The threshold public goods game is one of the best-known models of nonlinear public goods dilemmas. Cooperators and defectors typically coexist in this game when the population is assumed to follow the so-called structured deme model. In this paper we develop a dynamical model of a general N-player game in which there is no deme structure: individuals interact with randomly chosen neighbours and selection occurs between randomly chosen pairs of indi- viduals. We show that in the deterministic limit the dynamics in this model leads to the same replicator dynamics as in the structured deme model, i.e. coexistence of cooperators and defectors is typical in threshold public goods game even when the population is completely well-mixed. We extend the model to study the effect of density dependence and density fluctuation on the dynamics. We show analytically and numerically that decreasing population density increases the equilibrium frequency of cooperators till the fixation of this strategy, but below a critical density coop- erators abruptly disappear from the population. Our numerical investigations show that weak density fluctuations enhance cooperation, while strong fluctuations suppress it.