# American Institute of Mathematical Sciences

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

## Traveling wave solutions in an integro-differential competition model

 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.
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, 14 (1977), 1. [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. [3] S. Fedotov, Front propagation into an unstable state of reaction-transport systems,, Phys. Rev. Lett., 86 (2001), 926. 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. 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. 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. 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. 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. 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), 1. [10] K. Müller, "Investigations on the Organic Drift in North Swedish Streams,", Tech. Report 34, (1954). [11] K. Müller, Stream drift as a chronobiological phenomenon in running water ecosystems,, Ann. Rev. Eco. Sys., 5 (1974), 309. doi: 10.1146/annurev.es.05.110174.001521. [12] J. D. Murray, "Mathematical Biology I: An Introduction,", Third edition, 17 (2002). [13] J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", Third edition, 18 (2003). [14] A. Nilsson, Coleoptera from a Malaise-trap placed across a coastal stream in northern Sweden,, Fauna Norrlandica, 3 (1981), 1. [15] A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives,", Second edition, 14 (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. 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, (1991). [19] N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford University Press, (1997). [20] M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion,, Arch. Ration. Mech. Anal., 73 (1980), 69. doi: 10.1007/BF00283257. [21] D. Tilman and P. Kareiva, "Spatial Ecology,", Princeton University Press, (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. 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. doi: 10.1007/s002850200145. [24] L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models,", Ph.D thesis, (2011).

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##### References:
 [1] D. Aronson, The asymptotic speed of propagation of a simple epidemic,, in, 14 (1977), 1. [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. [3] S. Fedotov, Front propagation into an unstable state of reaction-transport systems,, Phys. Rev. Lett., 86 (2001), 926. 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. 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. 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. 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. 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. 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), 1. [10] K. Müller, "Investigations on the Organic Drift in North Swedish Streams,", Tech. Report 34, (1954). [11] K. Müller, Stream drift as a chronobiological phenomenon in running water ecosystems,, Ann. Rev. Eco. Sys., 5 (1974), 309. doi: 10.1146/annurev.es.05.110174.001521. [12] J. D. Murray, "Mathematical Biology I: An Introduction,", Third edition, 17 (2002). [13] J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", Third edition, 18 (2003). [14] A. Nilsson, Coleoptera from a Malaise-trap placed across a coastal stream in northern Sweden,, Fauna Norrlandica, 3 (1981), 1. [15] A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives,", Second edition, 14 (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. 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, (1991). [19] N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford University Press, (1997). [20] M. M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion,, Arch. Ration. Mech. Anal., 73 (1980), 69. doi: 10.1007/BF00283257. [21] D. Tilman and P. Kareiva, "Spatial Ecology,", Princeton University Press, (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. 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. doi: 10.1007/s002850200145. [24] L. Zhang, "Traveling Wave Solutions and Periodic Solutions for Several Classes of Nonlinear Population Models,", Ph.D thesis, (2011).
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