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We use the Liapunov–Schmidt (LS) technique of bifurcation theory to derive a low-dimensional model for laminar dispersion of a nonreactive solute in a tube. The LS formalism leads to an exact averaged model, consisting of the governing equation for the cross-section averaged concentration, along with the initial and inlet conditions, to all orders in the transverse diffusion time. We use the averaged model to analyze the temporal evolution of the spatial moments of the solute and show that they do not have the centroid displacement or variance deficit predicted by the coarse-grained models derived by other methods. We also present a detailed analysis of the first three spatial moments for short and long times as a function of the radial Peclet number and identify three clearly defined time intervals for the evolution of the solute concentration profile. By examining the skewness in some detail, we show that the skewness increases initially, attains a maximum for time scales of the order of transverse diffusion time, and the solute concentration profile never attains the Gaussian shape at any finite time. Finally, we reason that there is a fundamental physical inconsistency in representing laminar (Taylor) dispersion phenomena using truncated averaged models in terms of a single cross-section averaged concentration and its large scale gradient. Our approach evaluates the dispersion flux using a local gradient between the dominant diffusive and convective modes. We present and analyze a truncated regularized hyperbolic model in terms of the cup-mixing concentration for the classical Taylor–Aris dispersion that has a larger domain of validity compared to the traditional parabolic model. By analyzing the temporal moments, we show that the hyperbolic model has no physical inconsistencies that are associated with the parabolic model and can describe the dispersion process to first order accuracy in the transverse diffusion time.