controlling the large-scale dynamics of the system, we analyze the linear stability of the ordered nematic state in various special cases that best highlight a particular mechanism.

The ordered nematic state has constant density $\rho_0$, zero mean polarization density, $\bm\tau_0=0$, and a finite value for the nematic alignment tensor. Choosing a coordinate system with the $x$ axis pointing along the direction of broken nematic symmetry, the alignment tensor in the uniform nematic state has components $Q^0_{xx}=-Q^0_{yy}=Q_0/2$ and $Q^0_{xy}=Q^0_{yx}=0$, with $Q_0=\sqrt{\alpha_0/\beta}$ and $\alpha_0=\alpha(\rho_0)$. We now examine the linear stability of this state in various regions of parameters by considering the dynamics of small fluctuations, $\delta\rho({\bf r},t)=\rho({\bf r},t)-\rho_0$, $\delta\bm\tau({\bf r},t)=\bm\tau({\bf r},t)$ and $\delta Q_{ij}({\bf r},t)=Q_{ij}({\bf r},t)-Q_{ij}^0$. We will generally work in Fourier space by introducing Fourier transforms of the fluctuations as $\phi^\alpha_{\bf k}(t)=\int_{\bf r}e^{i{\bf k}\cdot{\bf r}}\delta\phi_\alpha({\bf r},t)$, where $\delta\phi_\alpha=\left(\delta\rho,\bm\tau,\delta Q_{ij}\right)$.

Dynamical Self-Regulation and Banding Instability {#subsec:self-reg}
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We first consider the linear dynamics of the system in the region just above the mean-field transition at $\rho_c$. For simplicity we only discuss spatial variations normal to the direction of broken symmetry, as these correspond to the most unstable modes, i.e., let $\mathbf{k}=k\mathbf{\hat{y}}$. Fluctuations in $\tau
_{x}$ and $\delta Q_{xy}$ then decouple and are 