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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
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