The classical theory of astrophysical discs (including Saturn’s rings, protoplanetary systems, and high-energy accretion discs around black holes) assumes circular orbital motion around a central mass. However, certain systems are known to contain eccentric forcing, necessitating a generaliza- tion to the shearing box model to include the oscillatory local geometry associated with this eccentricity. The hydrodynamic equations in this model are non-standard because of the use of time-dependent, non-orthogonal coordinates, and are known to lead to hydrodynamic instability involving the growth of internal waves. Here we present the results of the first ever local nonlinear simulations in an eccentric shearing box representing an elliptic disc with vertical structure. The nonlinear sat- uration of this parametric instability inherent to eccentric discs generates further self-regulating azimuthal zonal flows, and results in stable limit cycle behavior. We explore this energy pathway from the global eccentric mode into turbulence and finally the zonal flows, and discuss the viability of this instability to balance the eccentricity growth in systems exhibiting mean-motion orbital resonances such as the eccentric Lindblad resonance.