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Zonostrophic instability leads to the spontaneous emergence of zonal jets on a β-plane from a jetless basic-state flow which is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal-mean and eddy equations, and neglecting the eddyeddy interactions, defines the quasi-linear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system. Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. We characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of six smaller than that for QL zonostrophic instability. Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors till the flow matures into a state with jets which are significantly broader than the linear prediction, but have have similar spacing as NL jets.