Doi models for flows of concentrated solutions of homogeneous liquid crystal polymers (LCPs) are analyzed in the quadratic closure approximation. Our purpose is to clarify a remarkable near-equilibrium behavior of these equations which has apparently gone unnoticed; these results are important for any numerical or experimental interpretations of LCP flows based on Doi models near mechanical and nematic equilibria. To reveal this behavior, we analytically solve the linearized Doi nematodynamic equations; this calculation explicitly captures the coupling between the pure nematic and pure hydrodynamic linearized dynamics. The original Doi model without solvent viscosity is analyzed first: the low concentration (N<3) isotropic phase and the high concentration (N>8/3) prolate nematic phase yield well-posed linearized dynamic; at higher concentration both the isotropic phase (N>3) and the oblate nematic phase (N>3) yield catastrophic linearized dynamics, with exponential growthrates proportional to the amplitude of the wavevector of the linearized disturbance. This result implies that there is unbounded growth in vanishingly small length scales for data near these equilibria. We then explore three physical regularizations of the original Doi model: solvent viscosity, finite-range intermolecular interactions, and spatial inhomogeneity in LCP concentration. Each effect yields well-posed linearized dynamics of all now-nematic equilibria, with bounded growth rates. Solvent viscosity and spatial inhomogeneity alone are not sufficient to produce a finite wavelength instability cutoff of the high-concentration isotropic and oblate nematic equilibria, whereas a finite-range intermolecular potential alone yields a finite cutoff. The flow-orientation interactions for unstable nematic phases produce a spatial wavevector dependence of the instability from which we reveal a flow-induced spatially anisotropic (or directional) instability of the oblate phase.