  How long does it take for an initially advantageous mutant to establish
itself in a resident population, and what does the population composition look
like then? We approach these questions in the framework of the so called Bare
Bones evolution model Klebaner et al (2011) that provides a simplified approach
to the adaptive population dynamics of binary splitting cells. As the mutant
population grows, cell division becomes less probable, and it may in fact turn
less likely than that of residents. Our analysis rests on the assumption of the
process starting from resident population, with sizes proportional to a large
carrying capacity $K$. Actually, we assume carrying capacities to be $a_1K$ and
$a_2K$ for the resident and the mutant populations, respectively, and study the
dynamics for $K\to\infty$. We find conditions for the mutant to be successful
in establishing itself alongside the resident. The time it takes turns out to
be proportional to $\log K$. We introduce the time of establishment through the
asymptotic behavior of the stochastic nonlinear dynamics describing the
evolution, and show that it is indeed $\log K/\log \rho$, where $\rho>1$ is
twice the probability of successful division of the mutant at its appearance.
Looking at the composition of the population, at times $\log K/\log \rho +n, n
\in \mathbb{Z}_+$, we find that the densities (i.e. sizes relative to carrying
capacities) of both populations follow closely the corresponding two
dimensional nonlinear deterministic dynamics that starts at {\it a random
point}. We characterise this random initial condition in terms of the scaling
limit of the corresponding dynamics, and the limit of the properly scaled
initial binary splitting process of the mutant. The deterministic approximation
with random initial condition is in fact valid asymptotically at all times
$\log K/\log \rho +n$ with $n\in \mathbb{Z}$.
