From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step

Abstract

We consider a model for Darwinian evolution in an asexual population with a large but nonconstant populations size characterized by a natural birth rate, a logistic death rate modeling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population ($K\to\infty$) size, rare mutations ($u\to0$) and small mutational effects ($\sigma\to0$), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, for example, by Champagnat and Méléard, we take the three limits simultaneously, that is, $u=u_{K}$ and $\sigma=\sigma_{K}$, tend to zero with $K$, subject to conditions that ensure that the time-scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that require the use of completely different methods. In particular, we cannot use the law of large numbers on the diverging time needed for fixation to approximate the stochastic system with the corresponding deterministic one. To solve this problem, we develop a “stochastic Euler scheme” based on coupling arguments that allows to control the time evolution of the stochastic system over time-scales that diverge with $K$.