# Steady states in a non-conserving zero-range process with extensive rates as a model for the balance of selection and mutation

## TL;DR

This paper models the balance of selection and mutation using a non-conserving zero-range process, revealing phase transitions and connections to population dynamics and evolving networks, with implications for understanding fitness landscapes.

## Contribution

It introduces a novel zero-range process model with creation and annihilation, linking it to the house-of-cards model and network evolution, highlighting phase transitions in population and network structures.

## Key findings

- Single sites can host a macroscopic population fraction below a critical mutation rate.
- The model reproduces the house-of-cards distribution in the large density limit.
- Network link distributions mirror the zero-range process under certain conditions.

## Abstract

We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and removed from the system with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the mean-field geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model of particles is mapped to a model of population dynamics: the site label is interpreted as a level of fitness, the site-dependent creation rate is interpreted as a selection function, and the hopping process is interpreted as the introduction of mutants. In the limit of large density, the fraction of the total population occupying each site approaches the limiting distribution in the house-of-cards model of selection-mutation, introduced by Kingman. A single site can be occupied by a macroscopic fraction of the particles if the mutation rate is below a critical value (which matches the critical value worked out in the house-of-cards model). This feature generalises to classes of selection functions that increase sufficiently fast at high fitness. The process can be mapped to a model of evolving networks, inspired by the Bianconi--Barab\'asi model, but involving a large and fixed set of nodes. Each node forms links at a rate biased by its fitness, moreover links are destroyed at a uniform rate, and redirected at a certain rate. If this redirection rate matches the mutation rate, the number of links pointing to nodes of a given fitness level is distributed as the numbers of particles in the non-conserving zero-range process. There is a finite critical redirection rate if the density of quenched fitnesses goes to zero sufficiently fast at high fitness.