  Motivated by the goal of understanding the evolution of populations
undergoing selection, we consider branching Brownian motion in which particles
independently move according to one-dimensional Brownian motion with drift,
each particle may either split into two or die, and the difference between the
birth and death rates is a linear function of the position of the particle. We
show that, under certain assumptions, after a sufficiently long time, the
empirical distribution of the positions of the particles is approximately
Gaussian. This provides mathematically rigorous justification for results in
the biology literature indicating that the distribution of the fitness levels
of individuals in a population over time evolves like a Gaussian traveling
wave.
