Motivated by a better understanding of cancer evolution, we consider an individual-based model representing a cell population where cells divide, die, and mutate along the edges of a finite directed graph, representing the genetic trait space. The process starts with only one wild-type cell. Following typical parameter values in cancer cell populations, we study the model under power-law mutation rates, where mutation probabilities are parametrised by negative powers of a scaling parameter $n$, and the typical sizes of the population of interest are positive powers of $n$.

A wide variety of models with power-law mutation rates have been explored in the adaptive dynamics literature. A common result in this context is the convergence of the logarithmic frequencies of the different traits when time is scaled by a $\log n$ factor. Under a non-increasing growth rate condition, we succeed in describing not only the logarithmic frequencies but also the actual first-order asymptotics of the size of each subpopulation on the $\log n$ time scale, as well as in the random time scale at which the wild-type subpopulation, respectively the total population, reaches the size $n^t$. These results, derived using a meticulous martingale approach, allows for an exact characterisation of evolutionary pathways.

Selective mutations present a far greater challenge, as the martingale approach typically fails in this case. Together with Hélène Leman, we developed a novel methodology to determine the first-order asymptotics for the first selective mutant trait.