A maximum principle for diffusive Lotka–Volterra systems of two competing species

Using an elementary approach, we establish a new maximum principle for the diffusive Lotka–Volterra system of two competing species, which involves pointwise estimates 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.