American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

K. Müller, "Investigations on the Organic Drift in North Swedish Streams," Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954. Google Scholar

[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.  Google Scholar

[12]

J. D. Murray, "Mathematical Biology I: An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.  Google Scholar

[13]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.  Google Scholar

[14]

A. Nilsson, Coleoptera from a Malaise-trap placed across a coastal stream in northern Sweden, Fauna Norrlandica, 3 (1981), 1-9. Google Scholar

[15]

A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.  Google Scholar

[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.  Google Scholar

[17]

T. Roos, Studies on upstream migration in adult streamdwelling insects,, Inst. Freshwater Res. Drottningholm, (): 167.   Google Scholar

[18]

W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.  Google Scholar

[19]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford University Press, Oxford, 1997. Google Scholar

[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.  Google Scholar

[21]

D. Tilman and P. Kareiva, "Spatial Ecology," Princeton University Press, Princeton, New Jersey, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models," Ph.D thesis, Sichuan University, 2011. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

K. Müller, "Investigations on the Organic Drift in North Swedish Streams," Tech. Report 34, Institute of Freshwater Research, Drottningholm, Sweden, 1954. Google Scholar

[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.  Google Scholar

[12]

J. D. Murray, "Mathematical Biology I: An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.  Google Scholar

[13]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.  Google Scholar

[14]

A. Nilsson, Coleoptera from a Malaise-trap placed across a coastal stream in northern Sweden, Fauna Norrlandica, 3 (1981), 1-9. Google Scholar

[15]

A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.  Google Scholar

[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.  Google Scholar

[17]

T. Roos, Studies on upstream migration in adult streamdwelling insects,, Inst. Freshwater Res. Drottningholm, (): 167.   Google Scholar

[18]

W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.  Google Scholar

[19]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford University Press, Oxford, 1997. Google Scholar

[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.  Google Scholar

[21]

D. Tilman and P. Kareiva, "Spatial Ecology," Princeton University Press, Princeton, New Jersey, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models," Ph.D thesis, Sichuan University, 2011. Google Scholar

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