Methane and molecular hydrogen desorption from a methyl and hydrogen exposed Cu(001) surface is investigated. Both gaseous products are observed nearly more than 100 K The lower temperature desorption, at similar to 325 simultaneously within two temperature regimes separated by K, is believed to result from two processes which compete for adsorbed atomic hydrogen: methyl reduction and associative hydrogen desorption. The higher-temperature competitive desorption is initiated after the onset of thermal decomposition of remaining methyl species, at similar to 420 K. Kinetic simulations of the two presumed competing reactions are used to show observable and comparable methane and hydrogen evolution can occur in two temperature regimes, only with a precise balance of kinetic parameters, but fail to accurately reproduce the observed small differences in CH4 and H-2 peak desorption temperatures. It is concluded that either the utilized desorption kinetics are inaccurate at low H-(a) coverages or rapid desorption, or the same reactions are not competitive at higher temperatures and an alternative active mechanism for product evolution must exist.