  We consider the frog model with geometric lifetime (parameter $1-p$) on
homogeneous trees of dimension $d$. In 2002, \cite{alves2002-2} proved that
there exists a critical lifetime parameter $p_c\in(0,1)$ above which infinitely
many frogs are activated with positive probability, and they gave lower and
upper bounds for $p_c$. Since then, the literature on this model focussed on
refinements of the upper bound. In the present paper we improve the bounds for
$p_c$ \emph{on both sides}. We also provide a discussion comparing the bounds
of the literature and their proofs. Our proofs are based on coupling.
