  We study protoplanetary disc evolution assuming that angular momentum
transport is driven by gravitational instability at large radii, and
magnetohydrodynamic (MHD) turbulence in the hot inner regions. At radii of the
order of 1 AU such discs develop a magnetically layered structure, with
accretion occurring in an ionized surface layer overlying quiescent gas that is
too cool to sustain MHD turbulence. We show that layered discs are subject to a
limit cycle instability, in which accretion onto the protostar occurs in bursts
with an accretion rate of 10^{-5} solar masses / yr, separated by quiescent
intervals where the accretion rate is 10^{-8} solar masses / yr. Such bursts
could lead to repeated episodes of strong mass outflow in Young Stellar
Objects. The transition to this episodic mode of accretion occurs at an early
epoch (t < 1 Myr), and the model therefore predicts that many young
pre-main-sequence stars should have low rates of accretion through the inner
disc. At ages of a few Myr, the discs are up to an order of magnitude more
massive than the minimum mass solar nebula, with most of the mass locked up in
the quiescent layer of the disc at around 1 AU. The predicted rate of low mass
planetary migration is reduced at the outer edge of the layered disc, which
could lead to an enhanced probability of giant planet formation at radii of 1-3
AU.
