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Motivated by the goal of understanding the evolution of populationsundergoing selection, we consider branching Brownian motion in which particlesindependently move according to one-dimensional Brownian motion with drift,each particle may either split into two or die, and the difference between thebirth and death rates is a linear function of the position of the particle. Weshow that, under certain assumptions, after a sufficiently long time, theempirical distribution of the positions of the particles is approximatelyGaussian. This provides mathematically rigorous justification for results inthe biology literature indicating that the distribution of the fitness levelsof individuals in a population over time evolves like a Gaussian travelingwave.

A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate