# The Wright-Fisher model with efficiency

## TL;DR

This paper introduces a Wright-Fisher model incorporating resource efficiency, showing that inefficiency can confer a selective advantage, with the population dynamics converging to a generalized diffusion process.

## Contribution

It develops a new Wright-Fisher model with efficiency, extending the ancestral selection graph, and demonstrates inefficiency as a potential driver of selection in populations.

## Key findings

- Efficient individuals' frequency converges to a generalized Wright-Fisher diffusion.
- Inefficiency can promote selective advantage, not just a trade-off.
- The genealogy is described by a branching-coalescing process called the ASE-Graph.

## Abstract

In populations competing for resources, it is natural to ask whether consuming fewer resources provides any selective advantage. To answer this question, we propose a Wright- Fisher model with two types of individuals: the inefficient individuals, those who need more resources to reproduce and can have a higher growth rate, and the efficient individuals. In this model, the total amount of resource N, is fixed, and the population size varies randomly depending on the number of efficient individuals. We show that, as N increases, the frequency process of efficient individuals converges to a diffusion which is a generalisation of the Wright- Fisher diffusion with selection. The genealogy of this model is given by a branching-coalescing process that we call the Ancestral Selection/Efficiency Graph, and that is an extension of the Ancestral Selection Graph (Krone and Neuhauser (1997a), Krone and Neuhauser (1997b)). The main contribution of this paper is that, in evolving populations, inefficiency can arise as a promoter of selective advantage and not necessarily as a trade-off.