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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 Office), No. 1, Sec. 4, Roosevelt Road, Taipei, 10617

2. 

Department of Mathematics, National Taiwan University, and National Center for Theoretical Sciences (Taipei Office), 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.   Google Scholar

[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.  Google Scholar

[3]

Y.-S. Chiou, Travelling wave solutions for reaction-diffusion-advection equations,, Master Thesis, (2010), 1.   Google Scholar

[4]

P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems,, \emph{Institute of Math., 11 (1979).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Ikeda, Multiple travelling wave solutions of three-component systems with competition and diffusion,, \emph{Methods Appl. Anal.}, 8 (2001), 479.   Google Scholar

[8]

H. Ikeda, Travelling wave solutions of three-component systems with competition and diffusion,, \emph{Math. J. Toyama Univ.}, 24 (2001), 37.   Google Scholar

[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.   Google Scholar

[10]

H. Ikeda, Dynamics of weakly interacting front and back waves in three-component systems,, \emph{Toyama Math. J.}, 30 (2007), 1.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species,, \emph{SIAM J. Appl. Math.}, 29 (1975), 243.   Google Scholar

[17]

P. D. Miller, Nonmonotone waves in a three species reaction-diffusion model,, \emph{Methods Appl. Anal.}, 4 (1997), 261.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, \emph{Hiroshima Math. J.}, 30 (2000), 257.   Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[3]

Y.-S. Chiou, Travelling wave solutions for reaction-diffusion-advection equations,, Master Thesis, (2010), 1.   Google Scholar

[4]

P. de Mottoni, Qualitative analysis for some quasilinear parabolic systems,, \emph{Institute of Math., 11 (1979).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. Ikeda, Multiple travelling wave solutions of three-component systems with competition and diffusion,, \emph{Methods Appl. Anal.}, 8 (2001), 479.   Google Scholar

[8]

H. Ikeda, Travelling wave solutions of three-component systems with competition and diffusion,, \emph{Math. J. Toyama Univ.}, 24 (2001), 37.   Google Scholar

[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.   Google Scholar

[10]

H. Ikeda, Dynamics of weakly interacting front and back waves in three-component systems,, \emph{Toyama Math. J.}, 30 (2007), 1.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species,, \emph{SIAM J. Appl. Math.}, 29 (1975), 243.   Google Scholar

[17]

P. D. Miller, Nonmonotone waves in a three species reaction-diffusion model,, \emph{Methods Appl. Anal.}, 4 (1997), 261.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[20]

M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system,, \emph{Hiroshima Math. J.}, 30 (2000), 257.   Google Scholar

[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.  Google Scholar

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