The coupled Dirac-Einstein equations with a negative cosmological cons
tant for an open FRW universe are studied in detail. The corresponding
solutions admit bounces (hooked right arrow minimal radius) of the un
iverse such that the matter energy in any comoving 3-volume is either
increased or decreased during the bounce according to whether the boun
ce pressure of the spinor field is appropriately negative or not. If m
atter is generated (annihilated) during a bounce, the universe subsequ
ently becomes larger (smaller) than before the bounce. Therefore matte
r can be generated only during the growth of the universe, but it is a
nnihilated again during the subsequent shrinking phase, which together
with the growing phase forms a cosmic supercycle.