Using an elementary approach, we establish a new maximum principle for the diffusive Lotka-Volterra system of two competing species, which involves pointwise estimate of an elliptic equation consisting of the second derivative of one function, the first derivative of another function, and a quadratic nonlinearity. This maximum principle gives a priori estimates for the total mass of the two species. Moreover, applying it to the system of three competing species leads to a nonexistence theorem of traveling wave solutions.
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