  In this paper we will establish nonlinear a priori lower and upper bounds for
the solutions to a large class of equations which arise from the study of
traveling wave solutions of reaction-diffusion equations, and we will apply our
nonlinear bounds to the Lotka-Volterra system of two competing species as
examples. The idea used in a series of papers
\cite{NBMP-Discrete,JDE-16,CPAA-16,DCDS-B-18,NBMP-n-species,DCDS-A-17} for the
establishment of the linear N-barrier maximum principle will also be used in
the proof.
