# American Institute of Mathematical Sciences

September  2013, 12(5): 2083-2090. doi: 10.3934/cpaa.2013.12.2083

## The sign of the wave speed for the Lotka-Volterra competition-diffusion system

 1 Department of Mathematics, Tamkang University, 151, Ying-Chuan Road, Tamsui, Taipei County 25137 2 Department of Mathematics, National Taiwan Normal University, 88, S-4, Ting Chou Road, Taipei 11677

Received  March 2012 Revised  September 2012 Published  January 2013

In this paper, we study the traveling front solutions of the Lotka-Volterra competition-diffusion system with bistable nonlinearity. It is well-known that the wave speed of traveling front is unique. Although little is known for the sign of the wave speed. In this paper, we first study the standing wave which gives some criteria when the speed is zero. Then, by the monotone dependence on parameters, we obtain some criteria about the sign of the wave speed under some parameter restrictions.
Citation: Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
##### References:
 [1] C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitve reaction-diffusion model,, {Indiana Univ. Math. J.}, 33 (1984), 319. [2] R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretic approach,, {J. Differential Equations}, 44 (1982), 343. doi: 10.1016/0022-0396(82)90001-8. [3] J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, {J. Dyn. Diff. Equat.}, 23 (2011), 353. doi: 10.1007/s10884-011-9214-5. [4] Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model,, {Bulletin of Math. Biology}, 60 (1998), 435. doi: 10.1006/bulm.1997.0008. [5] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations,, {IAM J. Math. Anal.}, 26 (1995), 340. doi: 10.1137/S0036141093244556. [6] Y. Kan-on, Existence of standing waves for competition-diffusion equations,, {Japan J. Indust. Appl. Math.}, 13 (1996), 117. doi: 10.1007/BF03167302. [7] Y. Kan-on, Stability of monotone travelling waves for competition-diffusion equations,, {Japan J. Indust. Appl. Math.}, 13 (1996), 343. doi: 10.1007/BF03167252. [8] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, {Nonlinear Analysis, 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. [9] Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations,, {Hiroshima Math. J.}, 23 (1993), 193. [10] M. Mimura and P. C. Fife, A 3-component system of competition and diffusion,, {Hiroshima Math. J.}, 16 (1986), 189. [11] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, {Hiroshima Math. J.}, 30 (2000), 257. [12] M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion,, {Arch. Rational Mech. Anal.}, 73 (1980), 69. doi: 10.1007/BF00283257.

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##### References:
 [1] C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitve reaction-diffusion model,, {Indiana Univ. Math. J.}, 33 (1984), 319. [2] R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretic approach,, {J. Differential Equations}, 44 (1982), 343. doi: 10.1016/0022-0396(82)90001-8. [3] J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, {J. Dyn. Diff. Equat.}, 23 (2011), 353. doi: 10.1007/s10884-011-9214-5. [4] Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model,, {Bulletin of Math. Biology}, 60 (1998), 435. doi: 10.1006/bulm.1997.0008. [5] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations,, {IAM J. Math. Anal.}, 26 (1995), 340. doi: 10.1137/S0036141093244556. [6] Y. Kan-on, Existence of standing waves for competition-diffusion equations,, {Japan J. Indust. Appl. Math.}, 13 (1996), 117. doi: 10.1007/BF03167302. [7] Y. Kan-on, Stability of monotone travelling waves for competition-diffusion equations,, {Japan J. Indust. Appl. Math.}, 13 (1996), 343. doi: 10.1007/BF03167252. [8] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, {Nonlinear Analysis, 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. [9] Y. Kan-on and E. Yanagida, Existence of nonconstant stable equilibria in competition-diffusion equations,, {Hiroshima Math. J.}, 23 (1993), 193. [10] M. Mimura and P. C. Fife, A 3-component system of competition and diffusion,, {Hiroshima Math. J.}, 16 (1986), 189. [11] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, {Hiroshima Math. J.}, 30 (2000), 257. [12] M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion,, {Arch. Rational Mech. Anal.}, 73 (1980), 69. doi: 10.1007/BF00283257.
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