The deterministic mathematical model for two competing biological species is described by the Lotka-Volterra non-linear differential equations. These have been derived with the assumption of two linear functions to represent the diminutions in the Malthusian net growth rates resulting from the effects of self-inhibition and competition. Firstly, a generalized model assuming non-linear forms for these functions is proposed here, and is shown to lead to differential equations of the same form as the original L-V equations. In a previous paper presenting a competition model with time-varying rates, a certain integrable class of the appropriate non-autonomous differential equations was also reduced to the autonomous L-V equations. As these latter are thus seen to arise in cases other than the original case for which they have been derived, we give here six more exact solutions (six were previously given by the author). Secondly, we generalize the original L-V equations in another direction by considering immigration and emigration of one or both species at constant rates. For these equations we have obtained over fifty exact solutions, six of which are here given.