A new proof for the convergence of an individual based model to the Trait substitution sequence

TL;DR

This paper presents a new proof demonstrating that a stochastic individual-based model in adaptive dynamics converges to the trait substitution sequence, a process describing population evolution through successive trait fixations.

Contribution

It introduces a novel proof technique for the convergence of the model to the trait substitution sequence, simplifying and strengthening previous results.

Findings

Established convergence of the individual-based model to the trait substitution sequence.

Provided a new proof method using averaging techniques for adaptive dynamics.

Confirmed the 'invasion implies substitution' assumption in the model.

Abstract

We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and we view the process in the timescale of mutations. Using averaging techniques due to Kurtz (1992), we give a new proof of the convergence of the individual based model to the trait substitution sequence of Metz et al. (1992) first worked out by Dieckman and Law (1996) and rigorously proved by Champagnat (2006): rigging the model such that "invasion implies substitution", we obtain in the limit a process that jumps from one population equilibrium to another when mutations occur and invade the population.