  Understanding the temporal spread of gene drive alleles -- alleles that bias
their own transmission -- through modeling is essential before any field
experiments. In this paper, we present a deterministic reaction-diffusion model
describing the interplay between demographic and allelic dynamics, in a
one-dimensional spatial context. We focused on the traveling wave solutions,
and more specifically, on the speed of gene drive invasion (if successful). We
considered various timings of gene conversion (in the zygote or in the
germline) and different probabilities of gene conversion (instead of assuming
100$\%$ conversion as done in a previous work). We compared the types of
propagation when the intrinsic growth rate of the population takes extreme
values, either very large or very low. When it is infinitely large, the wave
can be either successful or not, and, if successful, it can be either pulled or
pushed, in agreement with previous studies (extended here to the case of
partial conversion). In contrast, it cannot be pushed when the intrinsic growth
rate is vanishing. In this case, analytical results are obtained through an
insightful connection with an epidemiological SI model. We conducted extensive
numerical simulations to bridge the gap between the two regimes of large and
low growth rate. We conjecture that, if it is pulled in the two extreme
regimes, then the wave is always pulled, and the wave speed is independent of
the growth rate. This occurs for instance when the fitness cost is small
enough, or when there is stable coexistence of the drive and the wild-type in
the population after successful drive invasion. Our model helps delineate the
conditions under which demographic dynamics can affect the spread of a gene
drive.
