  This paper is concerned with a reaction--diffusion system modeling the
fixation and the invasion in a population of a gene drive (an allele biasing
inheritance, increasing its own transmission to offspring). In our model, the
gene drive has a negative effect on the fitness of individuals carrying it, and
is therefore susceptible of decreasing the total carrying capacity of the
population locally in space. This tends to generate an opposing demographic
advection that the gene drive has to overcome in order to invade. While
previous reaction--diffusion models neglected this aspect, here we focus on it
and try to predict the sign of the traveling wave speed. It turns out to be an
analytical challenge, only partial results being within reach, and we complete
our theoretical analysis by numerical simulations. Our results indicate that
taking into account the interplay between population dynamics and population
genetics might actually be crucial, as it can effectively reverse the direction
of the invasion and lead to failure. Our findings can be extended to other
bistable systems, such as the spread of cytoplasmic incompatibilities caused by
Wolbachia.
