  We introduce and study a model in one dimension of $N$ run-and-tumble
particles (RTP) which repel each other logarithmically in the presence of an
external quadratic potential. This is an "active'' version of the well-known
Dyson Brownian motion (DBM) where the particles are subjected to a telegraphic
noise, with two possible states $\pm$ with velocity $\pm v_0$. We study
analytically and numerically two different versions of this model. In model I a
particle only interacts with particles in the same state, while in model II all
the particles interact with each other. In the large time limit, both models
converge to a steady state where the stationary density has a finite support.
For finite $N$, the stationary density exhibits singularities, which disappear
when $N \to +\infty$. In that limit, for model I, using a Dean-Kawasaki
approach, we show that the stationary density of $+$ (respectively $-$)
particles deviates from the DBM Wigner semi-circular shape, and vanishes with
an exponent $3/2$ at one of the edges. In model II, the Dean-Kawasaki approach
fails but we obtain strong evidence that the density in the large $N$ limit
retains a Wigner semi-circular shape.
