On the establishment of a mutant
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 $n\in \mathbb{Z}$.