Traveling wavetrains in generalized two-species predator-prey models and two-component reaction-diffusion equations are considered. The stabilities of the fixed points of the traveling wave ODEs (in the usual spatial variable) are found. For general functional forms of the nonlinear prey birthrate/death rate or reaction terms, Hopf bifurcations are shown to occur at various critical values of the parameters. The post-bifurcation dynamics is investigated for three different functional forms of the nonlinearities. The normal forms near the double Hopf points are derived using the method of multiple scales. The possible post-bifurcation dynamics resulting from the normal form comprises stable limit cycles and 2-period tori corresponding to periodic and quasiperiodic wavetrains. In principle, subcritical Hopf bifurcations may yield more complex behavior, although none has been observed. The diverse behaviors predicted from the normal forms in various parameter regimes are validated using numerical simulations and diagnostics.