distance dispersal processes of competing species in space. Using lower and upper traveling wave solutions, we show that the model has traveling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.

Keywords: competition, dispersal kernel, traveling wave solution., Integro-differential equation.

Mathematics Subject Classification: Primary: 92D40, 92D25, 47G20; Secondary: 34C37, 34K1.

Citation: Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417

References:

[1]	D. Aronson, The asymptotic speed of propagation of a simple epidemic, in "Nonlinear Diffusion" (eds. W. Fitzgibbon and H. Walker) (NSF-CBMS Regional Conf. Nonlinear Diffusion Equations, Univ. Houston, Houton, TX, 1976), Res. Notes Math., 14, Pitman, London, (1977), 1-23.
[2]	P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transition, Arch. Rational. Mech. Anal., 138 (1997), 105-136.
[3]	S. Fedotov, Front propagation into an unstable state of reaction-transport systems, Phys. Rev. Lett., 86 (2001), 926-929. doi: 10.1103/PhysRevLett.86.926.
[4]	M. A. Lewis, B. Li and H. F. Weinberger, Spreading speeds and linear determinancy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.
[5]	B. Li, M. A. Lewis and H. F. 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.
[6]	W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with application to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003.
[7]	F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772. doi: 10.1137/050636152.
[8]	B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.
[9]	V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wave-front speed problem, Phys. Rev. E., 65 (2002), 041109/1-041109/6.
[10]	K. Müller, "Investigations on the Organic Drift in North Swedish Streams," Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954.
[11]	K. Müller, Stream drift as a chronobiological phenomenon in running water ecosystems, Ann. Rev. Eco. Sys., 5 (1974), 309-323. doi: 10.1146/annurev.es.05.110174.001521.
[12]	J. D. Murray, "Mathematical Biology I: An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.
[13]	J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.
[14]	A. Nilsson, Coleoptera from a Malaise-trap placed across a coastal stream in northern Sweden, Fauna Norrlandica, 3 (1981), 1-9.
[15]	A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.
[16]	W. D. Pearson and R. H. Kramer, Drift and production of two aquatic insects in a mountain stream, Ecol. Monogr., 42 (1972), 365-385. doi: 10.2307/1942214.
[17]	T. Roos, Studies on upstream migration in adult streamdwelling insects,, Inst. Freshwater Res. Drottningholm, (): 167.
[18]	W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
[19]	N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford University Press, Oxford, 1997.
[20]	M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257.
[21]	D. Tilman and P. Kareiva, "Spatial Ecology," Princeton University Press, Princeton, New Jersey, 1997.
[22]	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.
[23]	H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.
[24]	L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models," Ph.D thesis, Sichuan University, 2011.

show all references

References:

[1]	D. Aronson, The asymptotic speed of propagation of a simple epidemic, in "Nonlinear Diffusion" (eds. W. Fitzgibbon and H. Walker) (NSF-CBMS Regional Conf. Nonlinear Diffusion Equations, Univ. Houston, Houton, TX, 1976), Res. Notes Math., 14, Pitman, London, (1977), 1-23.
[2]	P. W. Bates, P. C. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transition, Arch. Rational. Mech. Anal., 138 (1997), 105-136.
[3]	S. Fedotov, Front propagation into an unstable state of reaction-transport systems, Phys. Rev. Lett., 86 (2001), 926-929. doi: 10.1103/PhysRevLett.86.926.
[4]	M. A. Lewis, B. Li and H. F. Weinberger, Spreading speeds and linear determinancy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144.
[5]	B. Li, M. A. Lewis and H. F. 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.
[6]	W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with application to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003.
[7]	F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772. doi: 10.1137/050636152.
[8]	B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98. doi: 10.1016/j.mbs.2005.03.008.
[9]	V. Méndez, T. Pujol and J. Fort, Dispersal probability distributions and the wave-front speed problem, Phys. Rev. E., 65 (2002), 041109/1-041109/6.
[10]	K. Müller, "Investigations on the Organic Drift in North Swedish Streams," Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954.
[11]	K. Müller, Stream drift as a chronobiological phenomenon in running water ecosystems, Ann. Rev. Eco. Sys., 5 (1974), 309-323. doi: 10.1146/annurev.es.05.110174.001521.
[12]	J. D. Murray, "Mathematical Biology I: An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.
[13]	J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.
[14]	A. Nilsson, Coleoptera from a Malaise-trap placed across a coastal stream in northern Sweden, Fauna Norrlandica, 3 (1981), 1-9.
[15]	A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.
[16]	W. D. Pearson and R. H. Kramer, Drift and production of two aquatic insects in a mountain stream, Ecol. Monogr., 42 (1972), 365-385. doi: 10.2307/1942214.
[17]	T. Roos, Studies on upstream migration in adult streamdwelling insects,, Inst. Freshwater Res. Drottningholm, (): 167.
[18]	W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.
[19]	N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford University Press, Oxford, 1997.
[20]	M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257.
[21]	D. Tilman and P. Kareiva, "Spatial Ecology," Princeton University Press, Princeton, New Jersey, 1997.
[22]	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.
[23]	H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.
[24]	L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models," Ph.D thesis, Sichuan University, 2011.

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