While alternans in a single cardiac cell appears through a simpleperiod-doubling bifurcation, in extended tissue the exact natureof the bifurcation is unclear. In particular, the phase ofalternans can exhibit wave-like spatial dependence, eitherstationary or travelling, which is known as discordantalternans. We study these phenomena in simple cardiac modelsthrough a modulation equation proposed by Echebarria-Karma. Asshown in our previous paper, the zero solution of their equationmay lose stability, as the pacing rate is increased, througheither a Hopf or steady-state bifurcation. Which bifurcationoccurs first depends on parameters in the equation, and for onecritical case both modes bifurcate together at a degenerate(codimension 2) bifurcation. For parameters close to thedegenerate case, we investigate the competition between modes,both numerically and analytically. We find that at sufficientlyrapid pacing (but assuming a 1:1 response is maintained), steadypatterns always emerge as the only stable solution. However, inthe parameter range where Hopf bifurcation occurs first, theevolution from periodic solution (just after the bifurcation) tothe eventual standing wave solution occurs through an interestingseries of secondary bifurcations.