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July  2016, 15(4): 1451-1469. doi: 10.3934/cpaa.2016.15.1451

## Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species

 1 Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences (Taipei Office), No. 1, Sec. 4, Roosevelt Road, Taipei, 10617 2 Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences (Taipei Office), No. 1, Sec. 4, Roosevelt Road, Taipei, 10617

Received  July 2014 Revised  December 2015 Published  April 2016

In reaction-diffusion models describing the interaction between the invading grey squirrel and the established red squirrel in Britain, Okubo et al. ([19]) found that in certain parameter regimes, the profiles of the two species in a wave propagation solution can be determined via a solution of the KPP equation. Motivated by their result, we employ an elementary approach based on the maximum principle for elliptic inequalities coupled with estimates of a total density of the three species to establish the nonexistence of traveling wave solutions for Lotka-Volterra systems of three competing species. Applying our estimates to the May-Leonard model, we obtain upper and lower bounds for the total density of a solution to this system. For the existence of traveling wave solutions to the Lotka-Volterra three-species competing system, we find new semi-exact solutions by virtue of functions other than hyperbolic tangent functions, which are used in constructing one-hump exact traveling wave solutions in [2]. Moreover, new two-hump semi-exact traveling wave solutions different from the ones found in [1] are constructed.
Citation: Chiun-Chuan Chen, Li-Chang Hung. Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451
##### References:
 [1] C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems,, \emph{Hiroshima Math J.}, 43 (2013), 176. [2] C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 17 (2012), 2653. doi: 10.3934/dcdsb.2012.17.2653. [3] Y.-S. Chiou, Travelling wave solutions for reaction-diffusion-advection equations,, Master Thesis, (2010), 1. [4] P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems,, \emph{Institute of Math., 11 (1979). [5] N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system,, \emph{Nonlinear Anal. Real World Appl.}, 4 (2003), 503. doi: 10.1016/S1468-1218(02)00077-9. [6] L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species,, \emph{Jpn. J. Ind. Appl. Math.}, 29 (2012), 237. doi: 10.1007/s13160-012-0056-2. [7] H. Ikeda, Multiple travelling wave solutions of three-component systems with competition and diffusion,, \emph{Methods Appl. Anal.}, 8 (2001), 479. [8] H. Ikeda, Travelling wave solutions of three-component systems with competition and diffusion,, \emph{Math. J. Toyama Univ.}, 24 (2001), 37. [9] H. Ikeda, Global bifurcation phenomena of standing pulse solutions for three-component systems with competition and diffusion,, \emph{Hiroshima Math. J.}, 32 (2002), 87. [10] H. Ikeda, Dynamics of weakly interacting front and back waves in three-component systems,, \emph{Toyama Math. J.}, 30 (2007), 1. [11] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations,, \emph{SIAM J. Math. Anal.}, 26 (1995), 340. doi: 10.1137/S0036141093244556. [12] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, \emph{Nonlinear Anal.}, 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. [13] A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem,, \emph{Bull. Math}, 1 (1937), 1. [14] A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 171. doi: 10.3934/dcdsb.2011.15.171. [15] A. W. Leung, X. Hou, and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities,, \emph{J. Math. Anal. Appl.}, 338 (2008), 902. doi: 10.1016/j.jmaa.2007.05.066. [16] R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species,, \emph{SIAM J. Appl. Math.}, 29 (1975), 243. [17] P. D. Miller, Nonmonotone waves in a three species reaction-diffusion model,, \emph{Methods Appl. Anal.}, 4 (1997), 261. [18] P. D. Miller, Stability of non-monotone waves in a three-species reaction-diffusion model,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 129 (1999), 125. doi: 10.1017/S0308210500027499. [19] A. Okubo, P. Maini, M. Williamson and J. Murray, On the spatial spread of the grey squirrel in britain,, \emph{Proceedings of the Royal Society of London. B. Biological Sciences}, 238 (1989), 113. [20] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, \emph{Hiroshima Math. J.}, 30 (2000), 257. [21] M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations,, \emph{Japan J. Indust. Appl. Math.}, 18 (2001), 657. doi: 10.1007/BF03167410.

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##### References:
 [1] C.-C. Chen, L.-C. Hung, M. Mimura, M. Tohma and D. Ueyama, Semi-exact equilibrium solutions for three-species competition-diffusion systems,, \emph{Hiroshima Math J.}, 43 (2013), 176. [2] C.-C. Chen, L.-C. Hung, M. Mimura and D. Ueyama, Exact travelling wave solutions of three-species competition-diffusion systems,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 17 (2012), 2653. doi: 10.3934/dcdsb.2012.17.2653. [3] Y.-S. Chiou, Travelling wave solutions for reaction-diffusion-advection equations,, Master Thesis, (2010), 1. [4] P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems,, \emph{Institute of Math., 11 (1979). [5] N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system,, \emph{Nonlinear Anal. Real World Appl.}, 4 (2003), 503. doi: 10.1016/S1468-1218(02)00077-9. [6] L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species,, \emph{Jpn. J. Ind. Appl. Math.}, 29 (2012), 237. doi: 10.1007/s13160-012-0056-2. [7] H. Ikeda, Multiple travelling wave solutions of three-component systems with competition and diffusion,, \emph{Methods Appl. Anal.}, 8 (2001), 479. [8] H. Ikeda, Travelling wave solutions of three-component systems with competition and diffusion,, \emph{Math. J. Toyama Univ.}, 24 (2001), 37. [9] H. Ikeda, Global bifurcation phenomena of standing pulse solutions for three-component systems with competition and diffusion,, \emph{Hiroshima Math. J.}, 32 (2002), 87. [10] H. Ikeda, Dynamics of weakly interacting front and back waves in three-component systems,, \emph{Toyama Math. J.}, 30 (2007), 1. [11] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations,, \emph{SIAM J. Math. Anal.}, 26 (1995), 340. doi: 10.1137/S0036141093244556. [12] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, \emph{Nonlinear Anal.}, 28 (1997), 145. doi: 10.1016/0362-546X(95)00142-I. [13] A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem,, \emph{Bull. Math}, 1 (1937), 1. [14] A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited,, \emph{Discrete Contin. Dyn. Syst. Ser. B}, 15 (2011), 171. doi: 10.3934/dcdsb.2011.15.171. [15] A. W. Leung, X. Hou, and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities,, \emph{J. Math. Anal. Appl.}, 338 (2008), 902. doi: 10.1016/j.jmaa.2007.05.066. [16] R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species,, \emph{SIAM J. Appl. Math.}, 29 (1975), 243. [17] P. D. Miller, Nonmonotone waves in a three species reaction-diffusion model,, \emph{Methods Appl. Anal.}, 4 (1997), 261. [18] P. D. Miller, Stability of non-monotone waves in a three-species reaction-diffusion model,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 129 (1999), 125. doi: 10.1017/S0308210500027499. [19] A. Okubo, P. Maini, M. Williamson and J. Murray, On the spatial spread of the grey squirrel in britain,, \emph{Proceedings of the Royal Society of London. B. Biological Sciences}, 238 (1989), 113. [20] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, \emph{Hiroshima Math. J.}, 30 (2000), 257. [21] M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations,, \emph{Japan J. Indust. Appl. Math.}, 18 (2001), 657. doi: 10.1007/BF03167410.
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