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

TL;DR

This paper proves the convergence of a stochastic individual-based model of Darwinian evolution to the canonical equation of adaptive dynamics by simultaneously taking large population, rare mutation, and small mutational effects limits, using a novel stochastic Euler scheme.

Contribution

It introduces a new method involving a stochastic Euler scheme to handle simultaneous limits in adaptive dynamics models, overcoming previous technical challenges.

Findings

Convergence to the canonical equation of adaptive dynamics established.

Development of a stochastic Euler scheme for controlling long-term stochastic processes.

Handling of simultaneous limits in population size, mutation rate, and mutational effects.

Abstract

We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate modelling 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 $(u\to 0)$, and small mutational effects $(\sigma\to 0)$, proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, e.g. by Champagnat and M\'el\'eard, we take the three limits simultaneously, i.e. $u=u_K$ 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 requires 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 $K$.