  We consider a system of independent branching random walks on $\R$ which
start off a Poisson point process with intensity of the form
$e_{\lambda}(du)=e^{-\lambda u}du$, where $\lambda\in\R$ is chosen in such a
way that the overall intensity of particles is preserved. Denote by $\chi$ the
cluster distribution and let $\phi$ be the log-Laplace transform of the
intensity of $\chi$. If $\lambda\phi'(\lambda)>0$, we show that the system is
persistent (stable) meaning that the point process formed by the particles in
the $n$-th generation converges as $n\to\infty$ to a non-trivial point process
$\Pi_{e_{\lambda}}^{\chi}$ with intensity $e_{\lambda}$. If
$\lambda\phi'(\lambda)<0$, then the branching population suffers local
extinction meaning that the limiting point process is empty. We characterize
(generally, non-stationary) point processes on $\R$ which are cluster-invariant
with respect to the cluster distribution $\chi$ as mixtures of the point
processes $\Pi_{ce_{\lambda}}^{\chi}$ over $c>0$ and $\lambda\in
K_{\text{st}}$, where $K_{\text{st}}=\{\lambda\in\R: \phi(\lambda)=0,
\lambda\phi'(\lambda)>0\}$.
