being the Cartesian components of the vorticity and rate-of-strain tensors, respectively. In Eq., $\mathbb{P}_{i}^{\alpha\beta} = \delta^{\alpha\beta} - p^{\alpha}_{i}p_{i}^{\beta}$, is the projection operator, $\delta^{\alpha\beta}$ denotes the Kronecker delta, $\nabla_i^\alpha = \partial/\partial x_i^\alpha$, and $B = \left( a^2 - 1\right)/\left( a^2 + 1\right)$ is the measure of the swimmer’s nonsphericity [@kimkarrila] based on its aspect ratio $a$. For strongly elongated particles, $B\rightarrow 1$, while for spheres, $B=0$. Finally, each swimmer randomly changes its orientation with a rate $\lambda$, thus mimicking the run-and-tumble motion commonly exhibited by bacteria [@Berg1993]. We note here that we neglect the effects of rotational and translational diffusion on the particle’s dynamics, and random tumbling is thus the only source of stochasticity in our model.

The velocity field generated by a self-propelled particle sufficiently far away from its surface is often well-described by the field produced by a point dipole with the same position and orientation [@Lauga2009; @Drescher2011]. In a dilute suspension of microswimmers, where the particles are sufficiently separated from each other, we can approximate $\mathcal{U}^{\alpha}\left( {\bm x}_i \right)$ by a sum of dipolar contributions $$\begin{aligned}
 \mathcal{U}^{\alpha}\left( {\bm x}_i \right) = \sum_{j\neq i}^{N} u_{d}^{\alpha}({\bm x}_{i}; {\bm z}_{j}),\end{aligned}$$ where $$\begin{aligned}
{\bm u}_{d}({\bm x}_{i}; {\bm z}_{j}) = \frac{\kappa}{8\pi} \left[ 3
