Cardiac alternans is a heart rhythm instability that is associated with cardiac arrhythmias and may lead to sudden cardiac death. The onset of this instability, which is linked to period-doubling bifurcation and may be a route to chaos, is of particular interest. Mechano-electric feedback depicts the effects of tissue deformation on cardiac excitation. The main effect of mechano-electric feedback is delivered via the so-called stretch-activated ion channels and is caused by stretch-activated currents. Mechano-electric feedback, which is believed to have proarrhythmic and antiarrhythmic effects on cardiac electrophysiology, affects the action potential duration in a manner dependent on cycle length, but the mechanisms by which this occurs remain to be elucidated. In this study, a biophysically detailed electromechanical model of cardiac tissue is employed to show how a stretch-activated current can affect the action potential duration at cellular and tissue levels, illustrating its effects on the onset of alternans. Also, using a two-dimensional iterated map that incorporates stretch-activated current effects, we apply linear stability analysis to study the stability of the bifurcation. We show that alternans bifurcation can be prevented depending on the strength of the stretch-activated current.