Branching Random Walk in an inhomogeneous breeding potential

We consider a continuous-time branching random walk in the inhomogeneous breeding potential $\beta \mathrm{\mid }\mathrm{.}{\mathrm{\mid }}^p$, where $\beta >0$, $p\ge 0$. We prove that the population almost surely explodes in finite time if $p>1$ and doesn’t explode if $p\le 1$. In the non-explosive cases, we determine the asymptotic behaviour of the rightmost particle.