While alternans in a single cardiac cell appears through a simple
period-doubling bifurcation, in extended tissue the exact nature
of the bifurcation is unclear. In particular, the phase of
alternans can exhibit wave-like spatial dependence, either
stationary or travelling, which is known as discordant
alternans. We study these phenomena in simple cardiac models
through a modulation equation proposed by Echebarria-Karma. As
shown in our previous paper, the zero solution of their equation
may lose stability, as the pacing rate is increased, through
either a Hopf or steady-state bifurcation. Which bifurcation
occurs first depends on parameters in the equation, and for one
critical case both modes bifurcate together at a degenerate
(codimension 2) bifurcation. For parameters close to the
degenerate case, we investigate the competition between modes,
both numerically and analytically. We find that at sufficiently
rapid pacing (but assuming a 1:1 response is maintained), steady
patterns always emerge as the only stable solution. However, in
the parameter range where Hopf bifurcation occurs first, the
evolution from periodic solution (just after the bifurcation) to
the eventual standing wave solution occurs through an interesting
series of secondary bifurcations.