  We study the contact process with stirring on $\mathbb{Z}^d$. In this
process, particles occupy vertices of $\mathbb{Z}^d$; each particle dies with
rate 1 and generates a new particle at a randomly chosen neighboring vertex
with rate $\lambda$, provided the chosen vertex is empty. Additionally,
particles move according to a symmetric exclusion process with rate $N$. For
any $d$ and $N$, there exists $\lambda_c$ such that, when the system starts
from a single particle, particles go extinct when $\lambda < \lambda_c$ and
have a chance of being present for all times when $\lambda > \lambda_c$.
Durrett and Neuhauser proved that $\lambda_c$ converges to 1 as $N$ goes to
infinity, and Konno, Katori and Berezin and Mytnik obtained dimension-dependent
asymptotics for this convergence, which are sharp in dimensions 3 and higher.
We obtain a lower bound which is new in dimension 2 and also gives the sharp
asymptotics in dimensions 3 and higher. Our proof involves an estimate for
two-type renewal processes which is of independent interest.
