Maintaining diversity in structured populations.

We examine population structures for their ability to maintain diversity in neutral evolution. We use the general framework of evolutionary graph theory and consider birth-death (bd) and death-birth (db) updating. The population is of size N. Initially all individuals represent different types. The basic question is: what is the time $T_N$ until one type takes over the population? This time is known as consensus time in computer science and as total coalescent time in evolutionary biology. For the complete graph, it is known that $T_N$ is quadratic in N for db and bd. For the cycle, we prove that $T_N$ is cubic in N for db and bd. For the star, we prove that $T_N$ is cubic for bd and quasilinear ( $N\log N$ ) for db. For the double star, we show that $T_N$ is quartic for bd. We derive upper and lower bounds for all undirected graphs for bd and db. We also show the Pareto front of graphs (of size $N=8$ ) that maintain diversity the longest for bd and db. Further, we show that some graphs that quickly homogenize can maintain high levels of diversity longer than graphs that slowly homogenize. For directed graphs, we give simple contracting star-like structures that have superexponential time scales for maintaining diversity.