  We present large deviations estimates in the supremum norm for a system of
independent random walks superposed with a birth-and-death dynamics evolving on
the discrete torus with $N$ sites. The scaling limit considered is the
so-called \textit{high density limit} (see the survey \cite{franco} on the
subject), where space, time and initial quantity of particles are rescaled. The
associated rate functional here obtained is a semi-linearised version of the
rate function of \cite{JonaLandimVares}, which dealt with large deviations of
exclusion processes superposed with birth-and-death dynamics. An ingredient in
the proof of large deviations consists in providing a limit of a suitable class
of perturbations of the original process. This is precisely one of the main
contributions of this work: a strategy to extend the original high density
approach (as in \cite{Arnold,blount2,blount,francogroisman,Kote2,KoteHigh1988})
to weakly asymmetric systems. Two cases are considered with respect to the
initial quantity of particles, the power law and the (at least) exponential
growth. In the first case, we present the lower bound only on a certain subset
of smooth profiles, while in the second case, additionally assuming concavity
of the birth and the death functions and a constant initial profile, we provide
a full large deviations principle.
