Originally motivated by problems in physics, the Bak-Sneppen model is a simplified description of how the fitness levels of a population evolves over time due to natural selection, while also taking into account how the population members are distributed about in space. Experimental work has shown that it is a reasonable model for describing the fitness evolution of simple organisms such as E. Coli. From a mathematical point of view Bak-Sneppen is a Markov chain with transition rules that are very easy to describe, and it is well known that there is a stationary distribution of the chain that describes the long-time statistical properties of the population's fitness levels. Analyzing this stationary distribution is a notoriously difficult task however, and has led to an interesting but unproved conjecture for its asymptotic properties as the population size goes to infinity. I will describe some recent work in this direction that is joint with Ga-Yeong Lee and Mackenzie Simper.